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Why are snowflakes symmetrical? The title says it all. Why are snowflakes symmetrical in shape and not a mush of ice?
Is it a property of water freezing or what? Does anyone care to explain it to me? I'm intrigued by this and couldn't find an explanation.
| Not quite an answer but the first attempt to explain the shape was published by astronomer Johannes Kepler in 1611, the original is in Latin - "Strena Seu de Nive Sexangula" (A New Year's Gift of Hexagonal Snow). There is an English translation ("The Six-Cornered Snowflake") available at Amazon and elsewhere.
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Stopping Distance (frictionless) Assuming I have a body travelling in space at a rate of $1000~\text{m/s}$. Let's also assume my maximum deceleration speed is $10~\text{m/s}^2$. How can I calculate the minimum stopping distance of the body?
All the formulas I can find seem to require either time or distance, but not one or the other.
| Another equation of motion problem,very easy.
$$v^2 = u^2 + 2.a.s$$ where $$v = \textbf{final velocity} $$ , $$u = \textbf{initial velocity}$$ , $$a=\textbf{ acceleration or in this case negative acceleration}$$ , $$s = displacement$$ . This equation is time independent. Now, to find $s$ , put the values: $$s = \frac{-1000^2}{2.(-10)} \implies s = 50000~\text{m}$$ . Simple,right?
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Will tensile strength keep a cable from snapping indefinitely? Trying to secure a wall hanging using magnets; me and a coworker came up with an interesting question:
When the hanging is hung using 1 magnet, the weight of it causes it to quickly drag the magnet down and the hanging drops. Using n magnets retards this process; causing it to fall more slowly, but does there exist a number of magnets m such that their combined strength will prevent the hanging from slipping, entirely and permanently?
Because this doesn't make for a very good question; we worked at it and arrived at a similar one; but slightly more idealized:
A weight is suspended, perfectly still, from a wire in a frictionless vacuum. If the mass of the weight is too great; it will gradually distend the cable, causing it to snap and release the weight; but will a light enough weight hang there indefinitely, or will the mass of the weight (and indeed the cable) cause the cable to snap sooner or later?
| Ultimately, the weight will fall.
If the weight falls, it will collide with the floor and convert its gravitational potential energy into heat, resulting in an increase in entropy, so the process of the weight eventually falling is thermodynamically favored.
One possible (and somewhat ludicrous) mechanism is quantum tunneling. A bizarre result of this mechanism is that the time we expect to wait before the weight tunnels increases exponentially with the square root of the weight's mass. That means a light object will fall to the ground before a heavy object, by a long shot!
This is a somewhat silly mechanism for a macroscopic weight to fall, but it's a theoretical limit. My guess is that realistically the weight will fall because small statistical fluctuations in the wire and weight due to their thermal energies will eventually cause molecular-scale defects in crystal structure of the wire, evaporation of atoms from the wire, etc. When enough of these defects have accumulated the wire strength will be low enough that the weight falls. If you had a very light weight and you kept the box very cold and extremely-well mechanically isolated from all surroundings, it would take an incredibly long time for the weight to fall. Nonetheless, the second law of thermodynamics says fall it must, eventually.
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Haag's theorem and practical QFT computations There exists this famous Haag's theorem which basically states that the interaction picture in QFT cannot exist. Yet, everyone uses it to calculate almost everything in QFT and it works beautifully.
*
*Why? More specifically to particle physics: In which limit does the LSZ formula work?
*Can someone give me an example of a QFT calculation (of something measurable in current experiments, something really practical!) in which the interaction picture fails miserably due to Haag's theorem?
| Lubosh wrote: "... perturbative QFT clearly works...". No, it miserably fails in the initial approximation ("bare" particles, no soft radiation predicted) and in course of search of the solutions by iterations (infinite corrections to the initial approximation). That is why there are so many questions to it!
What is comparable with experimental data is a renormalized and IR summed up result (a "repaired solution") which is quite different from the original solution. And even after that there are conceptual and mathematical difficulties in the theory. Besides, there are non-renormalizable theories where attempts to "repair solutions on go" fail hopelessly.
QFT, as a human invention, suffers from severe problems. It is very far from a desired state and needs repairing. Some times renormalizations "work" but not always, and we are far from the statement "QFT has no problem". We should try other constructions. I disagree with the Lubosh's statement "this is not possible", especially if with help of renormalizations and IR contribution summation we go away from initially wrong approximation and obtain reasonable results. I believe we may start from a better initial approximation, eliminate those problems, and arrive at the final results directly. Denying such opportunities is not wise, to say the least.
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Renormalization and Infinites Measuring a qubit and ending up with a bit feels a little like tossing out infinities in renormalization. Does neglecting the part of the wave function with a vanishing Hilbert space norm amount to renormalizing of Hilbert space?
| I am maybe a bit uncertain what you are asking, but from what I understand the answer would be no. Renormalization is a procedure for absorbing infinities in an interacting field theory. A quantum bit is really just a state, but referred to in information theoretic terms. The two physics are not directly related as such. In a measurement if one considers it as a collapse there is a new normalization (renormalization?) of the system state, which is just the state vector which pertains to the measurement outcome.
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A reference request for real world experimental data I always use to wonder how the experimental physicists discover new particles every now and then whose dimensions/properties/mass/charge several order of magnitudes below that of anything that is visible/perceptible. So as engineers do, I guess they also set up a extremely complicated equipment and do some pretty complicated experiments and measure some physical quantity, say (pardon me if my example is poor) an voltage or a magnetic field intensity or could be anything depending on the experimental setup. Now they compare this measured voltage variation with that of what is expected theoretically and then go on to prove the hypothesis. ( This is the hardest thing I could imagine).
Now my request is where can i find a set of data (preferable a continuous variation of a physical parameter with respect to another... may be sampled at sufficient sampling frequency) along with the context of experiment (as minimal as possible but sufficient) so that i can carry out some processing of data in my own way so that i can verify the hypothesis or any such a thing. Simply put i need some really cool real world data ( in the form of signal).
Is any such thing available on the internet or where could I find one?
Please suggest me something which involves signal processing.
| You can download raw (or processed) data from various observational cosmology projects (e.g. COBE and WMAP) here:
http://lambda.gsfc.nasa.gov/product/map/current/
As a project, you could use this to re-compute the famous angular power spectrum of the cosmic microwave background radiation.
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Why are there only derivatives to the first order in the Lagrangian? Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded?
Is there a good reason for this or is it simply "because it works".
| This question actually needs a 2 steps answer:
*
*Why Lagrangian has only derivatives to the first order?:
Lagrangian has been defined in such a way, that problem to be solved would produce a second order derivative with respect to time when Euler-Lagrange equation is produced. It includes an implicit derivation of the momentum (notice time derivative after minus sign in $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot q}=0$) which in turn, is a first order derivative of position. It means that, acceleration is actually taken care when the full problem is setup. One can verify it by simply checking that for most cases Euler-Lagrange equation just turns to be $\frac{\partial L}{\partial q}-m \ddot q=0$ and if one defines$\frac{\partial L}{\partial q}=F$ it becomes Newton’s second law. Having said that, we need to move to the next step, which is,
*Why jerk (or any bigger time derivative) is not necessary?:
This question has already been replied (including one by me) here Why $F=ma$ and not $F=m \dot a$. The short answer is:
“… second order derivative is all one needs to differentiate natural states of motion from affected states of motion”.
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How to avoid getting shocked by static electricity? sometimes I get "charged" and the next thing I touch something that conducts electricity such as a person, a car, a motal door, etc I get shocked by static electricity.
I'm trying to avoid this so if I suspect being "charged" I try to touch something that does not conduct electricity (such as a wooden table) as soon as possible, in the belief that this will "uncharge me".
*
*Is it true that touching wood will uncharge you?
*How and when do I get charged? I noticed that it happens only in parts of the years, and after I get out of the car...
| Removing the radicals from the surface will protect you.
http://dx.doi.org/10.1126/science.1241326
Drink your anti-oxidants;-)
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Why is the relationship between atomic number and density not linear? What are the factors that affect the density of an atom?
| At an atomic scale, there are two things that go into the density, which is $\rho=\frac{\mbox{mass}}{\mbox{volume}}$. First, we have the number density, which is $n=\frac{\mbox{number of atoms}}{\mbox{volume}}$. Then we also have the mass per atom, $\mu=\frac{\mbox{mass}}{\mbox{atom}}$. It is easy to see by combining these equations that the density is then $\rho=n\mu$ - it depends on both number density and atomic weights.
While the mass per atom, $\mu$, goes up linearly (very roughly speaking) as you go up in atomic number, the number density does not go up linearly. This is because interactions between the actual atoms cause them to cluster closer together, so you get a higher value of $n$ in the equation above for some atoms, but a much lower one for others. For example, in that graph you linked to above:
Those spikes in density are caused by much higher number density, which is in turn caused by metallic attractions between the atoms. The deepest valleys are the noble gases which have virtually no mutual attractions whatsoever.
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Physical meaning of Legendre transformation I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
| See
http://en.wikipedia.org/wiki/Legendre_transformation#Applications
In theoretical physics, the basic or defining mathematical properties of the Legendre transformation are used to switch between one form of the energy - or "potential", as the generalized energies are called in thermodynamics - to another.
This is important to switch between the Lagrangian in abstract mechanics that depends on $x,v$ (positions and velocities) to the Hamiltonian, the true energy that depends on $x,p$.
In thermodynamics, the number of applications and "types of switches" is even higher. You may go from energy to enthalpy or Helmholtz free energy or Gibbs free energy by Legendre-transforming with respect to various variables. The transform goes back and forth. As the Wikipedia example explains, there are other useful variables that you may Legendre-transform with respect to, including the charge and voltage.
You may consider the Legendre transformation to be a "mere" redefinition of variables - but that's why it's so important in practice. In reality, the different ways to describe the system that differ by a Legendre transformation are "equally fundamental" or "equally natural" so it's often useful to be familiar with all of them and to know what is the relationship between them. The relationship is given by the Legendre transformation.
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How to explain the weak force to a layman? I'm trying to explain in simple terms what the weak interaction does, but I'm having trouble since it doesn't resemble other forces he's familiar with and I haven't been able to come up (or find on the web) with a good, simple visualization for it.
| The weak force "looks" different because in the first (and still most important) reincarnation we have encountered it - namely beta-decay (including the decay of the neutron) - the force seems to be a contact interaction: it has an extremely short range, essentially zero.
However, any phenomenon that differs from the indefinite existence of an object that moves in the same direction by the same speed forever requires a force to be explained. The force required for the beta-decay is the weak nuclear force.
While the decay seems to "directly" transform a neutron into a proton, electron, and antineutrion, a closer investigation of the force that began in the 1960s has demonstrated that this force is actually analogous to other forces, including electromagnetism, because its range is finite (nonzero). It's only limited because it's mediated by the W and Z bosons which are, unlike photons, massive. So the force doesn't get "too far".
However, in our modern description of the forces, electromagnetism and the weak force have to be described by a unified "electroweak" theory and they mix with one another. At distances much shorter than the range of the W/Z bosons, the electromagnetic and weak forces become equally strong and, in some proper sense, indistinguishable.
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What would happen if $F=m\dot{a}$? What would happen if instead of $F=m \frac{d^2x}{dt^2}$, we had $F=m \frac{d^3x}{dt^3}$ or higher?
Intuitively, I have always seen a justification for $\sim 1/r^2$ forces as the "forces being divided equally over the area of a sphere of radius $r$".
But why $n=2$ in $F=m\frac{d^nx}{dt^n}$ ?
| There is a deeper reason for $F~=~\frac{d^2x}{dt^2}$ Within the Galilean group it is an invariant with respect to all changes of frame $x’~=~x~+~vt$. The acceleration of a body is not something which can be made to vanish by boosting to another Galilean frame as
$$
F’~=~\frac{d^2x’}{dt^2}~=~\frac{d^2x}{dt^2}~+~\frac{d^2vt}{dt^2}
$$
where for constant $v$ the second term is clearly zero. The next higher derivative $dF/dt~=~mda/dt$, called a jerk” is also invariant, as are all $d^nx/dt^n$, but the acceleration contained in $T^2_p$ is the lowest element on the jet $T^n_p$ $n~\ge~2$ which is invariant. Further, odd powers of $n$ would not be time reverse invariant under $t~\rightarrow~-t$
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Why is quicksilver (mercury) liquid at room temperature? This is a nice question when you find it out, and I am really looking for a proper answer.
Take quicksilver (Hg) in the periodic table. It has one proton more than Gold (melting point 1337.33 K), and one less than Thallium (melting point 577 K). It belongs to the same group as Zinc (692.68 K) and Cadmium (594.22 K). All not very high melting points, but still dramatically higher than quicksilver (234.32 K). When his neighbors melt, quicksilver vaporizes (at 629.88 K).
What is the reason for this exceptional behavior of quicksilver ?
| Gold has its 6s valence shell unfilled, so is more reactive than Hg. thallium has a valence electron in the 6p shell which again is unfilled. The valence shell of mercury, 6s2 is filled, and is drawn closer to the nucleus because the proximity causes the electron to move at relativistic speeds thereby causing its rest mass to increase, which draws it closer yet to the nucleus. Thus it is even less reactive with other mercury atoms and thus has a lower melting point.
See Mercury
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Glass melting at near absolute zero? I read this report and summarise here but my question is - if quantum mechanics will make glass melt at temperatures near absolute zero and it is near absolute zero then wouldn't this be a huge issue for satellites and space shuttles?
http://www.sciencedaily.com/releases/2011/02/110202102748.htm
Scientists Use Quantum Mechanics to Show That Glass Will Melt Near Absolute Zero
ScienceDaily (Feb. 4, 2011)
Prof. Eran Rabani of Tel Aviv University's School of Chemistry and his colleagues at Columbia University have discovered a new quantum mechanical effect with glass-forming liquids. They've determined that it's possible to melt glass -- not by heating it, but by cooling it to a temperature near absolute zero.
This new basic science research, to be published in Nature Physics, has limited practical application so far, says Prof. Rabani. But knowing why materials behave as they do paves the way for breakthroughs of the future. "The interesting story here," says Prof. Rabani, "is that by quantum effect, we can melt glass by cooling it. Normally, we melt glasses with heat."
| A preprint has appeared for this:
Accepted for publication in Nature Physics., Thomas E. Markland, Joseph A. Morrone, B. J. Berne, Kunimasa Miyazaki, Eran Rabani, David R. Reichman, Quantum fluctuations can promote or inhibit glass formation
http://arxiv.org/abs/1011.0015
While the article does apply to glasses in general, it is not a danger to our spacecraft because their temperatures are nowhere near absolute zero. In fact, even intergalactic space is quite warm, compared to absolute zero, because of the cosmic microwave background radiation (which has a temperature of about 2.7 degrees Kelvin).
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What is the wavefunction of the observer himself? I am aware about different interpretations of quantum mechanics out there but would mostly like to see an answer from the perspective of Copenhagen interpretation (or relative quantum mechanics if you wish).
Let an observer being a man with brain consisting of molecules and atoms. According the basic principles of quantum mechanics each of these particles has a wave function.
The question is: is there a combined wave function of all those particles which constitute the observer? Can such wavefunction be (in theory) determined by the observer himself?
Since the observer cannot isolate himself from his own brain, this would mean that the wave function, at least the part which determines his thoughts is permanently collapsed (i.e. the measurement happens instantly once the state changes). Does this "permanently collapsed" wave function imply special physical properties of the observer's own brain?
Does knowing his own thoughts constitute a measurement? Which moment should be counted as the moment of the collapse of wave function when making measurements on own brain?
Pretend an observer tries to measure the wave function of his own brain by a means of an X-ray apparatus or other machinery and read his own thoughts. Would not his own knowledge of that measurement or its results invalidate the results thus making the whole measurement impossible?
Does the behavior of particles which constitute the observer's brain differ statistically (acoording his measurements) from the behavior of particles which constitute the brains of other people?
Is there a connection with quantum immortality here?
| Look in the mirror and you will see your wave functions squared ;-).
No, I am joking. In the mirror you see an inclusive picture, not elastic one.
The observer himself consist of so many degrees of freedom with so tiny differences between energetic levels that it is impossible to keep it in a pure state. There are permanent transitions leading to decoherence.
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Black holes in a head-on collision Assume two uncharged non-rotating black holes traveling straight at each other with no outside forces acting on the system. What is thought to happen to the kinetic energy of these two masses when they collide? Is the excess energy lost through gravitational radiation? What would the effect of these gravity waves be on matter or energy that they encounter?
| Some energy is lost to gravitational radiation. Some probably ends up in the final black hole (i.e., $m_{\rm final}$ could be greater than the sum of the two initial masses). Figuring out the proportions of these two is not trivial, I would imagine.
The gravitational waves striking matter would not have any terribly dramatic effect. Once you're a decent way away from the black holes, the strain amplitude is low. They'd shake things around a bit.
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Bose-Einstein condensate in 1D I've read that for a Bose-Einstein gas in 1D there's no condensation. Why this happenes? How can I prove that?
| It is necessary to clarify that a uniform, non interacting Bose gas (considered to be confined in a periodic box) in thermal equilibrium does not have a macroscopic occupation of the zero momentum mode if $d<3$. This is not quite accurate for $d=2$ as macroscopic occupation is achieved at T=0, or rather the critical temperature tends to zero in the limit of $N \to \infty$, $V \to \infty$, $N V = {\rm const}$.
This is however not the case if one has external potentials and makes no continuum approximation in the thermodynamics. Additionally attractive condensates $(a_s < 0)$ can form stable, self localised states (solitons) even without confinement in $d=1$. Such states satisfy the conditions for off diagonal long range order required for BEC.
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Mathematical background for Quantum Mechanics What are some good sources to learn the mathematical background of Quantum Mechanics?
I am talking functional analysis, operator theory etc etc...
| The most important background is the extension of linear algebra to infinite-dimensional vectorial spaces. So you introduce Banach and Hilbert spaces, $L^p$ and note that only $L^2$ (that's the space of quantum waves functions) is a Hilbert space.
You must study linear operator on $L^2$, and $l^2$: many attentions must be given to adjoint operators, hermitian and antihermitian operators, unitary operators, proiectors...After that, definition of the norm of a vector and a operator and limited and nonlimited operator (a limited operator is a continuous operator), and the Riesz theorem.
If you study Lebesgue's measure theory is better.
The definition of tensor product of many hilbert space (only a finite tensor product of hilbert spaces is a hilbert space again) is important too.
Last but not least, Fourier analisys, the notion of a complete and orthonormal basis, scalar products and generalized Fourier series; Green functions.
All these definitions and arguments can be found in this book for example: Reed M , Simon B , Methods Of Modern Mathematical Physics, that is mathematically very rigorous and accurate.
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Noether's theorem vs. Heisenberg uncertainty principle In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle.
Because both the principle and the theorem relate energy with time, momentum with space, direction with angular momentum. When this is a general fact then e.g. electrical charge and electrostatic potential(*) should be partners in an uncertainty relationship too. Are they?
I feel that these results look so basic and general that I hope that a pure physical reasoning (without math or only with a minimal amout of math) exists.
Also compare this question where again momentum and space are connected, this time through a Fourier transform.
(*) i.e. electric potential and magnetic vector potential combined.
| Expanding on Marek's comment, they are related, but not in a deep way. They are related by the notion from Hamiltonian mechanics that every dynamical variable can be interpreted as an infinitesimal generator of some canonical transformation, or the quantum mechnical notion that every Hermitian operator generates a unitary transformation.
The Heisenberg principle is true of any variable with a continuous spectrum and the infinitesimal generator of translations in that variable, just because these variables always have a nonzero commutator in every possible state. Position and momentum, angle and angular momentum, charge and phase, these are all conjugates in classical mechanics. The charge operator generates infinitesimal rotations in the phase of charged-particle wavefunctions, not changes in potential (you were probably thinking of the effect of a gauge transformation on a potential, but a global gauge transformation, the kind that gives you Noether's theorem for charge, does absolutely nothing to the potential).
The Noether theorem states that when translations of a certain variable are a symmetry, the infinitesimal generator of those translations is conserved. So translations in x, translations in angle, and translations in phase give conservation of momentum, angular momentum, and charge. But these generators obey the HUP with their conjugate variables.
The relationship is that both HUP and Noether talk about canonically conjugate pairs.
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Gravitational and gauge-gravitational anomalies in ${\cal N}=1$ $D=4$ supergravity coupled to a SUSY gauge theory with chiral matter When people talk about the first superstring revolution they often mention the miraculous cancellation of anomalies via the Green-Schwarz mechanism. My question is whether such a string-theoretic mechanism is also at work when the 4D gravitational and gauge-gravitational anomalies are tackled? In this context, would it be fair to say that a possible discovery of superpartners at the LHC, which automatically implies some version of ${\cal N}=1$ $D=4$ supergravity, imply that stringy couplings (higher order in $\alpha'$) must be present in the corresponding Lagrangian to cancel the anomalies? What type of coupling are those?
| In 4D, we can have an axion mechanism. We have the axion-gauge coupling $\int d^4x\, d^2\theta \, \Phi W^\alpha W_\alpha$. But there are no gauge or gravitational anomalies in MSSM!
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Resistance between two points on a conducting surface Suppose we have a cylindrical resistor, with resistance given by $R=\rho\cdot l/(\pi r^2)$
Let $d$ be the distance between two points in the interior of the resistor and let $r\gg d\gg l$.
Ie. it is approximately a 2D-surface (a rather thin disk).
What is the resistance between these two points?
Let $r,l\gg d$, (ie a 3D volume), is the resistance $0$ ?
Clarification:
A voltage difference is applied between two points a distance $d$ apart, inside a material with resistivity $\rho$, and the current is measured, the proportionality constant $V/I$ is called $R$. The material is a cylinder of height $l$ and radius $r$, and the two points are situated close to the center, we can write $R$ as a function of $l$, $r$ and $d$, $R(l,r,d)$, for small $d$.
The questions are then:
What is $$ \lim_{r \rightarrow \infty} \lim_{l \rightarrow \infty} R(l,r,d) $$
What is $$ \lim_{r \rightarrow \infty} \lim_{l \rightarrow 0} R(l,r,d) $$
| I'll tackle the 3D case. I am using the SI system. It should be noted that the electrical resistance of an electrical element (in given case a homogeneous medium) measures its opposition to the passage of an electric current (in given case direct current). The resistance of a homogeneous medium between two electrodes is defined as $$R=\frac{U}{I}=\frac{\rho\epsilon_0}{C}$$ where C is a capacitance between two electrodes.
Let's start with the assumption that instead of two points we have two conducting tiny spheres with a radius $r_0$. The distance between the centers of the spheres is $d$ and $r_0<<d$. To simplify the calculation let us assume that the charge on the spheres is distributed spherically symmetric. Then $$C=2\pi\epsilon_0r_0$$ Finally, required resistance:
$$R=\frac{\rho}{2\pi r_0}$$
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Should you really lean into a punch? There's a conventional wisdom that the best way to minimize the force impact of a punch to the head is to lean into it, rather than away from it.
Is it true? If so, why?
EDIT: Hard to search for where I got this CW, but heres one, and another. The reason it seems counter-intuitive is that I'd think if you move in the direction that a force is going to collide into you with, the collision would theoretically be softer. You see that when you catch a baseball barehanded; it hurts much more when you move towards the ball, rather than away from the ball, as it hits your hand.
| This movement causes the temper of the neck muscles, and strengthens the spine. The punching power is also distributed by the muscles. The soft flesh does not absorb any impact energy and is simply overwhelmed.
How much more energy is absorbed by the neck less energy will be transmitted to the brain.
Similarly a handball goalkeeper hit the ball. Thus the arm is much more tense. From personal experience he is liable to break the arm if he is distracted.
Best to do: avoid the punch ;)
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Cooling a cup of coffee with help of a spoon During breakfast with my colleagues, a question popped into my head:
What is the fastest method to cool a cup of coffee, if your only available instrument is a spoon?
A qualitative answer would be nice, but if we could find a mathematical model or even better make the experiment (we don't have the means here:-s) for this it would be great! :-D
So far, the options that we have considered are (any other creative methods are also welcome):
Stir the coffee with the spoon:
Pros:
*
*The whirlpool has a greater surface than the flat coffee, so it is better for heat exchange with the air.
*Due to the difference in speed between the liquid and the surrounding air, the Bernoulli effect should lower the pressure and that would cool it too to keep the atmospheric pressure constant.
Cons:
*
*Joule effect should heat the coffee.
Leave the spoon inside the cup:
As the metal is a good heat conductor (and we are not talking about a wooden spoon!), and there is some part inside the liquid and another outside, it should help with the heat transfer, right?
A side question about this is what is better, to put it like normal or reversed, with the handle inside the cup? (I think it is better reversed, as there is more surface in contact with the air, as in the CPU heat sinks).
Insert and remove the spoon repeatedly:
The reasoning for this is that the spoon cools off faster when it's outside.
(I personally think it doesn't pay off the difference between keeping it always inside, as as it gets cooler, the lesser the temperature gradient and the worse for the heat transfer).
| According the fastest way would be keep a flat rectangular thick strip in the coffee with an ice cube at the other end the heat will flow by conduction. And since the temperature difference between the two is large a lot of heat energy will be absorbed by the ice cube to melt.(Taking perhaps a colder object might be better).
This heat is deducted from you coffee and it is cooled down. I think it is quite similar to dropping some ice cubes in a coffee just without diluting the coffee.
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How fast does gravity propagate? A thought experiment: Imagine the Sun is suddenly removed. We wouldn't notice a difference for 8 minutes, because that's how long light takes to get from the Sun's surface to Earth.
However, what about the Sun's gravitational effect? If gravity propagates at the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent, so we could observe the Sun's disappearance more quickly.
What is the state of the research around such a thought experiment? Can this be inferred from observation?
| the fact that distortion travels 'as soon' as a mass is removed or not is not implied in any way by gravity being due to a distortion of spacetime. In fact distortions of spacetime are as limited to travel to the speed of light as any other physical influence.
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If we are willing to accept tachyonic string modes, are there valid projections other than the GSO projections? One reason superstring theory has often been touted as being an improvement over bosonic string theory is that we can impose GSO projections to remove the tachyonic mode. If we insist upon the no tachyon condition, we are left with only five superstring theories.
But what if we don't mind tachyonic modes? After all, they only signify the background we are expanding about is unstable. Can we have more string models then? Type 0 string theories are one example, but are there others? I realize we can't have supersymmetry in target space if we have negative energies, but there's no theorem stating that string theories have to be supersymmetric.
| In ten dimensions this question was answered long ago in the three papers listed below.
The first one found several new possibilities using orbifold projections including a theory without spacetime supersymmetry and with no tree-level tachyon, the latter two papers classified modular invariant spin structures in a fermionic formulation. Below ten dimensions there are many more possibilities.
1) String Theories in Ten-Dimensions Without Space-Time Supersymmetry.
Lance J. Dixon, Jeffrey A. Harvey, (Princeton U.) . PRINT-86-0244 (PRINCETON), Feb 1986. 18pp.
Published in Nucl.Phys.B274:93-105,1986.
2) Spin Structures in String Theory.
N. Seiberg, (Weizmann Inst. & Princeton, Inst. Advanced Study) , Edward Witten, (Princeton U.) . Print-86-0218 (PRINCETON), Feb 1986. 28pp.
Published in Nucl.Phys.B276:272,1986.
3) An O(16) x O(16) Heterotic String.
Luis Alvarez-Gaume, Paul H. Ginsparg, Gregory W. Moore, C. Vafa, (Harvard U.) . HUTP-86/A013, Feb 1986. 15pp.
Published in Phys.Lett.B171:155,1986.
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Do extra-dimensional theories like ADD or Randall-Sundrum require string theory to be true? What I mean is could it turn out that the world is not described by string theory / M-Theory, but that nevertheless some version of one of these extra-dimensional theories is true?
I have no real background in this area. I just read Randall and Sundrum's 1999 paper "A Large Mass Hierarchy from a Small Extra Dimension" (http://arxiv.org/PS_cache/hep-ph/pdf/9905/9905221v1.pdf). Other than the use of the term "brane" and a couple of references to string excitations at TeV scale, I don't see much about string theory, and I notice their theory only requires 1 extra dimension, not 6 or 7.
| In 1974 String theories are shown to require extra dimensions. An object similar to the graviton is found in superstring theories.
However these alternative dimensions or realities must be exactly the same as each other, even to the last grain of sand. For example There can not be different version of you in those universes, they must exactly be the same. To explain why, one must consider the butterfly effect, any small change can lead to large changes, leading to completely different worlds containing completely different contents. So you and I are unique to this universe alone or there are exact copy of us in other universes, much like effect of two parallel mirror
| {
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Electricity takes the path of least resistance?
Electricity takes the path of least resistance!
Is this statement correct?
If so, why is it the case? If there are two paths available, and one, for example, has a resistor, why would the current run through the other path only, and not both?
| If you turn on the water at your sink it comes out the nozzle, not the pipe.(unless you have a leak) Or in the case of a rocket if you ignite the fuel it comes out of the opening. These all have the path of least resistance, if you have two different paths the flow of energy will go through both of them until one of the paths has too much resistance, then the flow of energy will go through only one path. The same basically applies to electrical circuits.
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What is Fermi surface and why is this concept so useful in metals research? What is Fermi surface and why is this concept so useful in metals research?
Particularly, I can somewhat appreciate the Fermi energy idea - the radius of Fermi surface which is a sphere. But is there any quantitative use of more complicated Fermi surfaces?
| One more thing about geometrical properties of the Fermi surface. Its structure defines material transport properties like conductivity. It is actually equal to the integral of the mean free path along those wave vectors that define a Fermi surface.
Knowing this is very important. How do you sample the Fermi surface of a given metal? By means of the de Haas-van Alphen effect.
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Is it really possible for water to be held in a "cone shape" for a brief period of time? I just saw this "trick" where a cup of water is turned over onto a table without spilling (using a piece of cardboard. After removing the cardboard from underneath the cup, the person then removes the cup in a particular way (lifts straight up and twists) and lo and behold, the water stays in it's position as if the cup were still there!? (watch the video to fully understand) Is this really possible? If so, and the real question I'm looking to have answered is, how?
After further research it appears that deionized water is needed as well.
| We used to do the trick with the glass of water and a piece of paper when I was a child. You fill the glass, to the rim it is more successful, cover with a dry piece of paper or a bit of flat aluminum foil, put your hand flat over it, and upend it slowly. Slowly remove the hand. The water does not empty. The paper stays. The weight of the water is balanced by the ambient pressure and the suction that occurs at the top, as gravity pulls the water down and a vacuum is formed at the top of the glass.
At first glance I thought that conservation of the angular momentum given by the twist, same with a spinning top, would keep the water there ( it looks as if it is spinning in the video). But the suction from the removal of the glass (goes plop, I am an experimentalist after all) could not produce such an unperturbed shape, imho. So I vote that it is a trick, although the way the water collapses without any debris of bags is realistic, so there must be video manipulation too.
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Conjectures that have been disproved with extremely large counterexamples I would like to migrate this Math Question into physics. The question is:
*
*Are there conjectures in Physics which have been disproved with extremely large counterexamples? If yes, i would like to know some of them.
| Lots of properties that were found to hold locally (in space and time) turned out to be only local approximations.
Flat earth hypothesis - long journey.
Galilean transformations - breaks at large velocities.
Global curvature of spacetime, locally it is flat - large distances.
Spacetime is not expanding - breaks at large distances (Hubble's law)
Classical mechanics - breaks also at extremely small scales (QM).
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How does Newtonian mechanics explain why orbiting objects do not fall to the object they are orbiting? The force of gravity is constantly being applied to an orbiting object. And therefore the object is constantly accelerating. Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas? I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it?
| The force of gravity has little to do with friction. As dmckee says, what is happening is that the body falls, but precisely because it has enough momentum, it falls around the object towards which it gravitates instead of into it. Of course, this is not always the case, collisions do happen. Also systems of astronomical bodies are complicated and the combined effect of the action of several different bodies on one can destabilize trajectories that in a simple 2-body case would be stable ellipses. The result could be collision or escape of the body.
In the 2-body case however, the crucial aspect of gravity which guarantees the stability of the system is the fact that gravitation is a centripetal force. It always acts towards the center of the other gravitating mass. One can show that this feature implies the conservation of angular momentum, which means that if the 2-body system had some angular momentum to begin with, it will keep the same angular momentum indefinitely.
(Extra note, even in the 2-body case, there can be collisions and escape to infinity, the first if there is not enough angular momentum (for instance one body having velocity directed towards the other body, like an apple falling from a tree), the other if there is too much angular momentum, resulting in parabolic or hyperbolic trajectories.)
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Relationship between magnetic resonance linewidth and spin relaxation First of all, what is the mathematical relationship between measured linewidth (usually in units of magnetic field) and spin relaxation time? I see papers talk about spin relaxation times in terms of linewidths but I have no idea how to correlate the size of the linewidth to an actual time.
Second, is the measured linewidth in NMR or ESR experiments related to $T_2^{\ast}$ only. If you want just $T_2$, you would have to do spin-echo, right? If that is the case, how is $T_1$ determined?
| The theory is rather generic Fourier transform: if you have a perfectly non-decaying oscillation $e^{i\omega_0 t}$ (with real $k$) then the transform gives a perfectly sharp spectrum as a delta function $\delta(\omega - \omega_0)$. But if the excitation decays $e^{i\omega_0 t} e^{-k t}$ then we get $\delta(\omega - \omega_0 - i k)$, which has a real part that is broadened: $1/\left({(\omega-\omega_0)^2 + k^2}\right)$.
Now, usually this manifests as some sort of sweep time, as experimentally you would drive the system at some changing frequency $\omega$ and observe the driven amplitude.
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Are elementary particles ultimate fate of black holes? From the "no hair theorem" we know that black holes have only 3 characteristic external observables, mass, electric charge and angular momentum (except the possible exceptions in the higher dimensional theories). These make them very similar to elementary particles. One question naively comes to mind. Is it possible that elementary particles are ultimate nuggets of the final stages of black holes after emitting all the Hawking radiation it could?
| The short answer is no. Have a look at the wikipedia article on dissipation of black holes.
quote: Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of gamma rays.
The possibility of micro black holes from extra dimensions in some string models still has them dissolving thermodynamically into elementary particles as soon as they are formed.
Edit: Herein I have been replying to the question stated clearly in the last sentence:
Is it possible that elementary particles are ultimate nuggets of the final stages of black holes after emitting all the Hawking radiation it could?
Not to the different question that people seem to be replying to: "are black holes like elementary particles."
A yes answer to the latter, does not reply to the former, i.e. whether quarks and leptons are the nugget, what is left over, from a black hole. A yes answer to this last would offer the intriguing model of the snake eating its tail, maybe quite probable in some new more encompassing theory, but not foreseen now, at least from the answers given. If after shedding innumerable quarks leptons and photons and entropy on the way, a black hole ends up as an electron (for example) in an identifiable quantum mechanical history.By this last I mean something similar to a decay chain in nuclear cascades.
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Physics for mathematicians How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages of Lecture 1 of the book summarizes what I intend by physics from a mathematical stand point. I wanted to find out what are the other good sources for other branches of physics.
| You want the book by V.I. Arnold, Mathematical Methods of Classical Mechanics. It takes a very rigorous, axiomatic approach to Lagrangian and Hamiltonian mechanics, and it should be accessible to, and enjoyable by, a broad spectrum of mathematicians.
For more details see this review by Ian Sneddon, which also covers Walter Thirring's A course in mathematical physics, vol. 1: Classical dynamical systems.
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Do we take gravity = 9.8 m/s² for all heights when solving problems? Why or why not? Do we take gravity = 9.8 m/s² for all heights when solving problems?
| The approximation of 9.81 m/s^2 is a generalisation. The exact value is most likely different at a specific location, due to the distance from the centre of the earth to the point being evaluated.
The reference to "surface of the earth" is also a relative since the earth is known not to be perfectly round due to centrifugal forces making the radius greater at the equator.
Also, since the earth is spinning the same centrifugal forces have a slight influence on object mass at the evaluation point.
In metrology laboratories, the exact value for g is displayed for that exact location.
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Special Relativistic Time Dilation -- A computer in a very fast centrifuge Ok, I've stumbled onto what I think is a bit of a paradox.
First off, say you had some computer in a very fast(near light speed) centrifuge. You provide power to this computer via a metal plate on the "wall" of the centrifuge's container, so it works similar to how subways and streetcars are powered.
If the computer normally would consume 200 watts, how much power would it consume at say 1/2 of light speed? Would it consume 400 watts from our still viewpoint?
Also, what if you were to be capable of communicating with this computer? Would the centrifuge-computer receive AND transmit messages faster from our still viewpoint? I'm a bit lost in even thinking about it.
| If you want your computer to run fastest, leave it at rest ;-).
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List of freely available physics books I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics in general.
What are some freely available great physics books on the Internet?
edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around.
As a starter:
http://www.phys.uu.nl/~thooft/theorist.html
jump to list sorted by medium / type
Table of contents sorted by field (in alphabetical order):
*
*Chaos Theory
*Earth System Physics
*Mathematical Physics
*Quantum Field Theory
| Quantum Mechanics for Engineers
*
*http://www.eng.fsu.edu/~dommelen/quantum/
I stumbled across this book the other day when I was looking for a text on nuclear physics. It seems like a handy resource.
To quote the preface:
The book was primarily written for engineering graduate students who find themselves caught up in nano technology. It is a simple fact that the typical engineering education does not provide anywhere close to the amount of physics you will need to make sense out of the literature of your field. You can start from scratch as an undergraduate in the physics department, or you can read this book.
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Paradoxical interaction between a massive charged sphere and a point charge Suppose we have a sphere of radius $r$ and mass m and a negatively charged
test particle at distance d from its center, $d\gg r$. If the sphere is electrically neutral, the particle will fall toward the sphere because of gravity. As we deposit electrons on the surface of the sphere, the Coulomb force will overcome gravity and the test particle will start to accelerate away. Now suppose we keep adding even more electrons to the sphere. If we have n electrons, the distribution of their pairwise distances has a mean proportional to $r$, and there are $n(n-1)/2$ such pairs, so the binding energy is about $n^2/r$. If this term is included in the total mass-energy of the sphere, the gravitational force on the test particle would seem to be increasing quadratically with $n$, and therefore eventually overcomes the linearly-increasing Coulomb force. The particle slows down, turns around, and starts falling again. This seems absurd; what is wrong with this analysis?
| As a brainstorming answer, lets calculate the binding energy in another way:
suppose we have N electron in the sphere, the electrostatic energy to bring a new electron into the sphere is $Ne/R$. If we add a new electron, only the net electrostatic potential $Ne/R$.
the net potential on far-away electrons is
$N( e - m_{e} G )/R$
what is actually quadratic in $N$ is the energy required to bind together N electrons in the sphere, it is actually
$e/R + 2e/R + ... Ne/R = N(N-1)e/2R$
This also contributes to gravitational weight, but there will be a maximum capacity where the electrons will escape the sphere
the capacitance of a sphere is given by $4 \pi \epsilon R$. So in this case the binding energy is bounded by the capacitance of the conductor used for your sphere
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Does a photon interfere only with itself? I sometimes hear statements like:
Quantum-mechanically, an interference pattern occurs due to quantum interference of the wavefunction of a photon. The wavefunction of a single photon only interferes with itself. Different photons (for example from different atoms) do not interfere.
First of all -- is this correct?
If it is correct -- how do we explain basic classical interference, when we don't care about where the plane waves came from?
I heard that there are experiments with interference of two different lasers -- is this considered as a refutation of the statement? If it is -- how should one formally describe such a process of interference of different photons?
Finally -- such statements are usually attributed to Dirac. Did Dirac really say something like that?
| Lubos Motl's answer is correct, of course, but I would like to point out a physical system in which it appears that photons from different sources are interfering with each other.
Consider two external-cavity lasers, nominally identical but running independently. HeNe lasers will work just fine. On the first laser, mount one of the cavity mirrors on a piezoelectric stack so that the cavity length can be finely controlled. Combine beams from both lasers through a beamsplitter onto a detector, run the output of the detector to an amplifier, and run the output of the amplifier to the piezoelectric stack. Turn the system on, and the first laser will automatically phase-lock to the second laser if the two beams are well aligned. Beams split off from the two lasers will interfere, and can even be used to make holograms, using one beam for reference and the other for object illumination.
Any interference pattern formed using the two beams is comprised of single photon events, but it is not possible to determine which laser any particular photon is coming from without destroying the interference. IF we say that the interference is always due to a photon interfering with itself, then we are forced to accept that the wavefunction of each photon has its source in both lasers.
All this goes to say that a photon is not exactly a particle. Its definition is not simple!
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Does hot air really rise? "Heat rises" or "warm air rises" is a widely used phrase (and widely accepted phenomenon).
Does hot air really rise? Or is it simply displaced by colder (denser) air pulled down by gravity?
| Just one word for you, the convection. Have you heard about it?
Convective heat transfer, often referred to simply as convection, is
the transfer of heat from one place to another by the movement of
fluids. Convection is usually the dominant form of heat
transfer(convection) in liquids and gases
We are just too stupid to understand what it means, thinking that this is just a clever word for the nerds. How do you think hot masses are transferred if hot does not goes up and cool does not goes down?
Moreover, if warm air is not raising then what do you have, where does it go? You say that cool air is sucked into the hot place. What do you have then? Is all the air accumulated in one place in the space? I think that the density is overly high already when you heat the spot. That is why hot molecules start spreading out. This makes air less dense and, thus, lighter, and your mushroom raises up, as your child hydrogen baloon does. Yes, baloons physically raise up because they are lighter than the rest of the air and it is more efficient for the nature to have lighter objects at higher altitudes and have heavier masses at lower altitudes (nature does potential energy minimization).
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I need help with finding distance traveled How do I find the distance traveled of an object if the speed is not constant?
| Integrating velocity is OK, but usually I do simpler things to know the answer.
It depends on the context. Traveled you said?
An odometer is the ideal instrument. Cars, bycicles, pedestrians can use one.
I can use a GPS in cars, bykes, pedestrian, airplanes and sea turtles, etc, complemented by Google Maps.
Trucks have a record of the instant speed for audit purposes (I think), this way is more complicated because you will have to integrate.
A movie cam is sometimes useful to record and keep track of the space traversed. It is used in sports and dancers and to study the body motion.
In football games on TV sometimes they give us the distance that each player traversed. They have to know the angle of the playfield with the recording camera, identify the player .. and SUM to the previous data. A sumation is more used in the real world than integration because we take measures at time intervals and accumulate to previous data. An integral presume that we have a continuous flux of data.
If the object is fast compared to light speed then data must be relativistic corrected as the same if you pretend to measure the space traversed when you walk an escalator in relation to the floor of the escalator itself or the outer building.
How interesting that our minds have an automatic complicated answer.
Answering 'If you want to know the traversed space you must have to know the velocity' forgets that to know the velocity is more difficult (need to know more: the space and the time consumed at every moment)
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Is causality synonymous with continuity? In general relativity, we use the term "time-like" to state that two events can influence one another. In fact, in order for an event to physically interact with another one, they have to be inifnitely close both in time and space.
As far as I know (correct me if I'm wrong) this principal of "near action/causality" is conserved in all branches of modern physics and that is one of the reasons people are looking for "force carriers".
If this is the case, then would it be accurate to say that causality is simply a measure of continuity in all dimensions - and not only the time dimension?
(I don't know anything about continuum mechanics other than its name, but it may have something to do with what I'm asking)
| The answer is "no" if you take precise definitions of "continuity" and "causality". Then these are different concepts.
Indeed. The "background" for general relativity is a manifold which necessarily have some topological structure -- it locally "looks like" an Euclidean space and you just take standard Euclidean topology to your manifold. You can intuitively understand it with the Einsteinian picture of coordinate system, where you have freely falling observer with a clock at every point. That's what I understand under "continuity". But the structure of a manifold is not enough.
The causal structure -- is some "order" that you establish for the points of your manifold. This is the extra structure and it is not the same as the "continuity" in I way one usually understands it.
To put it simply -- you can always continuously move to causally disconnected points (or events) on your manifold.
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Why are Saturn's rings so thin? Take a look at this picture (from APOD https://apod.nasa.gov/apod/ap110308.html):
I presume that rocks within rings smash each other. Below the picture there is a note which says that Saturn's rings are about 1 km thick.
Is it an explained phenomenon?
| The rings have formed where they are because there is greater gravity there. The reason is the shape of the gas giant. It is far from circular. It is wider at its equator. Centripetal force causes a thicker equator. Therefore because the thickness is far wider at the equator, the gravitation at the equator is more than that at the poles. There is more mass pulling you down at that point. The particles of the rings fall into a gravity well.
This explanation is self evident because as can be seen from the above photo, the rings are so incredibly thin, and are PERFECTLY aligned with the equator.
Compare the reason why the moon only shows us one face. It must have more mass on one side. It is like a ship floating on the ocean. The ship has more mass in its hull that gravity pulls into the water.
On the gas giants, given time every asteroid has worked their way to the equator.
I would imagine being an asteroid, it would take a very long time of maybe thousands of years and collisions to achieve such a thin presence. It is very beautiful though!
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Why is the decibel scale logarithmic? Could someone explain in simple terms (let's say, limited to a high school calculus vocabulary) why decibels are measured on a logarithmic scale?
(This isn't homework, just good old fashioned curiousity.)
| It's just because sounds that the human ear is capable of hearing range over a very large range of amplitudes. If you talked about the power delivered to the ear, rather than the log of the power delivered to the ear, you would need to use numbers like $10^{12}$ to talk about airplane engines. So, rather than deal with that, we use logarithims, so that most of the numbers we deal with when talking about sounds vary over reasonable number ranges.
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What is the closest general-relativistic equivalent of a "time slice"? In a newtonian universe, one can talk of a "time slice", that is, the state of the universe at a given point in (global) time. In a "typical" classical universe, a time slice would contain enough information to fully compute the state of the universe at any other point in time, backwards or forwards.
So, disregarding what we know of quantum effects and talking purely of a general-relativistic universe, which concept is closest to a time slice?
| See my answer to this question, which is complementary to yours. The question there is what is a foliation, which as it turns out is the closest thing in GR to a time slice. In a word, foliation of spacetime is a choice of time slicing of spacetime. Unlike the idea of geodesics which are unique after choosing initial conditions, there is a large amount of freedom in choosing such foliations - at least locally (but globally, such time slicing might not even exist). According to GR, they are all equivalent.
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Why do tsunami waves begin with the water flowing away from shore? A sign of a tsunami is that the water rushes away from the shore, then comes back to higher levels. It seems that waves should be both + and - polarized and that some tsunamis should go in the opposite direction. That is the first indication of them would be that the water begins rising. However, other than situations very close to the source, it seems that the wave always begins with the water drawing away from the coast.
For example, the wikipedia article on tsunamis states that:
In the 2004 Indian Ocean tsunami
drawback was not reported on the
African coast or any other eastern
coasts it reached. This was because
the wave moved downwards on the
eastern side of the fault line and
upwards on the western side. The
western pulse hit coastal Africa and
other western areas.
The above is widely repeated. However, when you search the scientific literature, you find that this is not the case:
Proc. IASPEI General Assembly 2009, Cape Town, South Africa., Hermann M. Fritza, Jose C. Borrerob, "Somalia Field Survey after the December 2004 Indian Ocean Tsunami":
The Italian-speaking vice council,
Mahad X. Said, standing at the
waterfront outside the mosque upon the
arrival of the tsunami (Figure 10a),
gave a very detailed description of
the initial wave sequence. At first, a
100-m drawback was noticed, followed
by a first wave flooding the beach.
Next, the water withdrew again by 900
m before the second wave partially
flooded the town. Finally, the water
withdrew again by 1,300 m offshore
before the third and most powerful
wave washed through the town. These
drawbacks correspond to 0.5-m, 4-m,
and 6-m depths. The detailed
eyewitness account of the numerous
drawbacks is founded on the locations
of the offshore pillars.
So is there a physical reason why tsunamis, perhaps over longer distances, tend to be oriented so that the first effect is a withdrawal of the water?
| Dear Carl, the recession of the sea level is an inevitable consequence of the mass conservation: the extra water in the tsunami has to come from somewhere. It comes from both sides - from the region in front of the wave as well as behind the wave.
So if the sea level is most elevated somewhere, it must obviously be lowered at both sides, too.
Imagine that you have a wave packet given by the function
$$ f(x) = \exp(-ax^2) \cos (kx) $$
The maximum absolute value of $f(x)$ is at $x=0$, right? That's the tsunami. However, it's inevitable that the second highest absolute value which can still be seen is at the pair of the nearby local minima. The two maxima at even higher values of $|x|$ are already pretty much negligible.
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Coriolis effect on Tsunami The Japanese tsunami, moving at about 700 km/h, affected areas as distant as Chile's coast, 20 hours after the earthquake. How does the Coriolis force affect tsunami? Also, I saw an image of a boat caught within a large whirlpool. Is the whirlpool's rotation due to Coriolis force?
| Your adjective "large" in "large whirlpool" may be very misleading. If you mean hundreds of meters, no effect of the Coriolis force may be visible by a naked eye at this scale. The origin of the whirlpool had to be different.
Quite generally, tsunami is all about waves, and if one has a wave, molecules of water are moving back and forth, in circular patterns - and the radius of the circle doesn't exceed 10 meters - the height of a tsunami.
That's very different from the motion of water in rivers which flow along straight lines and the Coriolis force always acts in the same direction.
However, if the water molecules are making periodic circles, the effect of the Coriolis force is not too different from the case of the Foucault pendulum,
http://en.wikipedia.org/wiki/Foucault_pendulum
The plane in which the circle belongs tends to rotate by 360 degrees in 24 hours or so - which is actually comparable to the lifetime of a tsunami on the ocean. However, I think that the internal dynamics of the water tends to redirect all the motion in the waves so that it can only go up and down, or in the same circular patterns correlated with the direction of the wave.
My guess is that the impact won't be measurable at the end - because radially centered waves on the ocean is the only resulting pattern I can easily imagine - but I am not quite sure.
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What are the modern marvels in Physics that seem crazy but true? What are the modern marvels in Physics which are crazy but true? Ideas which are ridiculed and dismissed in the beginning but passed the test of time?
| Relativity and Quantum Theory. That whole division of the waters thing, of the infinite and infinitesimal, leaving our everyday world of beer and sandwiches in the middle. But I still don't understand mass, or gravity or momentum.
Cataclysm Theory and Continental Drift. Loads of stuff.
Some of the replies seem a bit shirty, but I'm glad to have found this blog. After all, we have a vicar in every parish and most of them willing to discuss theology; its high time science/scepticism offered the same. I promise not to ask about flying saucers or quantum wormholes, well maybe quantum wormholes...
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Can a nuclear reactor meltdown be contained with molten lead? If lead can absorb or block radiation, would it be possible to pump molten lead into a reactor core which is melting, so that it would eventually cool and contain the radiation?
Is there something that can be dumped into the core that will both stop the reaction (extremely rapidly) AND will not combine with radioactive material and evaporate into the atmosphere, thus causing a radioactive cloud?
| Meltdown containment: Sounds like an option would be boron mixed with wet wet cement pumped through long flexible tubing with a water flush to keep it moving and a pump replaceable and far away from the meltdown far away from being affected. Forget recovery (containment). Pile it high and thick over the top. Dig deep underneath and do the same. Bury it.
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For an accelerated charge to radiate, is an electromagnetic field as the source necessary? For an accelerated charge to radiate, must an electromagnetic field be the source of the force?
Would it radiate if accelerated by a gravitational field?
| If you mean an external EMF, the answer is "The radiation is determined with an external filed". The charge acceleration is proportional to the external field, and a single accelerated charge radiates.
If you mean the radiated field influence on the charge motion and subsequent radiation, the answer is "No" because the radiation is only expressed via external field. There is no need to invoke the proper field here.
In QM radiation of hard single photons happens discontinuously in time so QM is somewhat different.
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How much radiation exposure in the US was caused by the 105 nuke tests in the Pacific? Between 1947 and 1962 the US conducted 105 tests of nuclear weapons in the "Pacific Proving Grounds". I'm wondering how much radiation exposure resulted on the west coast of the US. These were part of the 1056 nuclear bombs that the US has ignited over the years (most underground, but two notably in Japanese cities).
So how much radiation exposure in the US was caused by the 105 nuke tests in the Pacific? Should the inhabitants of the West coast have taken iodine pills?
I'd also like to know how much fallout these bombs produced, as compared to the reactor steam releases in Japan, 2011.
| There are a lot of works contained in proceedings related to the IAEA, but one very interesting publication is a study of deposition of long lived isotopes as monitored at Tsukuba since 1956. The period of study includes all of the nuclear weapons testing period, and also capturing the Chernobyl accident. The report cited the total deposited activity (in Bq/m^2) for the year, and essentially shows that the deposition at it's highest for Cs-137 is ~2x10^3 Bq/m^2 for the year, in the early 60's (Bq=Becquerel=1 decay/second). After the testing stopped, this amount gradually decreased, spiking high for the year 1986 (Chernobyl), and then decreasing again.
You could make the crude approximation that the activity is uniformly dispersed in the atmosphere due to the prevailing winds causing dilution and mixing, so that sites world wide would note a similar deposition of radionuclides over time.
If one makes a further rough approximation that the Fukushima releases are about a tenth that of the Chernobyl releases, which is a number cited in the news media, one could expect about one tenth of 100 Bq/m^2 for long-lived isotopes, or 10 Bq/m^2 for the year. This may be incorrect, as the values cited for release of Cs-137 are pretty disparate in published reports, with some citing significantly more release of Cs-137 (up to and exceeding the Chernobyl levels). It's probably still a little too soon to tell.
Citation: Hirose, et. al: Analysis of the 50-year records of the atmospheric deposition of long-lived radionuclides in Japan, Applied Radiation and Isotopes, Volume 66, Issue 11, November 2008, Pages 1675-1678
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Why does nuclear fuel not form a critical mass in the course of a meltdown? A BWR reactor core may contain up to 146 tons of uranium. Why does it not form a critical mass when molten? Are there any estimates of the critical mass of the resulting zirconium alloy, steel, concrete and uranium oxide mixture?
| My guess would be that the moderator (normally the water, graphite already would have caused the accident) is missing and that is still vital in a compacted mass. The uranium oxide is not enriched to military grade, but I can't speak for MOx-elements.
Should be difficult to say without some decent figures about leftover UO_2, mass, previous distribution.
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What is the most energy efficient way to boil an egg? Starting with a pot of cold tap water, I want to cook a hard-boiled egg using the minimum amount of energy. Is it more energy efficient to bring a pot to boil first and then put the egg in it, or to put the egg in the pot of cold water first and let it heat up with the water?
| My cookbook says an egg should simmer in water for about 10 minutes. If you wait till the water simmers and then add the egg, the water will start boiling again in a few seconds, then needs to simmer for the 10 minutes. The heat of vaporization loss takes place for about 10 minutes.
If the egg is placed in cold water first, it'll cook in fewer than 10 minutes of simmering as the pot heats up to boiling, so the heat of vaporization loss will be less than 10 minutes.
| {
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Light bending surfaces and energy funneling properties In the last year there has been fuzz regarding metamaterials studied for the purpose of cloaking, and it seem to me they are somewhat glorified 2D waveguides, but in any case it seems a reasonable application of this to funnel radiation captured over a wide area of incidence into smaller area, and i thought that this could be a cheap way to produce ignition radiation for a fusion reactor.
Reasons why this wouldn't work, or not produce enough energy density? All stuff i've read about fusion ignition with lasers doesn't seem to depend too heavily on the coherence of the radiation
| Assuming that you have a source whose temperature is T, the best any optics system can do (in applying the energy from the source to a target) is in such a way that the target may be heated up to the temperature T. This is because of a basic law of thermodynamics: heat flows from hot things to cold things and never vice versa.
There's another way of saying this that applies to imaging optics; that is, to optical systems that produce images. I'll edit it in if I find it. But the mechanism you're describing is more general than an optics system.
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Quaternions and 4-vectors I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity:
$$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$$
Is it useful? Is it used? Does it bring anything? Or is it just funny?
| You've stumbled on a fertile area. Though not strictly what you were asking about, I can tell you that perhaps the most interesting relationship between orthogonal groups and quaternions comes from looking at spinors. As you may know, the symmetry group called $Spin$ double covers the group of rotations, and is the more relevant group for physics since spinors transform under this larger group. A useful example is the double cover $SU(2) \rightarrow SO(3),$ that is, $SU(2) = Spin(3)$ in the Euclidean signature.
Topologically, $SU(2)$ is a 3-sphere, which we can think of as the unit quaternions (remember, the norm is Euclidean, as pointed out by others). To understand the map $SU(2) \rightarrow SO(3),$ let $v$ be an imaginary quaternion (which we can think of as a 3-vector), and let $q$ be in $SU(2).$ Then since multiplication of quaternions preserves the norm,
$$\overline{q} v q$$
has the same norm as v, and you will note that it is still imaginary. Thus, the action $v\stackrel{q}{\mapsto} \overline{q}vq$ of $SU(2)$ on $R^3$ (the imaginary quaternions) is by a rotation. Furthermore, $q$ and $-q$ act the same way. So we have described a double cover of $SU(2)$ onto 3-dimensional rotations.
Maybe this is not what you immediately inquired about or discovered, but it's probably worth knowing.
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Where's earths death bulge, destroying everything in it's path? I was watching a BBC documentary on space last night. It was talking about gravity, and it said that the reason we only ever see one side of the moon, is because the earths gravity is strong enough to actually stretch the moon into a longer shape, creating a 7m bulge on the moon.
This bulge travelled around the moon. It was described on the documentary as something along the lines of, if you were there, you would see the ground make a 7m high wave as it rotated. The bulge then acted a brake, and the moons spin slowly ground to a halt until where we are today when it's spin is very very slow so it looks like we only see one side.
So where is this giant death bulge tidal wave on earth? Shouldn't we been seeing the ground rising up a few meters in relation to the suns position, demolishing and killing everything in its path?
I assume the documentary grossly simplified what actually happened, or I misunderstood! I'm a physics noob, please go easy on me, chances are I'm not going to understand any formulas that use anything but addition or subtraction.
| It would be good if you read an article about tides, and the bulge the sun/moon combination makes on the earth, seas and crust. Earth tides are interesting, the whole crust moves reaching 55 cms at places.
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If the universe were a fractal Inflation seems to solve many of the problems of cosmology like horizon problem, flatness problem etc. Now suppose, I am a devil's advocate and tries to find holes in this beautiful theory. I argue that the early universe were having a fractal geometry. For any generic initial conditions suppose it was indeed a fractal. Then obviously no amount of stretching can make it smooth and flat locally. How would one reconcile this picture with inflation?
| The universe is a fractal according to inflation, or very nearly. The product of inflation is very close to a scale-invariant spectrum, which is a fractal spectrum, in that rescaling the fluctuations gives almost the same result. This fractal-ness is amplified to the current filamentous fractal structure of galaxies and dark-matter distribution.
When people say "the universe is a fractal" they don't mean nearly scale-invariant mass distribution, which is a consensus view. What they mean is that the distribution is zero density over large distances, to resolve Olber's paradox as suggested by Mandelbrot in "The fractal geometry of nature". This type of idea is not correct, because the universe is homogenous at large scale.
Stretching by inflation will smooth out anything to the special type of inflation initial condition fractal, no matter how differently fractal, because all the structure falls out of the very small cosmological horizon. During inflation, the universe is like an inside-out black hole, we are surrounded by a horizon, and everything falls away from the center into the horizon, leaving a smooth middle together with a fractal (scale invariant) fluctuations.
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Gravity and the Standard Model Gravity is ignored in the SM. The proton rest mass is ~0.938 GeV/$c^2$. LHC protons will move with 7 TeV energy, presumably with a relativistic mass about 7,450 times rest mass. A cosmic ray with the highest energy was detected at about $6\times 10^{21}$ eV. If it was a proton, its relativistic mass would have been about $6.4\times 10^{12}$ times rest mass. My question is, at what energy levels would it be necessary to include gravity in the SM? I recognize this isn't an issue for forseeable particle accelerators, but astronomical events like supernova are capable of generating extremely high energy particles.
| This question is a bit open ended. There are theories and maybe some experimental hints of extra large dimensions and black hole or AdS amplitudes with heavy ion collisions. The jury is of course still out, but there might be some gravity physics creeping into the standard model + QCD in the TeV energy range and above.
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Representation of quantum transformations as matrices I was reading Quantum Computation explained to my mother, which makes the following claim
Postulate 2 A closed physical system
in state V will evolve into a new
state W , after a certain period of
time, according to W = UV where U is a
n × n unit matrix of complex numbers.
Here, V is a column matrix with n rows. Can anybody justify this assumption?
| If we assume time evolution preserves the Hilbert space norm, then Wigner had shown it can only be a linear unitary transformation, or an antilinear antiunitary transformation. If time evolution is continuous in time, it can't possibly be antiunitary as there's no continuous deformation of the identity operator to an antiunitary transformation.
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What future technologies does particle physics and string theory promise? What practical application can we expect from particle physics a century or two from now? What use can we make of quark-gluon plasmas or strange quarks? How can we harness W- and Z-bosons or the Higgs boson? Nuclear physics has given us nuclear plants and the promise of fusion power in the near future. What about particle physics? If we extend our timeframe, what promise does string theory give us? Can we make use of black holes?
| Here are some possible applications of neutrinos
Submarine neutrino communication
Demonstration of Communication using Neutrinos
Searching for cavities of various densities in the Earth's crust with a low-energy electron-antineutrino beta-beam
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Definitions and usage of Covariant, Form-invariant & Invariant? Just wondering about the definitions and usage of these three terms.
To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are synonyms?
"Invariant" on the other hand refers to physical quantities?
Would you ever use "invariant" when talking about a law? I ask as I'm slightly confused over a sentence in my undergrad modern physics textbook:
"In general, Newton's laws must be replaced by Einstein's relativistic laws...which hold for all speeds and are invariant, as are all physical laws, under the Lorentz transformations." [emphasis added]
~ Serway, Moses & Moyer. Modern Physics, 3rd ed.
Did they just use the wrong word?
| This is a good question because I think physicists nowadays don't understand the difference between form invariant and covariant.
The equations of physics are form invariant under a Lorentz transformation, but they're not co-variant as in they don't vary with the Lorentz transformation.
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Why there's a whirl when you drain the bathtub? At first I thought it's because of Coriolis, but then someone told me that at the bathtub scale that's not the predominant force in this phenomenon.
| The whirl happens in the draining tube, whose optimal solution to drain the bathtub is a laminar flow allowing for some rotation in the tube. What you see in the surface is the match between the solution of flow in the tube and the solution of flow in the surface.
Angular momentum of the flow gets modified a lot as the tube twists and twists, sometimes even siphoning up and down.
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How might a resonant antenna and black body radiation interact? How does an antenna behave when it is cooled so that its black-body radiation is emitting energy at its resonant frequency?
Edit: To clarify, its not how they're related in general, but how might thermal radiation and resonance interact with each other when their spectra are aligned well?
Edit: Also, I'm sure that the thermal radiation spectra that have a significant peaks are associated with incredibly high temperatures, and peak at incredibly small wavelengths, rendering such an antenna completely impractical to build. Still, I'm still interested in the theoretical concept.
| They're not related. The black-body radiation as well as the resonance curve may look like "bumps" but they are very different bumps mathematically. The black body radiation gets emitted at all frequencies, and the "uncertainty of the frequency" is maximized, in some sense. On the other hand, resonances are peaked around a particular frequency.
Resonance curves are about matrix elements between pure states; thermal curves are traces over the whole Hilbert spaces so they arise from mixed stated. That's why the exponentials only appear in the thermal curves.
So the only thing they share is that they produce intensities as a function of frequency - but many other things in physics do the same thing - and in both cases, complex numbers are useful ($E_0-i\Gamma/2$ for resonances and imaginary time $i\beta$ in the thermal case) - but complex numbers are useful across physics.
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Alternative career paths in physics? What do you guys think of alternative career paths in physics away from academia free from the usual academic shackles?
Examples: Garrett Lisi who spends his time surfing and skiing while not working on E8.
Or Daniel Bedingham who is able to support himself part-time as an investment banker when not working on quantum mechanics.
Or Julian Barbour who supports himself by translating Russian journal articles into English when not working on quantum gravity.
Is it feasible to count upon Foundational Questions Institute to support such unconventional career paths using Templeton money?
| Before closure, thanks for informing me of the foundational question institutes ' existence :) .
According to the link :
It has run two worldwide grant competitions. The first competition provided US $2M to 30 projects
Any researcher who can compete is free to try for those 30 grants. On the other hand academic and research posts run into the hundreds.
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Does the foam on top of boiling maple sap affect the rate of evaporation? This is a serious question from someone engaged in evaporating large quantities of water to turn sap into syrup at this time of year.
Probably some background will help. When sap boils vigorously it creates quite a bit of foam, which will overflow the evaporator (incidentally filling the building with a pleasing maple caramel smell as it burns on the side of the evaporator). When the foam gets too high we touch it with a bit of lard and the foam level drops (surface tension - I know). However, it is tempting for me to give a good swipe so that the foam almost disappears (instead of just dropping). The old-timers however contend that I should just reduce the foam to the point where it isn't overflowing any more. They say that it will take longer to boil away the water if I eliminate the foam.
I fail to see how the foam will improve evaporation (although it seems to me that it might slow it down).
Edit: by request ( @georg ) , a link to the evaporator in question https://sites.google.com/site/lindsayssugarbush/_/rsrc/1240515239201/Home/2005-03-30--12-25-21.jpg
| I'm new at making syrup but the foam issue seems to relate to boiling water in a pot. A pot of hot water will rapidly boil with a lid, in this case a layer of foam, while an uncovered pot will barely form bubbles at the bottom of the pot. The foam seems to form a layer of insulation that allows the sap to reach a higher temperature to increase the rate of evaporation.
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Measuring acceleration of earth due to its fall around the sun Every orbiting of a satellite around a mass is nothing else but a constant fall - and therefore acceleration - towards this mass. In a way it is a "falling around" that mass.
My question Is it possible to measure this acceleration on earth due to its "falling around" the sun?
| The answer depends in part on what you mean by "measure". You can certainly calculate the acceleration using the laws of kinematics and dynamics, as Lawrence B. Crowell points out in his answer. Does that, in your mind, count as "measuring the acceleration"?
Here's a much more direct recipe that I think would uncontroversially count as measuring the acceleration. Pick a set of fixed stars, and measure the velocity of the Earth relative to them using the Doppler effect. Any one measurement only gives you one component of ${\bf v}$, but if you measure a few, you can get the full vector. Do this twice, at two different times, and calculate $\Delta{\bf v}/\Delta t$.
With current technology it would be very easy to do this to the required accuracy. In fact, astronomers are making precisely these required measurements all the time -- not with the purpose of measuring the acceleration, but for a wide variety of other reasons. In fact, astronomers often deliberately subtract out the time-varying Doppler shifts due to Earth's changing velocities, because what they're interested in are (often much smaller) changes due to the actual motions of the stars.
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How many Gs would a driver of the Bugatti Veyron experience on the Ehra-Lessien track when cornering before record attempt? Apologies if too specific.
Watched a documentary National Geographic Megafactories Bugatti Veyron
I told an colleague (engineer) about the need to warm the car up and then unleash it on the long stretch to be able to make it to 405km/h and mused that the curve before the straight probably had to be taken at 250km/h to do so. He immediately replied that the G-force of going around a(ny) curve at 250km/h would kill the driver... I find that very hard to believe but do not have the skills nor knowledge to refute it. It would be cool to get him this app http://www.dynolicious.com/ and a trip in a Veyron ;)
Here is the Ehra-Lessien track and here is the actual curve where some of the clever people here could glean the radius - I think 350m
Can someone show some math and a graph or so?
I have been looking at http://en.wikipedia.org/wiki/Formula_One_car#Lateral_force
Turn 8 at the Istanbul Park circuit, a
190° relatively tight 4-apex corner,
in which the cars maintain speeds
between 265 and 285 km/h (165 and 177
mph) (in 2006) and experience between
4.5g and 5.5g for 7 seconds—the longest sustained hard cornering in
Formula 1.
which supports my claim - but would love some more information on this specific matter
Here is a link on what an Apex means in motoring
And here are some formulas
| Neglecting friction, the force experienced is the Centrifugal Force $F=\frac{mv^2}{r}$ (it would be less if you included friction since the car actually slips) vectorially added to the orthogonal gravitational force $F_g=mg$, i.e. $F = m\sqrt{\left(\frac{v^2}r\right)^2 + g^2}$ where $g = 9.81 \frac{m}{s^2}$1. Divide this by $F_g$ to obtain a result in Gs. Remember to use SI-units, i.e. divide a km/h speed by 3.6 (1000 km/m / 3600 s/h) to obtain m/s.
For the 350 m curve and 285 km/h that yields about 2.08 gs only, to obtain the 5.5 gs mentioned a radius of about 118 m is required, or some higher velocity (490 km/h for the 350 m curve).
1) Thanks J.H. for this important correction!
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Are the basic postulates of QM the only set of postulates that can give rise to a sensible semi-probabilistic physical theory? Are the basic postulates of QM, such as complex Hilbert space, unitary evolution, Hermitian operator observables, projection hypothesis etc., the unique and only set of postulates that gives rise to a semi-deterministic and semi-probabilistic theory, in which the time evolution is non-degenerate? By non-degenerate, I mean different initial states never produce the same final state probabilities, which in QM is guaranteed by unitarity.
Phrased in another way, is it possible to prove from some general principles, such as semi-determinism, semi-reversibility (not for collapse), causality, existence of non-compatible observables etc., that a physical theory must satisfy these postulates? In particular, is it possible to prove that complex numbers, or a mathematical equivalent, must be fundamental to the theory?
I haven't studied anything about foundational issues of QM, so feel free to point out if I'm being a crackpot. I suppose this question may be similar to something like "can you prove that gravity must be a metric theory entirely from the equivalence principle?", whose answer is no, but I'll be glad if it turns out to be otherwise.
| No, in fact our current postulates allow certain ambiguity in the description of the same physical system (or at least of the possible set of measurements we can extract from them), suggesting that there might be a more concise underlying theory (or set of postulates) that groups such descriptions under a equivalence class
look at the answer to this question for an example.
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What reflective media do laser shows use? I am having a hard time in finding out what exact light media laser shows use. I am trying to build a laser show myself. I know that the laser light is reflected off these particles in such a way that that it makes the laser line "viewable" in all directions
Can somebody explain to me how exactly do the collection of particles make it viewable in all directions and what exact conditions are necessary? Does the angle of the incoming laser light matter? Does the size of the particles matter? Does the uniformity of how the particles are dispersed matter? Would water vapor work?
I have tried using a fog machine, but the red laser that I am using only reflects off of the fog particles in a way that makes it viewable only from a certain perspective. This would not be a good show to the people standing in one side of the room vs. another.
| Note that there is a huge difference between standard particles (elementary, atoms, or molecules) and droplets. Droplets are quite macroscopic and having a definite spherical shape, they would act as nontrivial optical medium (think about rainbows caused by water droplets) which could selectively prefer some directions. So fog is definitely not recommended.
On the other hand, if you have some gas at room temperature the scattering should be effectively classical and in all directions. So my recommendation would probably to use some non-lethal ;) gas.
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Where does the energy go, when light is blocked by polarisation I've been looking around about LCD monitors, and how they polarise light. When a pixel needs to be black, the light is "twisted" so that it can't go through the polarising sheet in front.
What happens to this light? Does it relfect back into the screen? Surely that would mean the screen would get quite hot if you leave it on a black image for a while? (but I've never noticed that happen).
What happens to the energy (from the electricity used to create the light) when it's blocked by polarisation.
| It is possible to "block" light based on its polarization in a number of ways. In the situation you are describing, where the light hits a polarizing filter, it is simply absorbed by the filter. The filter does indeed heat up, and in fact if you put your hand near the screen you can usually feel that it is quite warm.
It is also possible to have a polarized mirror, which either reflects or transmits light based on its polarization. In this case the mirror doesn't absorb any* of the energy, so it doesn't heat up.
*In the ideal case. In reality, of course, there is some small amount of absorption, but it can usually be ignored.
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How to measure the spin of a neutral particle? If a charged particle with charge $q$ and mass $m$ has spin $s \neq 0$ we can measure an intrinsic magnetic moment $\mu = g \frac{q}{2m}\hbar \sqrt{s(s+1)}$. This is how spin was discovered in the first place in the Stern-Gerlach Experiment.
But for a neutral particle $\mu = 0$, so we cannot measure the spin of the particle in the same manner. But it is said, that e.g. the Neutron or the Neutrino both have a spin $s=1/2$. How was or can this be measured?
| another way to measure 'spin' is through scattering experiments as the scattering cross sections depend on the spin-spin interactions- the most famous example is two types of hydrogen molecules called ortho and para -hydrogen whose scattering with neutrons yielded different cross sections-see any standard textbook on nuclear physics.
I do agree that nucleons/elementary particle's spin quantum states are not related to specific charges on the particles- neutral ones can have 'particle clouds' inside it and they can contribute to the spin state.
some comments atribute spins to the quarks inside the neutron- but spin was available before the 'quarks' were formulated!
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What is the difference between $|0\rangle $ and $0$? What is the difference between $|0\rangle $ and $0$ in the context of $$a_- |0\rangle =0~?$$
| $|0\rangle$ is just a quantum state that happens to be labeled by the number 0. It's conventional to use that label to denote the ground state (or vacuum state), the one with the lowest energy. But the label you put on a quantum state is actually kind of arbitrary. You could choose a different convention in which you label the ground state with, say, 5, and although it would confuse a lot of people, you could still do physics perfectly well with it. The point is, $|0\rangle$ is just a particular quantum state. The fact that it's labeled with a 0 doesn't have to mean that anything about it is actually zero.
In contrast, $0$ (not written as a ket) is actually zero. You could perhaps think of it as the quantum state of an object that doesn't exist (although I suspect that analogy will come back to bite me... just don't take it too literally). If you calculate any matrix element of some operator $A$ in the "state" $0$, you will get 0 as a result because you're basically multiplying by zero:
$$\langle\psi| A (a_-|0\rangle) = 0$$
for any state $\langle\psi|$. In contrast, you can do this for the ground state without necessarily getting zero:
$$\langle\psi| A |0\rangle = \text{can be anything}$$
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Can a disk like object (like UFO's) really fly? UFOs as shown in movies are shown as disk like objects with raised centers that emit some sort of light from bottom. Can such a thing fly?
My very limited knowledge in physics tell me that a disk like object may not be able to maneuver unless it has thrusters on sides and simple light can not be enough to make any object go up in the air.
Is it possible?
| The lift works on the craft body by interaction between the curren on which flows through the body surface and the magnetic field of the body center. ”The law on the left Fleming."
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What is a Kustaanheimo-Stiefel transformation? What is a Kustaanheimo-Stiefel transformation? Which applications has it in physics? Can you point me to a reference, where this transformation is explained?
| http://arxiv.org/abs/0803.4441
The Kustaanheimo-Stiefel transform turns a gravitational two-body problem into a harmonic oscillator, by going to four dimensions. In addition to the mathematical-physics interest, the KS transform has proved very useful in N-body simulations, where it helps handle close encounters. Yet the formalism remains somewhat arcane, with the role of the extra dimension being especially mysterious. This paper shows how the basic transformation can be interpreted as a rotation in three dimensions
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Is it possible to know the exact values of momentum and velocity of a particle simultaneously? I know that by Heisenberg's Uncertainty Principle that it is not possible to know the exact values of position and momentum of a particle simultaneously, but can we know the exact values of momentum and velocity of a particle simultaneously? I would think the answer would be no because even if we were 100% certain of the particle's position, we would be completely unsure of the particle's momentum, thus making us also completely unsure of the particle's velocity. Does anyone have any insight into this?
| The argument that Heisenberg's Uncertainty Principle prohibits that we can know the exact values of momentum and velocity of a particle simultaneously is already discredited in the old textbook by Feynman on Quantum Electrodynamics.
Two observables can be simultaneously determined if the operators commute. For velocity and momentum, the operators commute $[\hat{p},\hat{v}]=0$; they do even in the Dirac wavefunction theory with its Zitterbewegung effects.
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Supergravity calculation using computer algebra system in early days I was having a look at the original paper on supergravity by Ferrara, Freedman and van Nieuwenhuizen available here. The abstract has an interesting line saying that
Added note: This term has now been shown to vanish by a computer calculation, so that the action presented here does possess full local supersymmetry.
But the paper was written in 1976! Do you have any info what kind of computer and computer algebra system did they use? Is it documented anywhere?
| (This is not really an answer, but here I have not yet enough reputation to post comments. If someone wants to move this to a comment, I won't object.)
1976 is not a particularly early date for computer calculations: Fermi, Pasta, and Ulam used computer simulations in the early 50s for their 1955 paper.
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Why does a rotating tire use the static, rather than the dynamic coefficient of friction? The explanation I have heard of the difference between static and dynamic friction is that static friction is stronger because bonds form when one object is put on top of another object and these have to be overcome to get the movement started.
For a rotating tire, although the point on the ground will be stationary for an instant, it would seem that bonds wouldn't have time to form. So, why isn't the dynamic coefficient of friction used?
| I am not sure why you are rejecting the static friction on the basis on long the parts are in contact. A "bond" is not a chemical bond that might take time, but rather an interaction between adjacent molecules, or atoms. It propagates at the speed of light, so there is plenty of time for the adjacent molecules to "bond" when sufficiently close enough.
In real life though, pairing down the tire/road contact into a friction coefficient is the wrong approach. It is a non-linear contact, where the higher the normal load the wider the contact patch is and the distribution of contact pressures changes. In addition, some parts have micro sliding as only 1 point in the contact patch is truly stationary.
There is something called the "Pacejka Magic Formula" which is a well established model of a tire contact, and there are newer ones out there which minor and major refinements to it. In the end, it depends on what you want out of it, in order for you to decide what contact/traction model to use.
[ref: Magic Formula]
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Why do they store gold bars with the narrow side down? I watched on TV as they where showing gold bars stored in bank vaults and I noticed that they always stack them with the narrow side down and the wide side up. Like this:
So there has to be a mechanical reason why is that. Any ideas?
| The primary reason I have usually heard is that this makes it easier to lift them.
Pure gold is quite smooth, and very dense, and would be near impossible to lift from the other side.
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What are constraints on a "purity" operator in quantum mechanics? Consider the normalized state, written in some orthonormal basis as:
$$\psi = A |0\rangle + B |1\rangle$$
Let's define a "purity operator" for a basis as any operator whose expectation value gives 1 for a pure state in this basis, and 0 for the most mixed state in this basis. Inbetween states should give between 0 and 1, although the specific value doesn't matter.
One possible example (please note my question is on the general case though, this specific example is just to aid discussion), is
$$\langle \mathcal{O} \rangle = 1 - 4 \frac{|A||B|}{|A|+|B|}$$
What mathematically prevents such measurements in quantum mechanics?
| From the example you've given, it's clear that you're using the wrong terms to describe what you want. Purity and mixedness apply to density operators and not to state vectors -- if your system is described by a state vector $|\psi\rangle$, it is already pure.
What you seem to want to know is whether there is an operator that, in a particular basis, has an expectation value that lets you know to what degree the state is in a superposition of basis states. I think the uncertainty operator might work for you here. If $S$ is the operator whose eigenstates are $|0\rangle$ and $|1\rangle$, then the operator $(S-\langle S\rangle)^2$ has zero expectation value for the basis states and a non-zero value for all superpositions. You could scale this in some way to give you a value between 0 and 1.
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What is meant by positive and negative gravity/energy/spacetime-curvature? I have recently come across some cosmological assertions (based on empirical data) about the universe being self contained in the sense that it is entirely capable of coming into existence from a zero-energy initial state. This is based on the observation that at grand scale the positive and negative gravity/energy etc. cancel out each other.
What do the terms positive and negative actually mean in this context?
| No answer I'm afraid but I'd add my naive request to that of Mumtaz. It's true that I did hear this "negative energy" claim from a popular science TV program but since it was made by Stephen Hawking himself one cannot dismiss it. He seemed to suggest that in the co-emergence of space and mass-energy to form the universe, the space component embodied the negative energy required to balance the positive mass-energy component.
I find this difficult as I understand "space" to mean separation of material bodies, which under the attractive force fields (gravity, strong and weak nuclear forces etc.) represents positive potential energy (give or take a cosmological constant).
Perhaps Hawking means that potential energy in space thus affords a sink of kinetic and radiant energy and might thus be described as a negative energy reservoir. The "out of nothing" condition would imply however that this reservoir can be somehow be exactly filled by absorbing all kinetic and radiant energy in a final state of separation. Such a theory should surely have something to say about the approach to that endpoint. Does it?
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Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation) We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is
$$\nabla^2\frac{q}{r}~=~-4\pi q~ \delta^3(\vec{r}).$$
My question is, what is the Laplacian of $\frac{1}{r^2}$ (at the origin!)? Is there a charge distribution that would cause this potential?
| Vladimir's answer is off by factor of 2. The laplacian is $\nabla^2(\frac{1}{r^2}) = \frac{4}{r^4}$ A potential that falls of as $\frac{1}{r^2}$ is a dipole (In general, if it falls off as $r^{-n}$ its an ($2^{n-1}$)-pole, e.g $\frac{1}{r^3}$ bheaviour is quadrupole, etc).
Is this a dirac delta? To find out, check : $$\int_{\mbox{All space}}\nabla^2(\frac{1}{r^2})d^3r $$ $$=16\pi\int_0^\infty \frac{1}{r^2}dr\neq 1$$ Yup, integral diverges, so it is NOT a delta function.
I think your confusion is regarding the nature of a delta function. If something blows up at the origin it does not mean it is necessarily a delta function.
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Testing General Relativity Ever since Einstein published his GR theory in 1916, there have been numerous experimental tests to confirm its correctness--and has passed with flying colors.
NASA and Stanford have just announced that their Gravity Probe B activity has confirmed GR's predicted geodetic and frame-dragging effects. Are there any other facets of GR that need experimental verification?
| Sure there are. The theory has been tested within only a teeny tiny part of the range of its predictions. For example it predicts gravitational redshift in the range of 0% (no redshift) to 100% (black hole), but experiments to date have shown a maximum gravitational redshift less than 0.01%. It matters less how many tests of GR are done than how extensively those tests cover the range of what GR predicts.
While we have little experimental data to definitively show that GR is the correct theory of gravity, we do know that it leads to major problems for physics, like its breakdown at gravitational singularities, its incompatibility with quantum mechanics, and the black hole information loss paradox. A competing theory of gravity that is confirmed by all experimental tests of GR to date need not have any of those problems, indicating that a lot more testing of GR is warranted.
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Far-field intensity from scattering of small particles Howdy, I'm building a simulation for looking at the light field underwater. In order to verify my simulation, I'm looking for some data showing the far-field intensity that comes from single scattering from many small particles in suspension. I suspect Mie theory plays a part here, but I'm having a hard time finding some results, rather than doing all the derivations myself.
In other words, I want to know the power distribution on a plane after a beam of light has been scattered by a bunch of small particles through a volume. I know Oregon Medical has a nice online simulation that produces scattering phase functions (http://omlc.ogi.edu/calc/mie_calc.html), but that doesn't give me the power on a plane - only the scattering profile from individual particles. I'm fine with only a single scattering result.
I want to do initial verification using a fixed particle size. Having a hard time finding a reference with this data. Help?
| I was able to find experimental and simulated data for the plane intensity from multiple scattering of small (1, 5 and 10 $\mu$m spheres in the Thesis of Edouard Berrocal. His thesis can be downloaded here.
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Is this a weather phenomenon or an instrumental artifact? The radar image of the midwest provided by Weatherunderground at 10:30 PM Central time, May 8 2011 has odd patterns.
Are these patterns real? Perhaps caused by large scale convection over cities? Or are they artifacts of radar placement?
Here is the image that I am referring to, where green indicates light and yellow moderate rain:
| No proposed urban rainfall effect is dominant enough to cause that picture. But limited radar range, with radars being located in larger cities would easily explain it. The urban rainfall enhancement effect would have to be pretty extreme for it to be otherwise.
| {
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Protons' repulsion within a nucleus Do the protons inside the nucleus repel each other by the electrostatic force? If they do, why doesn't the repulsion drive the protons apart so that the nuclei get disintegrated?
| There is an electrostatic repulsion between the protons in the nucleus. However, there is also an attraction due to another kind of force besides electromagnetism, namely the so-called "strong nuclear interaction".
The strong nuclear interaction ultimately boils down to the forces between the "colorful" quarks inside the protons - and neutrons. It is mediated by gluons, much like electromagnetism is mediated by photons, described by Quantum Chromodynamics (QCD), much like electromagnetism is described by Quantum Electrodynamics (QED), and it acts (almost) equally on protons and neutrons.
The attractive strong nuclear interaction inside the nuclei is 1-2 orders of magnitude stronger than the repulsive electrostatic interaction which is what keeps the nuclei together despite the repulsive electrostatic force.
| {
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Feedback on the paper, 'CCC-predicted low-variance circles in CMB sky and LCDM' by V. G. Gurzadyan and R. Penrose Ref: CCC-predicted low-variance circles in CMB sky and LCDM
To all cosmology / theoretical physics / related or similar researchers and academics,
Are there some updates concerning the issue of these concentric circles observed in the CBM in view of the new Planck data? What do the new Planck data mean for the particular issues discussed in the paper?
Warmest gratitude in advance.
| Since the first paper that I associate the circles with the voids.
in the paper, about Fig 4
The distortions could be the result of
... or more likely, ...Indeed, the
presence of giant voids could
particularly influence such images
Gurzadyan V.G., Kocharyan A.A. (2009)
The concentric features reminds me a X-ray crystal diffraction pattern. It can trace a spatial volumetric regularity present in the creation of the photons of the CMB.
goggle for images of "X-ray crystal" and compare with Figs 4 and 5.
I think that this viewpoint is not allowed by the current mainstream cosmological model.
If you want to understand what it can trace, or not, one must ignore any preconceived model and start with the data analisys and discover if it was properly done, and then answer to the question: can it trace a volumetric regularity ?
If the answer is 'yes' or a 'may be' I can assure that a cosmological model already exists.
(not the CCC one)
| {
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Why is there an absolute entropy? Why is there an absolute entropy? Given any non-discrete probability distribution, we don't really have an absolute entropy because the entropy depends on the parametrization of the distribution (e.g. Beta vs. Beta-prime) which was arbitrarily chosen. Another way to put it is that instead of entropies we only have Kullback–Leibler (KL) divergences aka relative entropies. So, why isn't there a physics analogue to KL-divergence? Just as we have relativistic velocity, which has some properties, why don't we also have relative entropies, which have some properties? Instead of saying the absolute entropy of the universe increases, why don't we say that the relative entropy given our prior belief of the universe increases?
Alternatively, what is the relation between entropy and number of microstates when the physical system is continuous, and how do we "count microstates"?
| The measure isn't arbitrary. In classical mechanics, the symplectic structure of phase space defines the Liouville measure. In quantum mechanics, the Hilbert space norm plays this role.
| {
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How to calculate the effect of roof items on gas mileage? I have a kayak and a bike. I routinely put them on top of my car and drive 60-70 mph for hundreds of miles. I am curious how much this affects my gas mileage.
| The best way: just do it. Go out and drive on the highway for some distance with the kayak and bike on top of your car, and measure the amount of fuel used, then do the same trip without the items on top of your car and measure the amount of fuel used in that case. The trip would have to be long enough that you can get a fairly accurate measurement of how much fuel you used.
What you could do is fill up your fuel tank immediately before setting out each time, then fill it up again when you get back, and the amount of gas you need to buy will tell you how much you used. Just make sure to use the same gas station, and preferably the same pump, for every measurement.
If you want to calculate it theoretically, that's a whole different story. Presumably the kayak and bike on top of your car increase the drag force and thus reduce its fuel efficiency, but as far as I know there's no way to do a calculation without a complex aerodynamic simulation.
| {
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Is Shor's algorithm a demonstration of the many worlds interpretation? David Deutsch is very fond of pointing out Shor's integer factorization algorithm is a demonstration of the many worlds interpretation. As he often asked, where else did all the exponentially many combinations happen?
Are there any other alternative interpretations of quantum mechanics which can explain Shor's algorithm, and the Deutsch-Jozsa and Simon's algorithm?
| Absolutely no! In the computational basis given by $\{|x\rangle \}_x$, it certainly looks like there is a massive parallelism going on. This is the wrong, wrong, wrong way of thinking about it. Everything clears up once you realize the "correct" preferred basis for Simon's algorithm is actually $\left \{ \frac{1}{\sqrt 2} [ | x \rangle \pm | x+s \rangle ] \right\}_x$, and the correct preferred basis for Shor's algorithm is actually $\left\{ 2^{-n/2} \sum_i e^{2\pi i kx/p}|x\rangle \right\}_{k\in Z_p}$. In the correct preferred basis, there is no parallelism at all!!!
In general, the computational basis is NOT the preferred basis. This observation clears up all confusions. Everytime it appears there is parallelism going on, it's really a sign you're working in the wrong "preferred basis"!!!
| {
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5MHz RF pulse frequency analysed in software Is there software available that can analyse a 5MHz RF pulse to give a plot of frequency spectrum.
The signal data is visible on a LCD screen or a print out could be obtained.
| Thanks for your solutions they have confirmed my initial thoughts.
A 5MHz pulse of 3 or 4 cycles will have a bandwidth of 1 or 2 MHz but it is the profile of the spectrum that I need.
I also need a windows utility to digitise the pulse, so I may try Getdata or Dagra either can produce a file for input to excel but the latter can produce a file for MATLAB.
| {
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Is the wave function objective or subjective? Here is a question I am curious about.
Is the wave function objective or subjective, or is such a question meaningless?
Conventionally, subjectivity is as follows: if a quantity is subjective then it is possible for two different people to legitimately give it different values. For example, in Bayesian probability theory, probabilities are considered subjective, because two agents with access to different data will have different posteriors.
So suppose two scientists, A and B, have access to different information about the same quantum system. If A believes it has one wavefunction and B believes it has another, is one of them necessarily "right" and the other "wrong"? If so then the wavefunction is objective, but otherwise it must contain some subjective element.
| Consider an EPR experiment where an entangled pair of electrons is created. One of them hits a detector which finds its spin to be up. The other hits a detector at some distance. The first detector sends a light signal to the vicinity of the distant detector. That signal arrives before the other electron. Near that second detector sit two observers. The first is able to see the light signal and the second detector, the second observer is only able to see the detector but is shielded from the light signal. In the split second after the light signal arrives but before the second electron arrives, the wave function collapses for the first observer but not for the second. When the electron arrives, it collapses for the second observer too. But during that short interval the wave function is subjective in the sense that it is different for the two observers. Empirically both observers' observations agree with their version of the wave function: For the first observer the spin is always up or down as predicted by the light signal, for the second it is 50% of the time up and 50% down, agreeing with the uncollapsed version of the wave function.
| {
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What happens if you connect a hot resistor to a cold resistor? Kind of an extension to this question:
If you heat up an object, and put it in contact with a colder object, in an ideal insulated box, the heat from one will transfer to the other through thermal conduction and they will eventually reach an equilibrium temperature at the midpoint, correct?
Now if you have a hot resistor (electrical component) and a cold resistor, and connect them by their leads, so that they make a circuit:
there will be the same conduction and radiation heat transfers. But also, the hotter resistor will have a larger noise current, right? So will there additionally be a transfer of electrical energy from one resistor to the other? Would completing the circuit allow them to reach equilibrium temperature faster than if they were just touching through an insulator with the same thermal conductivity?
| 4kTBR is an approximation.
For very high frequencies or very low temperatures, quantum effects kick in. That's what limits the amount of power transferred, not the RF properties of real resistors.
Look it up on Wikipedia - Johnson Noise
| {
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Graduate Physics Problems Books Need to brush up on my late-undergrad and early-grad physics and was wondering if anyone can recommend books or lecture notes (hard copy, or on-line) that also have solutions.
Two that I have come across are:
Princeton Problems in Physics with Solutions - Nathan Newbury
University of Chicago Graduate Problems in Physics with Solutions - Jeremiah A. Cronin
Spacetime Physics - Taylor & Wheeler (favorite book on special relativity; has a lot of problems with solutions at the back; a lot of the problems really enforce the material and discuss paradoxes)
If possible, please also provide a reason why you like the books as opposed to just listing them.
| "Problem Book on Relativity and Gravitation" - A. Lightman, W. Press, R. Price, S. Teukolsky
| {
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Inflating a balloon (expansion resistance) I am doing a quick calculation on how to calculate the pressure needed to inflate a perfectly spherical balloon to a certain volume, however I have difficulties with the fact that the balloon (rubber) has resistance to stretching and how this affects the pressure needed. It has to do with the E-modulus of the material I think, but I can not think of a proper way to calculate it?
| The complete stress tensor, while accurate, is largely unnecessary for solving this problem, as it is a thin walled pressure vessel
Assuming the balloon is spherical, the strain can just be calculated from the current and initial radii.
$$\epsilon=\frac{r}{r_0}-1$$
The stress can be found using the modulus of elasticity:
$$\sigma=E\,\epsilon$$
The thin wall pressure equation can get you to pressure, if you know the thickness, by balancing outward pressure inside with the inward tension along a great circle of the sphere:
$$\pi\,r^2\,P=2\,\pi\,r\,\sigma\,t$$
$$P=\frac{2\,\sigma\,t}r$$
Because balloons get thinner as they stretch, the thickness will actually vary. Rubber typically has a poisson's ratio of 0.5 meaning it keeps a constant volume while being deformed. We can then calculate the thickness in terms of the radius:
$$t\,r^2=t_0\,{r_0}^2$$
$$t=t_0\,\left(\frac{r_0}{r}\right)^2$$
Putting them all together:
$$P=\frac{2\,E\,\left(r-r_0\right)\,t_0\,r_0}{r^3}$$
To see what this looks like, we can make a generic plot:
As you can see, there is a maximum pressure after which it becomes easier and easier to inflate the balloon. We can solve for this maximum pressure by equating the derivative with zero, solving for r, and plugging back in:
$$0=\frac{dP}{dr}=2\,E\,t_0\,r_0\left(\frac1{r^3}-3\frac{r-r_0}{r^4}\right)$$
$$r=\frac32\,r_0$$
$$P_{max}=\frac{8\,E\,t_0}{27\,r_0}$$
Of course this assumes a constant modulus of elasticity, which never holds true for a large enough deformation.
| {
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Why does the road look like it's wet on hot days? Often, I'll be driving down the road on a summer day, and as I look ahead toward the horizon, I notice that the road looks like there's a puddle of water on it, or that it was somehow wet. Of course, as I get closer, the effect disappears.
I know that it is some kind of atmospheric effect. What is it called, and how does it work?
| Mirage is an optical phenomenon very common in sunny days. It's caused by the redirection of the reflected light rays form the object, in other words, is a real physical phenomenon and not just an optical illusion. The sun light in the direction of the road gets refracted do to the temperature gradient (continuous change) of the layers of air. This refraction causes the redirection of the sun rays and finally it gets reflected (total reflection) from the layers of air near the surface of the road. This phenomenon can also be observed when the road surface is very cold and in this case the reflected images are upside down.
| {
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Helmholtz decomposition in the plane Prove or disprove the following proposition:
For any smooth plane vector field $\mathbf{H}=\left(H_x,H_y\right)$, there exist scalar potentials $\phi$, $\psi$ such that
$H_x=\frac{\partial \phi }{\partial x}+\frac{\partial \psi }{\partial y}$
$H_y=\frac{\partial \phi }{\partial y}-\frac{\partial \psi }{\partial x}$
| We can certainly find a $\psi$ that solves the following partial differential equation
$\frac{\partial ^2\psi }{\partial x^2}+\frac{\partial ^2\psi }{\partial y^2}=\frac{\partial H_x}{\partial y}-\frac{\partial H_y}{\partial x}$
It then follows that
$\frac{\partial }{\partial y}\left(H_x-\frac{\partial \psi }{\partial y}\right)-\frac{\partial }{\partial x}\left(H_y+\frac{\partial \psi }{\partial x}\right)=0$
This shows that the vector field $(H_x-\frac{\partial \psi }{\partial y},H_y+\frac{\partial \psi }{\partial x})$ is curl-free/solenoidal. If the domain is {\bf simply connected} it follows that there's a scalar field $\phi$ such that
$H_x-\frac{\partial \psi }{\partial y}=\frac{\partial \phi }{\partial x}$
$H_y+\frac{\partial \psi }{\partial x}=\frac{\partial \phi }{\partial y}$
or
$H_x=\frac{\partial \phi }{\partial x}+\frac{\partial \psi }{\partial y}$
$H_y=\frac{\partial \phi }{\partial y}-\frac{\partial \psi }{\partial x}$.
If the domain isn't simply connected then "curl-free/solenoidal vector field" doesn't always imply "the vector field is conservative".
| {
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Does a magnetic field do work on an intrinsic magnetic dipole? When you release a magnetic dipole in a nonuniform magnetic field, it will accelerate.
I understand that for current loops (and other such macroscopic objects) the magnetic moment comes from moving charges, and since magnetic fields do no work on charges ($F$?perpendicular to $v$) it follows that the work done on the dipole (that caused its gain in kinetic energy) must have come from somewhere other than the magnetic forces (like electric forces in the material).
However, what about a pure magnetic moment? I‘m thinking of a particle with intrinsic spin. Of course, such a thing should be treated with quantum mechanics, but shouldn't classical electrodynamics be able to accommodate a pure magnetic dipole? If so, when I release the pure dipole in a nonuniform B-field and it speeds up, what force did the work? Is it correct to say that magnetic fields DO do work, but only on pure dipoles (not on charges)? Or should we stick with "magnetic forces never do work", and the work in this case is done by some other force (what?)?
Thanks to anyone who can alleviate my confusion!
| See Appendix B on page 47 and further of this article:
Note that the failure of the “rest mass” m to be constant resolves a paradox concerning
what one is taught in elementary physics courses: On one hand, one is (correctly) taught
that an external magnetic field can “do no work” on a body, so a body moving in an external
magnetic field cannot gain energy. On the other hand, one is (also correctly) taught that a
magnetic dipole released in a non-uniform external magnetic field will gain kinetic energy.
Where does this kinetic energy come from? Equation (B6) shows that it comes from the
rest mass of the body.
| {
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How do laser rangefinders work when the object surface is not perpendicular to the laser beam? I find the functioning of a laser rangefinder confusing.
The explanation usually goes like this: "you shine a laser beam onto the object, the laser beam gets reflected and gets back to the device and time required for that is used to calculate the distance".
Okay. But the object surface can be uneven and not perpendicular to the laser beam so only a tiny fraction of beam energy is reflected back to the device. And there's plenty of other radiation around, sunlight included.
How does a rangefinder manage to "see" that very weak reflected signal in a reliable manner?
| Simply what it means is if you can see the laser dot on the wall the rangefinder can see it 1000s better! Don't think of it as a reflection coming off the fur of a bear at 300 yards. That seems stupid but if you aim a laser at a bear at 300 yards and had a telescope, you would and could see the dot on the bears fur. That's all the rangefinder does. It sees the dot, calculates time of flight to the bear then tells you 300 yards...simple.
| {
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Can heat be transfered via magnetic field in a vacuum? Say you want to store hot coffee in a container surrounded by a vacuum. To remove all sources of conductive energy loss the container is suspended in the vacuum by a magnetic field and does not have a physical connection to the sides of the vacuum chamber,
My question is would the magnetic field be a path for energy to be conducted out of the suspended container?
Another way to look at this question would be two magnets are suspended in a vacuum with their poles aligned. A heat source is attached to one of the magnets. Would the second magnet show a corresponding increase in temperature, excluding radiated heat transfer?
| I would say the first point to stress here is that heat by definition is the transfer of energy. 'Heat loss' can occur via convection, conduction or radiation. Clearly, the first two can be essentially reduced to zero by surrounding your coffee (or whatever) with a perfect vacuum. Creating a perfect vacuum is pretty much impossible though.
All objects at temperatures above absolute zero will radiate heat via emission of electromagnetic waves. (For more details, see this wiki article on black-body radiation.) This is a matter-of-fact, with or without a vacuum, hence why flasks don't keep your coffee warm indefinitely. A reflective layer (normally some sort of foil) is used to contain some of this radiated energy but it works better for some wavelengths, and worse for others.
To answer your question directly: no - The magnetic field used to suspend the vessel will not interfere with the energy being radiated, nor indeed does it constitute a medium through which heat can excape via any other means.
| {
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Where is spin in the Schroedinger equation of an electron in the hydrogen atom? In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom.
We are told that the Pauli Exclusion principle is a consequence of two electrons not being able to share the same wave equation.
However, in our derived equation, we did not have anything including the spin. We defined $\psi (r,\theta,\phi)$ as $\psi_{n,l,m} (r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi)$, where $Y_{l,m}(\theta,\phi) = f_{l,m}(\theta) e^{i m \phi}$. We then were given well-defined $R_{n,l}$ and $f_{l,m}$ that satisfied the partial differential equations in the Schroedinger Equation.
Nowhere in our final $\psi$ do we find anything that varies depending on a fourth degree of freedom, not to mention one that behaved as $m_s$ should.
Am I missing the point of the Pauli Exclusion Principle? Is there a part of the solutions for $\psi$ that I am not understanding?
EDIT:
I am referring to an $H^-$ ion, where there are two electrons each with their own wave equation. If we imagine the case where both have the same quantum numbers n,l,m, but different spin $m_s$, would not their wave equations be exactly the same, and therefore not allowed?
| A description of electron spin and the Pauli exclusion principle needs to go beyond the Schrödinger-equation to the spinor-valued Dirac-equation.
I don't remember my atomic physics course very well but at the level of your analysis I think you just add the rule of the two parallel spin orbitals explicitly.
For a discussion on this topic the wikipedia-page is helpful.
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