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<p>Does a surface in front of a radiator (not in contact) have a significant effect on the room's temperature or heating rate?</p>
<p>Some time ago I had a discussion about it, and despite none of us knowing anything about thermodynamics, we were pretty sure of our opposite intuitions (just like in politics, hehe).</p>
<p>I guess a quick way to determine it was to do the experiment in question, but since it may be quite straight forward for somebody who knows about the topic, and to avoid turning on the radiator in the middle of the summer, maybe it's better to just ask the experts here :-)</p>
<p>Just in case the answer depends very much on the setting, this particular set-up is a radiator of 0.9m x 0.5m at about 50ºC, just like <a href="http://www.theradiatorfactory.com/catalogue/ex/Twine_Pinch_LR.pdf" rel="nofollow">one of these</a>, on a room of 10m x 5m x 2.5m at an initial temperature of about 16ºC. The objects in front of the radiator (right next to it but not in contact) would be:</p>
<ul>
<li><em>a chair</em> (made of thin synthetic leather (PVC) and aluminum), similar to <a href="http://www.theofficeleader.com/images/thumbs/0009576_200.jpeg" rel="nofollow">this one</a></li>
</ul>
<p>or</p>
<ul>
<li><em>a wooden 0.5m x 1.5m x 0.05m framed mirror</em> (like <a href="http://shard3.1stdibs.us.com/archivesE/1stdibs/062112/huniford_jb//11/B.jpg" rel="nofollow">this one</a>, but on an aluminum stand with wheels)</li>
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<p>I have a time series of kW where each sample is measured at regular intervals (10 seconds). Could anyone explain to me how could I calculate the total power consumed (kWh) over an hour?</p>
<p>Thanks</p> | 6,013 | [
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<p>When an object's speed increases, its (relativistic) mass increases. Are new atoms created inside the object by its increased speed? or is its "gravitational charge" increased by its increased speed, without more atoms?</p> | 6,014 | [
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<p>How do I properly construct the <a href="http://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime" rel="nofollow">electromagnetic tensor in curved space-time</a>? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is right (if there is one).</p>
<p>First way:</p>
<ol>
<li><p>Compute the magnetic field $B$ from the curl of the magnetic vector potential $A$:
$$ \mathbf{B} = \nabla \times \mathbf{A}. $$</p></li>
<li><p>Place the resulting components directly in the contravariant electromagnetic tensor definition in cylindrical coordinates:
$$ F^{\mu\nu} = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & B^z \\
0 & 0 & 0 & -B^r \\
0 & -B^z & B^r & 0
\end{pmatrix}. $$</p></li>
</ol>
<p>Second way:</p>
<ol>
<li><p>Define the electromagnetic four-potential ($\phi$ is zero in my problem):
$$ A^\alpha = (\phi, \mathbf{A}). $$</p></li>
<li><p>Lower the four-potential index by contracting it with my covariant metric tensor.</p></li>
<li><p>Compute the electromagnetic field components with the formula
$$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$
I replaced the ordinary derivatives with covariant derivatives.</p></li>
<li><p>Raise the indexes of this covariant electromagnetic field tensor to compare with the first way.</p></li>
</ol>
<p>The problem is that I can't seem to have the same results with both methods, which says pretty clearly that I am doing something wrong. Is there something fundamentally wrong in taking these steps?</p> | 6,015 | [
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<p>What is sector decomposition and how can it be used to 'disentangle' UV and IR divergences?</p>
<p>I have read about it in the paper <a href="http://arxiv.org/abs/1011.5493" rel="nofollow">SecDec: A general program for sector decomposition</a>,<br>
but I have no idea if, with a suitable change of variable $x=1/q$ on each coordinate, we can change the IR divergences to UV divergences or change the UV divergences to IR ones and then apply sector method to disentangle the divergences.</p> | 6,016 | [
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<p>You may have seen the story of the <a href="http://www.cnn.com/2011/TECH/mobile/07/18/iphone.skydive/index.html?hpt=us_t2">iPhone which was dropped from perhaps 13,500</a> feet by a skydiver - it survived.</p>
<p>This made me wonder how to work out the terminal velocity for something like that. Obviously calculating terminal velocity for a sphere can be relatively straightforward, but with a flattened oblong, what factors come into play?</p>
<p>End on will be fast, flat will be slow, but is there a stable configuration?</p> | 6,017 | [
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] |
<p>Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?</p> | 6,018 | [
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<p>We know that spiral galaxies spin in a way such that we have to assume that dark matter is responsible for the extra mass required to do so.</p>
<p>My question is, can Lagrangian points (L1 and L2) be used to describe a galaxy's rotation instead?</p>
<p>Can we explain that objects far away from the center of the galaxy have higher velocity because they are at the L2 Lagrangian point of a Lagrangian system which consists of a) the galaxy's super massive black hole at its center, b) a part of its spiral arm c) the far away object in question?</p>
<p>(I'm a computer engineer interested in physics. Please excuse my ignorance) </p> | 6,019 | [
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<p>I am not sure about the pressure existing in fluids.If the vertical pressure acts due to gravity and the atmospheric pressure since it also changes due to height also acts due to gravity then why is there horizontal pressure in a liquid. If there is then is it also equal to the vertical pressure at the point?</p>
<p>If gravity is suddenly stopped then do we still have water pressure and atmospheric pressure? In air is pressure equal in all directions? An interesting theory one of my friends explained is that if there is a box of air in space with no gravity and they have a velocity in one direction and due to elastic collisions there is pressure in all directions. But is it so that pressure in liquids act only due to moving molecules colliding randomly if there is no gravity?</p>
<p>Any explanations would be appreciated.</p> | 6,020 | [
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] |
<p>Simple question: Why is charge $q$ outside symmetrically distributed?<br>
The material is a conductor.<br>
<img src="http://i.stack.imgur.com/8Tutg.png" alt="enter image description here"></p> | 6,021 | [
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<p>As a preface, I am not a physicist. I'm simply interested in abstract physics and fundamental principles of the universe and such. As such, if you can provide an answer for the layman (as non-academic and unjargonized as possible), it would be very, very appreciated so that I can actually understand it.</p>
<p>Everything I ever learned about physics seemed to be built off of an assumption that the universe and everything in it behaved deterministically. So it should always be a (theoretical) possibility that, given perfect knowledge of every particle and force in the universe at a given moment, we can calculate with 100% accuracy what the state of the universe will be in the next moment.</p>
<p>This of course assumes omniscience and unlimited computational capacity, which is why I said this is only a theoretical possibility. However, we can define our closed system to be much smaller -- say, a bottle full of nitrogen and helium -- and apply this principle more directly. And it seems like this assumption is absolutely necessary for scientific experiments to even take place or have any validity, since without this kind of determinism, the observations and the results inferred from them can't ever actually be trusted.</p>
<p>I don't understand quantum mechanics very well, but it seems like this theory breaks this assumption completely. From what I understand, there is no way to predict what the state of the particle will be at the next moment. The most I can know is that, given that a particle is in state <code>A</code>, it will next be in state <code>B</code> or state <code>C</code>. There is absolutely no way to know for sure, and the only way to find out is to observe it actually change. Furthermore, observations of this kind don't yield any insight into what other particles in state <code>A</code> will do.</p>
<p>So, in classical physics, laws used to look like this:</p>
<pre><code>A -> B [A implies B]
</code></pre>
<p>But with quantum physics, all of this is gone, and our laws can at best look something like this:</p>
<pre><code>A -> ((B v C) v D) v E [A implies B, or C, or D, or E, or ...]
</code></pre>
<p>How does this not break everything physics is built upon? The implications of this are seriously troubling to me, and I feel like it destroys everything I thought I knew. Can anyone explain how this works in slightly lower-level terms, or show how it's still possible for the theories and laws of classical physics to hold any weight?</p> | 6,022 | [
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<p>My Question is as follows.</p>
<p>What is the effect of refractive index of an object for imaging (Photographs by high speed camera) on its size and shape information incurred from image?</p>
<p>Lets say ,</p>
<p>I keep the camera focal length, aperture, distance between camera & object, light intensity of the background constant.
Put two objects of same diameter 'd' with different refractive index n1 and n2 in front of the camera one at a time.
Size incurred from image of those object is of diameter d1 & d2. </p>
<p>My question is will d1 & d2 be the same ? or it will differ? if it differs how can I co-relate analytically/ theoretically or by ray tracing projection?</p> | 6,023 | [
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<p>In order to prove the Bernoulli’s principle ($\frac{p}{\rho} + \frac{1}{2}u^2+\phi = constant$ ), I have to use the <a href="http://en.wikipedia.org/wiki/Euler_equations_%28fluid_dynamics%29" rel="nofollow">Euler equation</a>: $\frac{Du}{Dt} = -\frac{1}{\rho}\nabla p + g$.</p>
<p>I know how to prove it, but I didn't understand what does it mean and say (Euler equation)?</p>
<p>please explain me.</p>
<p>any help appreciated!</p> | 6,024 | [
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<p>I think this is a very straightforward question but I don't see it right now. In Tong's notes on String theory (<a href="http://www.damtp.cam.ac.uk/user/tong/string/four.pdf" rel="nofollow">http://www.damtp.cam.ac.uk/user/tong/string/four.pdf</a>) section 4.2.3 he defines the weight of an operator under $\delta z=\epsilon z$, $\delta \bar{z}=\bar{\epsilon}\bar{z}$ in equation 4.16. Then at the end of the next page he uses the Ward identity 4.12 for that transformation. That is
$$\delta O=-\mathrm{Res}[\epsilon zT(z)O(w)]=-\epsilon( hO(w)+w\partial O(w))$$
So
$$\epsilon zT(z)O(w)=...+\frac{\epsilon( hO+z\partial O)}{z-w}+...$$
But the I don't see how he gets the term $(z-w)^{-2}$ which gives
$$T(z)O(w)=...+\frac{ hO}{(z-w)^{2}}+\frac{\partial O}{z-w}+...$$
The naive thing is to divide by $\epsilon z$ but that doesn't give the result.</p> | 6,025 | [
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<p>Suppose, I somehow happen to dig a hole diametrically through the Earth (neglecting all constraints like molten core etc ) ... and I throw an object from one side of the hole, will it reach the other end? </p> | 76 | [
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] |
<p>In QM I have the state $\lvert 00 \rangle \langle 00 \rvert$. Can anyone tell me what this would look like as a matrix? I know that
$$ \lvert 00 \rangle = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}. $$</p> | 6,026 | [
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<p>For the Noether theorem for pseudoeuclidean 4-spacetime a-current $J_{a}^{\mu}$ is equal to</p>
<p>$$
J_{a}^{\mu} = \frac{\partial L}{\partial (\partial_{\mu}\Psi_{k})}Y_{k, a} - \left( \frac{\partial L}{\partial (\partial_{\mu}\Psi_{k})}\partial_{\nu}\Psi_{k} - \delta_{\nu}^{\mu}L \right) X_{a}^{\nu},
$$</p>
<p>$$
\quad Y_{k, a} = \left(\frac{\partial F_{k}(\Psi , \omega)}{\partial \omega^{a}}\right)_{\omega = 0}, \quad X_{a}^{\nu} = \left(\frac{\partial f^{\nu}(x , \omega)}{\partial \omega^{a}}\right)_{\omega = 0},
$$</p>
<p>$F_{k}$ - function of field transformation, $f^{\nu}$ - function of coordinates transformation, $ \omega^{a}$ - parameter of transformation.</p>
<p>The charge is equal to
$$
\partial_{\mu}J_{a}^{\mu} = 0 \Rightarrow Q_{a} = \int J_{a}^{0} (t, \mathbf r)d^{3}\mathbf r .
$$</p>
<p>How to "reduce" the number of dimentions in this theorem for euclidean 3-space (i.e. get the formulation of theorem for euclidean space)?</p> | 6,027 | [
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<p>I believe the purpose of a tuning fork is to produce a single pure frequency of vibration. How do two coupled vibrating prongs isolate a single frequency? Is it possible to produce the same effect using only 1 prong? Can a single prong not generate a pure frequency? Does the addition of more prongs produce a "more pure" frequency?</p>
<p>The two prong system only supports a single standing wave mode, why is that?</p> | 399 | [
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<p>Could somebody please explain something regarding the Nordstrom metric?</p>
<p>In particular, I am referring to the last part of question 3 on <a href="http://www.hep.man.ac.uk/u/pilaftsi/GR/example3.pdf" rel="nofollow">this sheet</a> -- about the freely falling massive bodies. </p>
<p>My thoughts: The gravitational effects would be significant since for a massive body, the geodesic is timelike. There woud thus be a $\eta^{\mu\delta}\partial_\delta \phi \dot x^\beta \dot x_\beta$ is not of the form $f(\lambda)\dot x^\mu$ so the affine parametrization does not eliminate this term containing the gravitational potential $\phi$.</p>
<p>Does this argument make any sense at all? Also, what more can I say about the geodesics of such massive particles?</p>
<p>Thanks.</p> | 6,028 | [
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<p>Photovoltaic panels are rated under standard conditions: eg a $100\mathrm{W}$ panel if irradiated with $1000\mathrm{W}/\mathrm{m}^2$ at $25^\circ\mathrm{C}$.</p>
<p>When the cost effectiveness of photovoltaics is discussed one is often presented with daily irradiation maps averaged yearly or possibly monthly. This is fine for cost effectiveness calculations.</p>
<p>But in designing the power conversion electronics or effects of grid injection one cannot work with average values - they must be able to sustain peak powers that are possibly larger than the nominal power of the panels, right?</p>
<p>If so, by what factor are these electronics oversized? I cannot find any similar maps of the average daily peak irradiance.</p>
<p>Given the daily solar energy incident ($\mathrm{kW hr /m^2\cdot day}$) averaged either annually or monthly, can one place an upper bound on the daily peak irradiance ($\mathrm{W / m^2}$)?</p>
<p>I am particularly interested in Mediterranean countries.</p> | 6,029 | [
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<p>A standard example of a problem involving torque is opening a door - the same force F applied far from the hinge causes a larger angular acceleration than if applied close to the hinge.</p>
<p>I always had trouble getting an intuitive explanation for this - when you're close to the hinge and you're pushing the door with all your strength, it barely moves - so where is all the force going? To the hinge perhaps?</p>
<p>Let's model the door with a rod attached to a hinge. Suppose the rod's length is $L$ with mass $M$, distributed evenly. Its moment of inertia is $I = ML^2/3$. You push the rod with force $F_{applied}$ at distance $d$, giving it an angular acceleration $a$. Using $T=Ia$ we get $F_{applied}*d = Ia$, or $F_{applied}=Ia/d = aML^2/3d$.</p>
<p>Now, if you consider each mass element (particle) of mass $m$ in the rod individually, it experiences a tangential force $F_{tan} = m a_{tan}= m a r$, where $r$ is its distance from the hinge, and so the total tangential force summed over all particles is $
F_{tan\ total}=\int_0^L (M/L)\ ar\ dr = aM/2L$.</p>
<p>But according to Newton's 2nd law, $F_{tan\ total} =$ Sum of all external forces in the tangential direction. So $F_{tan\ total} = F_{applied} + F_{some\ other}$. </p>
<p>What is this other force, $F_{some\ other} = aM/2L - aML^2/3d$, which depends on d (your point of application of the force)? Where does it come from?</p> | 6,030 | [
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<p>From Jackson, problem 10.3:</p>
<blockquote>
<p>A solid uniform sphere of radius $R$ and conductivity $\sigma$ acts as a scatterer of a plane-wave beam of unpolarized radiation of frequency $\omega$, with $\omega R /c \ll 1$. The conductivity is large enough that the skin depth $\delta$ is small compared to $R$.
(a) Justify and use a magnetostatic scalar potential to determine the magnetic field around the sphere, assuming the conductivity is infinite. (Remember that $\omega \neq 0$.)</p>
</blockquote>
<p>We'd like to show $\nabla \times {\bf B} = \nabla \cdot {\bf B}=0.$ </p>
<p>I have two questions. First, what does it mean for a plane wave to have a definite frequency and be unpolarized? For example, are there many sources out of phase, all at a given frequency, radiating with varying amplitudes and in all directions? If this is true, then is it possible for the superposition of all these waves to vary faster than the original frequency due to interference?</p>
<p>Assuming that the above question is resolved and that we can make the long-wavelength approximation so the magnetic field is roughly constant over the sphere, why does</p>
<p>$$\nabla \times {\bf B} =0\neq \frac{1}{c^2}\frac{\partial {\bf E}}{\partial t}$$
(assuming that the electric field is just like an oscillating dipole and has a term free from powers of $\omega$)?</p> | 6,031 | [
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<p>Or, perhaps, the question is in which circumstances can I couple it, and of these, which are the simplest. </p>
<p>For instance, I think that you can not have a massive Dirac fermion and just couple the left part of it to the electromagnetic field: you trigger some vector-axial current and then trigger the anomaly and spoil renormalizability, do you? </p>
<p>And, is the problem different if the fermion is massless, or if we just use a Weyl left fermion without ever adding the right handed counterpart? </p>
<p>(EDIT: this last paragraph could be a source of confusion, I am afraid... Of course in the massive case I still should have both left and right Weyl fermions, but with different coupling to the abelian field, and even one of the couplings could be zero. I am interested on answers for both cases, massive and massless fermions. Pure Majorana mass is of minor importance, but it is fine for completeness :-)</p> | 6,032 | [
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<ul>
<li>particles with real-mass have time-like kinematics ($ds^2 > 0$).</li>
<li>particles with zero-mass have light-like kinematics ($ds^2 = 0$).</li>
<li>particles with imaginary-mass have space-like kinematics ($ds^2 < 0$) (tachyons).</li>
</ul>
<p>So the question is pretty simple:</p>
<p><strong>What would be the kinematics of a particle with both non-zero real and imaginary parts?</strong></p> | 6,033 | [
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<p>In Special Relativity, the Lorentz Group is the set of matrices that preserve the metric, i.e. $\Lambda \eta \Lambda^T=\eta$.</p>
<p>Is there any equivalent in General Relativity, like: $\Lambda g \Lambda^T=g$? </p>
<p>(We could at least take locally $g\approx\eta$, so we recover the Lorentz group, but I don't know whether we could extend this property globally.)</p>
<p>Why does Group Theory have much less importance in General Relativity than in QFT and particle physics?</p> | 890 | [
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<p>Probability of interaction between two particles tends to wane with increasing energy. Technically, the cross section of most interactions falls off with increasing velocity.</p>
<p>$$\sigma(v) \propto \frac{1}{v}$$</p>
<p>This begs a fun question. Since interaction probability diminishes with increasing relative velocity, if you impart <em>enough</em> energy to a particle, might it just mostly pass through solid matter? Might solid matter pass through solid matter (with some "radiation damage" of course)? The above relation, of course, is lacking a great deal. We are really interested in the mean path length, as well as the linear rate of energy deposition. Let's consider the problem in the context of two chunks of solid matter moving at each other really fast (like space jousting). We have several requirements for a survivable experiment.</p>
<ul>
<li>Average path length for any given nucleus & electron from spaceship A moving through spaceship B must be much greater than spaceship B's length</li>
<li>The energy deposition as a result of the passing must be small enough such that they don't explode like a nuclear bomb right after passing</li>
</ul>
<p>The question also becomes highly relativistic, and I want to hear commentary from people who have knowledge of interactions in high energy accelerators.</p>
<p>Let's say you're on the Star Trek Enterprise, and the captain proposes an alternative to navigating the densely packed matter in the approaching galaxy by increasing speed to just under the speed of light, and not worrying about obstacles (because you'll pass through them). What arguments would you use to convince him this might not be the best idea?</p>
<hr>
<p>EDIT: this video seems to make the claim that super high energy <em>protons</em> may pass through the entire Earth. Do the answers here contradict the claim?</p>
<p><a href="http://www.youtube.com/watch?v=aTBvPxQIFts">http://www.youtube.com/watch?v=aTBvPxQIFts</a></p> | 6,034 | [
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<p>On one hand, I think they should be equal since the external force and internal force are equal in equilibrium. On the other hand, I don't see anything related between them, the inside pressure is hold by rubber strength, while ground pressure is equal to gravity, e.g a lightweight steel wheel can hold very high internal pressure but with low ground pressure</p>
<p>I have no idea, does the pressure inside the tire equal to ground pressure? Please explain it in detail</p> | 6,035 | [
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<p>sorry if I sound little noobish. Though I have a fairly good understanding of physics, I sometimes don't understand the electrical aspects.</p>
<p>Say there is a capacitor. This capacitor is expected to act as a storage buffer. By extension, the capacitor will have a "charge" interface and a "discharge" interface. There may or may not be an electrical circuit between these interfaces and the capacitor. The expected behavior of the system is, electrical energy may be input to the system via the charge interface, which will charge the capacitor, and energy may be simultaneously drawn also, via the discharge interface, which will draw energy from the capacitor, and this process can happen so long as the energy stored within the capacitor is within its maximum and zero. The actual path of the energy may be from the interface, through the circuit, to the capacitor, and back through the circuit to the other interface; or part of the energy may be routed from one interface to the other by the circuit, and the net energy difference between the two interfaces be actually sent to, or drawn from the capacitor.</p>
<p>Though I may think that this is possible, I'm not aware if any such system exists currently. The home inverter seems to be doing quite the same thing, but both the cycles don't happen at the same time though.</p>
<p>Edit: Yeah, a diagram will help me also to explain better what I have in mind.</p>
<p><img src="http://i.stack.imgur.com/cXRGC.jpg" alt="Energy Storage Buffer"></p>
<p>The two interfaces are part of the circuit which shields the capacitor. This circuit may work in two possible ways, which I've mentioned as flow 1 and flow 2. In Flow 1, "all" of the energy which flows into/out of the system, does so through the capacitor. In Flow 2, the circuit redirects part of the energy flow in one interface to the other interface, and only the net difference between the energy flows is actually transmitted to/from the capacitor.</p>
<p>Hope this makes it more clear. Now, let me restate my question. Is such a system possible, importantly, such a circuit possible. Are any systems available today, which do exactly the same thing. And your own views on this is really welcome.</p> | 6,036 | [
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<p>What exactly is meant when one uses the word <a href="http://en.wikipedia.org/wiki/Sector" rel="nofollow"><em>sector</em></a> in Particle Physics?</p>
<p>As in, the <a href="http://en.wikipedia.org/wiki/Hidden_sector" rel="nofollow"><em>Hidden Sector</em></a> or the <a href="http://en.wikipedia.org/wiki/Standard_Model#Electroweak_sector" rel="nofollow"><em>Electroweak Sector</em></a>.</p>
<p>Does it refer to a specific part of the Lagrangian?</p>
<p>Or does it refer to the range of energies at which certain phenomena are expected / observed to occur?</p> | 6,037 | [
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<p>To my understanding, practical resonance occurs when the amplitude is at a maximum. Is this correct? </p>
<p>Also I have looked all over for a derivation of the formula for angular frequency of practical resonance but I can't seem to find a clear one. Is anyone able to show me or provide a link? </p> | 6,038 | [
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<p>I've been reading Max Tegmark's book: Our Mathematical Universe. It's very interesting, but I wanted to know more about one particular thing. The book simplifies things and I know inflation theories to be varied and complex, but I will briefly describe what Max was saying and hopefully someone can pick up one what I'm talking about.</p>
<p>Max describes the inflation period as containing a non-diluting, inflating substance, where the energy used to inflate it causes it's mass to increase through extreme negative pressure. And gravity repulses this substance, accelerating its growth because the negative pressure causes negative gravity.</p>
<p>So where did this energy come from to create all this new mass? Well he says that the gravitational force provided this energy, and that to balance the energy it created negative energy in the gravitational field. Every time the gravitational field accelerates something it gains negative energy apparently.</p>
<p>So what does this mean? It clearly means that for all or nearly all energy in the universe there must be an equal amount of negative energy in the gravitational field (or anywhere else that can have negative energy). But what is this energy doing? Surely this means the universe could cancel out all energy and return to nothing or nearly nothing?</p>
<p>So can someone please explain what this negative energy actually is.</p>
<p>Thank you.</p> | 6,039 | [
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<p>Consider the following metric which is 5 dimensional (2-parameter) spherically symmetric Kaluza-Klein solution</p>
<p>$$-\left(\frac{1-m/r}{1+m/r}\right)^{2/\alpha}dt^2+(1+\frac{m}{r})^4\left(\frac{1-m/r}{1+m/r}\right)^{2(\alpha-\beta-1)/\alpha}(dr^2+r^2d\Omega^2)+\left(\frac{1-m/r}{1+m/r}\right)^{2\beta/\alpha}dx_5^2$$</p>
<p>where $x_5$ is the periodic fifth coordinate. In this <a href="http://www.sciencedirect.com/science/article/pii/0550321383904625" rel="nofollow">paper</a> (page 15) I read</p>
<blockquote>
<p>the inertial mass of the star can be determined by the asymptotic behavior of this metric, assuming no interior singularities, and is equal to $M_{in}=\frac{(1+\beta)m}{\alpha}$</p>
</blockquote>
<p>What should I understand as "inertial mass" of a star here? How can I see that this formula represents inertial mass?</p>
<p>In the same page, just a little below it says</p>
<blockquote>
<p>the gravitational mass of the stars can be determined by the asymptotic form of $g_{00}$ and is given by $M_g=m/\alpha$</p>
</blockquote>
<p>In this case, what should I understand as "gravitational mass" here? Again, how can I see that this formula represents gravitational mass?</p> | 6,040 | [
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<p>I thought that I understood the centrifugal force earlier, but I can't seem to grasp how it interacts when considering that everything is relative?</p>
<p>Let's imagine that you are the only one in the entire universe, and that you are spinning with high angular velocity, with the rotation axis pointing in the same direction that your eyes are directed. Surely you would feel the centrifugal force pulling your feet and head apart, wouldn't you?</p>
<p>A problem with this, though, is the following: Since you are the only object in the universe, there's no way to tell if you're rotating. Your angular velocity isn't even defined, since you aren't rotating in relation to anything else.</p>
<p>How, then, can one know how what the centrifugal force is? Is it defined in relation to all the other mass in the universe, in such a way that it's negligible in classical mechanics problems?</p> | 6,041 | [
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<p><a href="http://en.wikipedia.org/wiki/Hawking_radiation" rel="nofollow">Hawking Radiation</a> is formed when particle, anti particle pairs formed by the uncertainty principle are separated by the event horizon of a black hole. It seems like an equal amount of particles and anti-particles should end up on each side of the event horizon. So why don't the particles and anti-particles annihilate with a new partner once separated from their original partner by the event horizon? Thus canceling out any radiation released.</p> | 6,042 | [
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<p><img src="http://i.stack.imgur.com/HrAKL.png" alt="">the red dots represent a side view of path traveled, F is downward force and the tool used here is a pen placing parallel to the coin</p>
<p>I have newly started to study mechanical physics. based on study, I conduct a simple experiment. But unfortunately I am unable apply the laws in reality.</p>
<p>Experiment:
I placed a coin (2 Indian Rupee coin) with radius $r$ positioned flat part of coin parallel to base of my laptop. Also placed at edge of base such that $x$ (mm) of diameter of coin is supported by laptop while $2 r - x$ is free, unsupported with the coin in a balanced position. Now at $x+y\, <\, 2 r$" measured along diameter of coin toward the free end I applied a random force perpendicular to laptop base. </p>
<p>Now my question is , how can we compute(formulate) distance $z$ traveled by coin measured from center of coin at start point to center of coin at the place it stopped in terms of known variables mentioned below.</p>
<p>My attempt:
To me the known values are: $r$,$x$,$y$,density of coin($\rho$),width(or height) of coin($w$), time it took to stop($t$).</p>
<p>Here for simplicity I have not considered the torque generated, air resistance, and visualized the traveled path as a simple parabolic path. Now I can measure the distance traveled
$z = v_0t+\frac{1}{2}gt^{2}$. where g is acceleration due to gravity.</p>
<p>But how can I find the initial velocity $v_0$. I know $v_0$ is not zero, since the situation here is not equivalent to a freely falling scenario.</p> | 6,043 | [
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<p>I've probably read it somewhere in Sakurai but I cannot recall it at the moment. So how does one really measure the proton's spin? I mean the proton's spin and not its constituents. </p>
<p>Do you measure it using a Stern-Gerlach type of setup?
How about atoms and other stuff, how is their spin measured?</p> | 211 | [
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<p>"In the presence of chaos, even small fluctuations
(including quantum fluctuations) can be amplified to produce large uncertainties in later
behavior"(<a href="http://arxiv.org/pdf/gr-qc/9210010v2.pdf" rel="nofollow">http://arxiv.org/pdf/gr-qc/9210010v2.pdf</a>)</p>
<p>Is there some experimental evidence for the amplification of the quantum fluctuations in a classic domain (typically m, s, kg)?</p> | 6,044 | [
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<p>BQP is the set of problems solvable in polynomial time for a given error tolerance, and it is suspected to be larger than P (and BPP, which is probably equal to P). However, inability for the gates to act perfectly, etc would require error-checking overhead. What is the overhead cost in the algorithm? In particular, does either the time or the number of q-bits overhead grow more than polynomially in the problem size (if it did then BQP would be altered)?</p> | 6,045 | [
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<p>Is linear momentum conserved in any direction? More specifically, if you project all momentum vectors in a system onto another vector, will momentum be conserved?</p>
<p>I know that momentum is conserved along the $x$ and $y$ axes, so I'm expecting this to be true, but I have yet to see a rigorous proof.</p> | 6,046 | [
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<p>Below is a picture of Giant Water Lily. Scientific Name: Victoria Amazonica. Leaves of some of these could be as big as 3 m diameter and carry a weight of 45kg spread evenly and can support a child. Now the problem: </p>
<p>Suppose that a leaf of such flower with a child is floating freely on water. Child crawls along the edge of the leaf until it arrives back to the starting point. In other words, he makes a full circle in the reference frame of the leaf. Question: </p>
<p>What is the total angle $\theta $ that the leaf turns through in time child crawls? (in the reference frame of water). Assume that the leaf is a large rigid circular disk. Ignore air and water resistance. </p>
<p><strong>Edit:</strong> </p>
<p>ftiaronsem's solution is absolutely correct if we assume that the leaf can only freely rotate about its geometrical center. However i was keeping in an eye that the leaf is not connected to the ground and can freely move in any direction.</p>
<p>Data given: </p>
<p>$m$ (mass of child)<br>
$M$ (mass of the leaf)</p>
<p><img src="http://i.stack.imgur.com/Wflmi.jpg" alt="Giant Water Lily"></p> | 6,047 | [
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<p>I want to consider conformal maps on suitable compactifications of $\mathbb{R}^{n}$. I know that a special conformal transformation: $$x_i\mapsto\frac{x_i-x^{2}b_i}{1-2b\cdot x+b^{2}x^{2}}$$ can be written as a composition of a spherical inversion, a translation (by $b$) and another inversion about the same circle. Since the conformal group is generated by special conformal transformations, translations, dilations and rotations, it should be possible to write the spherical inversion: $$x_i\mapsto\frac{x_i}{x^{2}}$$ as a composition of these maps. I can't, however, think of such a representation. How can it be obtained?</p> | 6,048 | [
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<p>in recent questions like "<a href="http://physics.stackexchange.com/questions/2041/how-are-classical-optics-phenomena-explained-in-qed-snells-law">How are classical optics phenomena explained in QED (Snell's law)?</a>" and "<a href="http://physics.stackexchange.com/questions/1898/do-photons-gain-mass-when-they-travel-through-glass">Do photons gain mass when they travel through glass?</a>" we could learn something about <strong>effective properties</strong> of matter interacting with a force field in terms of the <a href="http://physics.stackexchange.com/questions/1898/do-photons-gain-mass-when-they-travel-through-glass/1899#1899">path integral</a> and <a href="http://physics.stackexchange.com/questions/1898/do-photons-gain-mass-when-they-travel-through-glass/1903#1903">quasiparticles</a>.</p>
<p>Surely, both approaches must be equivalent but come from a different philosophy. Widely used is the quasiparticle approach in <a href="http://en.wikipedia.org/wiki/Solid-state_physics" rel="nofollow">solid state physics</a> e.g. calculating dispersion relations of <a href="http://en.wikipedia.org/wiki/Phonon" rel="nofollow">phonons</a>.</p>
<p>I would really like to know if there are <strong>simple examples</strong> for <strong>explicit calculations</strong> of the <strong>properties of photon-quasiparticles</strong> coming from a rigorous approach like a matter description via QED and finding an effective action e.g. using the <strong>Wetterich equation</strong> (see e.g. <a href="http://arxiv.org/abs/hep-ph/0611146" rel="nofollow">Introduction to the functional RG and applications to gauge theories</a>).</p>
<p>Any calculations and/or references would be very nice.<br>
Thank you in advance, sincerely,</p>
<p>Robert</p> | 6,049 | [
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<p>How is the following relation true</p>
<p>$$\tau = \large\frac{I}{g} \times \alpha$$</p>
<p>where $\tau$ is torque,</p>
<p>$I$ is moment of inertia,</p>
<p>$g= 9.8ms^{-2}$,</p>
<p>and $\alpha=$ angular acceleration.</p> | 6,050 | [
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<p>I know the ground state of hydrogen is unaffected by the Stark effect to first order. And I also know that the 1st excited state is split from 4 degenerate states to 2 distinct, and 1 degenerate state like this:</p>
<p><img src="http://i.stack.imgur.com/zO0Zt.png" alt="Hydrogen splitting"></p>
<p>But I don't quite understand why. I imagine it is something to with (anti)symmetry of the wavefunctions and selection rules.</p>
<p>Can anyone explain?</p> | 6,051 | [
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<p><a href="http://en.wikipedia.org/wiki/Apsis#The_perihelion_and_aphelion_of_the_Earth" rel="nofollow">Earth's perihelion</a> passed about nine hours ago. How accurately do we know the moment of closest approach of the Earth to the center of the sun? How do we make this measurement?</p> | 6,052 | [
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<p>Except for Mercury, the planets in the Solar System <a href="http://en.wikipedia.org/wiki/Table_of_planets_in_the_solar_system#Planets">have very small eccentricities</a>.</p>
<p>Is this property special to the Solar System? <a href="http://en.wikipedia.org/wiki/Extrasolar_planet#Orbital_parameters">Wikipedia states</a>:</p>
<blockquote>
<p>Most exoplanets with orbital periods of 20 days or less have near-circular orbits of very low eccentricity. That is believed to be due to tidal circularization, an effect in which the gravitational interaction between two bodies gradually reduces their orbital eccentricity. By contrast, most known exoplanets with longer orbital periods have quite eccentric orbits. (As of July 2010, 55% of such exoplanets have eccentricities greater than 0.2 while 17% have eccentricities greater than 0.5.<a href="http://en.wikipedia.org/wiki/Table_of_planets_in_the_solar_system#Planets">1</a>) This is not an observational selection effect, since a planet can be detected about equally well regardless of the eccentricity of its orbit. The prevalence of elliptical orbits is a major puzzle, since current theories of planetary formation strongly suggest planets should form with circular (that is, non-eccentric) orbits.</p>
</blockquote>
<p>What is special about the Solar System that orbits of planets here are nearly circular, but elsewhere they are moderately or highly eccentric?</p> | 6,053 | [
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] |
<p>The Navier-Stokes fluid dynamics equations, said that, as Sir William Thomson (or Lord Kelvin) predicted:</p>
<ol>
<li><p>When two smoke-rings are moving in the same direction, with the same speed, one behind the other, the 'leading' ring will slow down and enlarge, while the 'following' one will get smaller and speed up, passing through the ring in front, and this will keep on happening until they fade away. It was experimentally proven by Kelvin.</p></li>
<li><p>When two smoke-rings move towards each other, rather than colliding and annihilating in a smokey mess, they actually slow down and enlarge, never meeting, just getting larger until they fade away. It was experimentally proven by Kelvin. </p></li>
</ol>
<p>My questions:</p>
<ol>
<li>How does phenomena number 1 happen?</li>
<li>How does phenomena number 2 happen?</li>
</ol> | 6,054 | [
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<p>The basic thermodynamics problem is stated as follows.</p>
<blockquote>
<p>The nebula contains a very tenuous gas of a given number density (atoms per volume) that is being heated to a given temperature. What is the gas pressure?</p>
</blockquote>
<ol>
<li><p>What are the basic assumptions that should be taken in solving this problem? There is no sealed container, obviously, but if nebulæ were to allowed to expand indefinitely then it would be an isobaric process, would it not? Seen how that would not allow us to determine the pressure (?), then some sort of constraint is to be placed. If we are to assume that it is, in fact, an isochoric process, would it be a simple matter of finding molar mass from given number density and plugging it in the formula of Ideal Gas Law ($p = nRT / V$) assuming the volume of 1 metre cubed?</p></li>
<li><p>Given the numbers ($1 × 10^{8}$ atoms per m$^3$, 7500K) what should be realistic order of magnitude for the answer (in $Pa$ or $atm$) for the purposes of assessment as many certainly would not have intuitive grasp of your average nebular pressures?</p></li>
</ol>
<p>In general, for questions like this (advanced question from introductory chapter) is it detrimental to overthink the problem, that is: do I look for more difficulty than I should?</p> | 6,055 | [
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<p>Is it possible to create, and sustain, an electrified silicon gas vortex?<br>
If it is possible: would it produce an electromagnetic field?<br>
And how would that field affect the vortex? </p> | 6,056 | [
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<p>I'm interested to know the colour of starlight - particularly in rgb terms, I'm pressuming it's either white, or very close to white, but I'm interested to know how close. </p>
<p>To make it slightly more specific - I'm interested in the colour of stars as observed from the ground with the naked eye. And I'm interested in the ones at deep night (so dawn and dusk arn't interfering) - but I am interested in any changes that may happen at or near the horizon. </p>
<p>So in summary - is the colour of a star, observed from the ground with the naked eye, white (or so close to white as it would be considered white in a 24-bit colour pattern)? </p> | 6,057 | [
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<p>I'm a bit puzzled about an excercise in which I have to find the expectation values for position and momentum. Normally this should be pretty easy but in this case I just don't get the point.
Wavefunction is: $$ \psi(x) = \frac{1}{\sqrt{w_0 \sqrt{\pi}}} e^{\frac{-(x-x_0)^2}{2(w_{0})^{2}}+ik_0 x} $$</p>
<p>In a) you have to find the momentum rep. of this and in b) they ask you to find the expectation values of position and momentum. </p>
<p>Normally I would just compute the integral but in the solution they state, that "By inspection, it is easy to see that the expectation values for position and momentum are: $x_0 $ and $\hbar k_0 $" and I really don't know how to find these values. If anyone could briefly explain what they meant with "easy" I would be really happy.</p> | 6,058 | [
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<p>To write the electromagnetic field tensor in CGS units I just have to kick off the c-s from the SI tensor right?
I know this is a stupid question but I need a reliable answer.</p> | 6,059 | [
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<p>I try to simulate a solar system with planets (with random mass) placed randomly around a sun with a mass $X \times \text{solar mass}$.</p>
<p>The simulation is going well when I use real data (sun,earth,moon for instance), but now I'd like to simulate randomly generated system.</p>
<p>My problem is that I didn't succeed in calculating linear velocity of planet.</p>
<p>On internet, I only found formulas to calculate linear velocity when we know the angular velocity, this mean knowing the time the planet make to do a revolution , which I don't want to determine.</p>
<p>I want, knowing only the distance and the two mass (and direction of velocity vector), be able to calculate the linear velocity vector.</p>
<p>I don't really have more information to provide, if you need something, just ask for it.</p> | 6,060 | [
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<p>Are there any examples in the history of physics where a phenomenon was considered by the physics community to be not explainable by classical physics and needed a quantum explanation whereas some time later it was noticed that this claim was wrong (perhaps because for instance one "over-idealized" the system, neglected boundary effects or did some other mistakes when "proofing" that there is no classical explanation), i.e. that the phenomenon has indeed a classical explanation? Let me call those effects "pseudo quantum effects" for short. </p>
<p>Are there such pseudo quantum effects which were today common misconceptions (i.e. where people think that one needs a quantum description but doesn't really do it...)</p> | 6,061 | [
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<p>It seems there is a new theorem that changes the rules of the game in the interpretational debate on QM:</p>
<p><a href="http://www.nature.com/news/quantum-theorem-shakes-foundations-1.9392">http://www.nature.com/news/quantum-theorem-shakes-foundations-1.9392</a></p>
<p>Does this only leave Bohm,Everett and GRW as possible candidates?</p> | 6,062 | [
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<p>Suppose, we know the length of the shadow of an object at some known time.</p>
<p>Can we use the this information to find position of the object (the longitude )? </p> | 6,063 | [
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<p>I've noticed that ads for postdoctoral positions emphasize the skill set that one must have for a particular position. That said, what are the areas of research to avoid because they give you few transferable skills and hence limit your range of possible postdoctoral positions?</p>
<p>For example, String Theory might be an area to avoid because it gives you virtually no experience with using scientific software packages that might essential for post docs in other areas. Any others?</p>
<p>(My motivation for asking this question is that I think it might be nice to roam into other interdisciplinary areas and fields post-PhD, rather than restrict oneself to a certain field for life.)</p> | 6,064 | [
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<p>This question is inspired by this question/answer pair: <a href="http://physics.stackexchange.com/questions/22789/is-this-formula-for-the-energy-of-a-configuration-of-3-fluids-physically-reasona/22823#22823">Is this formula for the energy of a configuration of 3 fluids physically reasonable?</a></p>
<p>Consider three immiscible fluids forming contact surfaces, where none of the three can make a lubrication layer for the other two (the surface energy between fluid 1 and fluid 2 is not decreased by putting a thin layer of fluid 3 inbetween them, and likewise for the other two permutations). In this case, if all the contact surface tensions are positive, you have a minimum energy when all the surfaces are flat.</p>
<p>But there is a separate line tension for the 3-fluid interface itself. Can this line tension be negative? If it is negative, the line would like to wriggle, but the surface tension will require that the wriggles straighten themselves out as quickly as possible, to make the surface energy least. In this case, the minimizing energy configuration seems to be a very rapidly wriggling curve which is only infinitesimally different at the atomic scale from straight line. This suggests that a negative line-tension always renormalizes to exactly zero line-tension at long wavelengths. Is this correct?</p>
<p>Are there experimental or computational 3-phase line interfaces with a negative line tension? Do they renormalize to a zero line tension limit? Does this mean that zero line tension is a common observation?</p> | 6,065 | [
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<p>What processes occur when a meteor enters earth's atmosphere and then what will be speed of meteor when it encounters air resistance?</p> | 6,066 | [
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<p>As history of thermodynamics say, it was a mystery that what is the required condition for a given energy conversion to take place? Like there are two possible events each conserving energy but only one is chosen. So, in order to resolve this Clausius introduced a quantity called <a href="http://en.wikipedia.org/wiki/Entropy" rel="nofollow">entropy</a> which was given by $\int dq/T$. But can I know the reason for which Clausius chose this integral or quantity, why not any other quantity (changes in whom, positive or negative, would decide the occurrence of a given event)? I hope there lies an explanation to this which does not use statistical mechanics.</p> | 6,067 | [
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<p>I have the following minus sign problem: </p>
<p>Consider an infinitesimal Lorentz transformation for which $\Lambda^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\lambda^{\mu}_{\nu}$, where $\lambda^{\mu}_{\nu}$ is infinitesimal small. Define the vector fields $M_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}$. Show that acting on $x^{\mu}$, we have</p>
<p>$\frac{1}{2}\lambda^{\rho\sigma}M_{\rho\sigma}(x^{\mu})=\lambda^{\mu}_{\nu}x^{\nu}$</p>
<p>If i make the derivations:</p>
<p>$\frac{1}{2}\lambda^{\rho\sigma}M_{\rho\sigma}(x^{\mu})=\frac{1}{2}\lambda^{\rho\sigma}(x_{\rho}\partial_{\sigma}-x_{\sigma}\partial_{\rho})x^{\mu}= \lambda^{\rho\sigma}x_{\rho}\partial_{\sigma}x^{\mu}=\lambda^{\rho\sigma}x_{\rho}\delta^{\mu}_{\sigma}=\lambda^{\rho\mu}x_{\rho}=-\lambda^{\mu\rho}x_{\rho}=-{\lambda^\mu}_{\rho}x^{\rho}$</p>
<p>I can't see how lose the minus sign.. Probably trivial, but it keeps me busy. </p>
<p>Correction: orignal question had in the last step $\lambda^\mu_\rho$ which should be ${\lambda^\mu}_\rho$</p> | 6,068 | [
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<p>There are a number of <a href="http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations#Exact_solutions_of_the_Navier.E2.80.93Stokes_equations" rel="nofollow">exact solutions</a> to the <a href="http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations" rel="nofollow">Navier-Stokes equations</a>. How many exact solutions are currently known? Is it possible to enumerate all of the solutions to the Navier-Stokes equations?</p> | 1,027 | [
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<p>Apparently it is to be shown that for electrons under an external magnetic field, in the limit as $B\to 0 $
$$
\chi = \frac{dM}{dB} \approx \frac{n\,\mu^{*^2}}{k\,T}\,\frac{f_{1/2}(z)}{f_{3/2}(z)}
$$</p>
<p>So this is the Pauli paramagnetism problem and I understand the effect and the physics, but cannot seem to get the requested ratio of $f_{\nu}(z)$ functions.</p>
<p>Here is how I have been doing it:</p>
<p>With the grand canonical ensemble formalism I have
$$
\langle N_{\pm} \rangle = \int_0^{\infty} d\epsilon\, g(\epsilon)\,\frac{1}{\exp^{\beta(\epsilon-\mu\pm \mu_B\,B)}}
$$
where
$$
g(\epsilon) = \frac{2}{\sqrt{\pi}}\,\left(\frac{2\,\pi\,m}{h^2}\right)^{3/2}\;V\,\sqrt{\epsilon}
$$
and is the density of states from translational motion.
I then say that
$$
\langle M \rangle = \mu_B\,(\langle N_- \rangle - \langle N_+ \rangle ) = \cdots = Const*V*[f_{1/2}(\beta\,\mu+\mu_B\,B)-f_{1/2}(\beta\,\mu-\mu_B\,B)]
$$</p>
<p>This is fine I think except that when I calculate $\chi$ I will get fermi-dirac functions with -1/2 and not the ratio I need.</p>
<p>$$
\chi = \lim_{B\to 0} \frac{\partial\,M}{\partial\,B} = \cdots ?
$$</p> | 6,069 | [
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<p>The cross section of a nuclear interaction is a measurement of the probability of that interaction occuring. These probabilities are typically presented in terms of barns ($10^{-28}$ m$^2$) as a function of incident particle energy. You can look up values <a href="http://atom.kaeri.re.kr/ton/index.html" rel="nofollow">here</a> with a bit of searching. Here's an example:</p>
<p><img src="http://i.stack.imgur.com/PN4r5.jpg" alt="enter image description here">
Figure 1: Total neutron absorption cross section of $^{238}$U</p>
<p>I believe that it is a general rule that total interaction cross section decreases with increasing incident particle energy. This phenomenon is clearly evident in the neutron absorption cross-section shown above because there are no complications from the coloumb repulsion.</p>
<p>In undergraduate university, I never got far enough along in physics to calculate these values. In graduate school, I took nuclear engineering classes and used these values, but the discussion of their origins was cursory. I understand that the peaks correspond to the energy levels of stable or metastable states of the target nucleus, but I never quite understood:</p>
<p>why does the total interaction cross section decrease with increasing incident particle energy?</p>
<p>I always imagined that this was a quantum phenomenon resulting from the increased localization of the incident particle in the directions perpendicular to the path of travel owing to the larger energy. Were my musings correct?</p> | 6,070 | [
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<p>There have already been a lot of questions on this site on diffraction but I still believe this one might be slightly different. In electromagnetic waves, diffraction and any other phenomenon of wave propagation can be solved by Huygen's principle, a geometric construction that asks us to consider all points on a wavefront as secondary sources of wavefront.<br>
A justification of this treatment is provided by Feynman:-<br>
<img src="http://i.stack.imgur.com/ZE8YJ.png" alt="enter image description here">
<img src="http://i.stack.imgur.com/JQBvy.png" alt="enter image description here"></p>
<blockquote>
<p>I understood this completely. But then diffraction is a very general
phenomenon. <strong>What if we talk about mechanical waves (sound waves) then
this treatment by superposing the fields produced by opaque and a
hypothetical plug is no longer valid, but the diffraction details are
similar. Why then <em>on an intuitive level</em> is there diffraction, details
of which can be worked out by considering secondary sources?</strong></p>
</blockquote> | 6,071 | [
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<p>I have a monochromatic ribbon beam with $E(x)e^{i(kz-\omega t)}$ being the electric field's amplitude. I want to show that the lowest order approximation in terms of plane waves is</p>
<p>$\mathbf{E}(x,z,t)=\mathbf{\epsilon} \int{d\kappa A(\kappa) e^{i(\kappa x+kz-\omega t)}}$</p>
<p>where $\mathbf{\epsilon}$ is the polarisation direction and $A(\kappa)$ is the Fourier transform of $E(x)$ around $\kappa=0$.</p>
<p>From the result I can understand/identify that the Fourier kernel is $e^{i(\kappa x)}$ but usually when you use a Fourier transformation you go from $f(x)\rightarrow F(\omega)$, from one variable to another that is, but here all of a sudden you just add one new variable in the transformed field. How is that possible?</p>
<p><strong>EDIT</strong> There is a new approach to this, but there is a tiny little point that I don't get. Consider the following geometry.</p>
<p><img src="http://i.imgur.com/pQRP6Wf.jpg" alt=""></p>
<p>$d=x_1\sin{\theta},\; \sin{\theta}=\dfrac{x_0}{r_{01}}\approx \dfrac{x_0}{z},\; d\approx \dfrac{x_0x_1}{z}$</p>
<p>Consider the travelling wave</p>
<p>$e^{ikr}=e^{ik(r_{01}-d)}$</p>
<p>Why $r$ becomes $(r_{01}-d)$?</p>
<p>I beieve that $r$ describes the optical path. It feels like the two rays have an optical difference of $d$, but I am not sure if this is the answer.</p>
<p>Any ideas would be more that welcomed!</p>
<p>Solution found <a href="https://www.google.gr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDoQFjAA&url=http://cfmriweb.ucsd.edu/ttliu/be280a_05/BE280A_05_US2_3s.pdf&ei=MExxUYXyC8iDtAa37ICAAQ&usg=AFQjCNHPq69VnOsLnTaJ-_Pg6jZCux8UCA" rel="nofollow">here</a></p> | 6,072 | [
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<p>I have been playing around with <a href="http://earth.nullschool.net/#current/wind/isobaric/10hPa/orthographic=-1.45,54.02,482" rel="nofollow">this nice visualisation</a> and noticed that in the mid-stratosphere (P=10hPa), the airflow around the equator and south of it is in the opposite direction to the earth's rotation [airflow relative to a fixed point on the surface], as one would expect.</p>
<p>In the northern hemisphere though (centered around Svalbard) the air flow is in the same direction as the rotation of the ground below and is at high velocity. This seems somewhat counter-intuitive - what is the reason for it?</p>
<p><img src="http://i.stack.imgur.com/wP5T9.jpg" alt="enter image description here"></p> | 6,073 | [
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<p>As I've been taught lately in my mechanics course: </p>
<blockquote>
<p>the wheel has a unique property: at every moment of motion, the
touching point between the wheel and the ground is not in movement and
therefore no work is done by the friction force.</p>
</blockquote>
<p>Now, many of those problems are solved by using the 2nd Newton law and its rotational analog. </p>
<p>For instance consider having a wheel with a mass $m$ and a radius $R$ rolling on a slope that creates an angle of $\theta$ and we want to calculate its acceleration then we can start by writing: $$ma=mg\sinθ−F_f$$ </p>
<p>and the analog equation for torque: $$F_f R=I\alpha$$.</p>
<p>where $F_f$ is the frictional force.
Now, the first equation is the 2nd Newton law applied on the centre of mass of the wheel, and as we see, one of the forces is the external frictional force. Now, though the touching point is not in movement at the moment, the center of mass is, and in the equation we assume there is a friction force on the center of mass and therefore work is done indeed. Now, after thinking about this for a while, I've come to the conclusion that this makes sense, cause if we see the wheel as point of mass located in the center, then energy is not preserved because some of it is transfered to the spin and that's why we have the second equation. </p>
<blockquote>
<p>The question I'm having trouble with is whether the "work" of the
friction force on the center of mass is equal to the energy transfered
to the spin of the wheel?</p>
</blockquote> | 6,074 | [
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<p>I understand the derivation and calculation of de Broglie wavelengths, but what exactly does the wave of a particle represent? What does the wavelength of a particle mean? I'm assuming it's not the wavelength of its probability wave because then there would be no uncertainty in momentum, unless it represents the average wavelength.</p>
<p>The idea of wave-particle duality for EMR makes sense to me in that it can be seen as oscillating changes in energy (a wave), and as photons from the photoelectric effect. But with particles, I don't understand what quantity would be oscillating in their wave.</p>
<p>Thanks!</p> | 6,075 | [
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<p>This is a very simple question, which will most likely yield a prompt response from someone who knows more than I. The reason I ask:The quarks that we can detect (as they interact electromagnetically) are much lighter than the particles that they make up. From what I know, we say that gluons augment the energy level of the quarks when they combine to form baryons, yet if there are quarks that are made up of partial charges, how can we rule out the possibility of another type of quark made up of partial neutrals(or just a uniform neutral quark)(?). One could say that neutral doesn't exist, and it is merely a combination of partial charges that sum to zero. Yet neutrinos exist as leptons, which are independent of quarks. Is it possible that there is a relationship between the gluon, gauge bosons, and lepton pairs that all has to do with the instant something such as a beta decay occurs subsequent to an ephemeral appearance by a W boson? And that they would all turn out to be the same thing just for the instant the gauge boson acts as a mediator? </p> | 6,076 | [
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<p>I will be a grad student in condensed matter theory starting this fall. As an undergrad, I did the basic physics and math courses as well as a few grad classes (qft, analysis, solid state physics etc.) </p>
<p>When I start reading research papers, I often feel overwhelmed because there is so much that I don't know and I find it hard to decide which points to gloss over and which points to spend time on and understand more thoroughly (which in my case, would probably require supplementary reading of textbooks or related papers) </p>
<p>What are some things to keep in mind while reading a paper so that: </p>
<ol>
<li><p>I get a general overview of the paper and I more useful insights into parts</p></li>
<li><p>I can do the above reasonably fast (say, finish reading at least 1 paper a week for a start)</p></li>
</ol>
<p>You don't have to be specific to condensed matter theory papers when you answer. </p> | 115 | [
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<p>Here's a puzzle I have been pondering over.</p>
<p>If we have two extremal black holes with the same charge, the electrostatic repulsion between them ought to cancel the gravitational attraction between them. Without any net attraction or repulsion between them, how close can we bring these two black holes to each other without merging? A peculiarity of the metric seems to suggest the event horizon is always infinitely far away for extremal black holes $\int_R^r dr' \frac{1}{r'-R} = \infty$. Does this give enough elbow room for both black holes to get arbitrarily close to each other without merging?</p> | 6,077 | [
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<p>Why do only big rocks (planets) have satellites, and not small ones? Why doesn't cosmic dust orbit rocks that are many times heavier than the dust grains? If dust is still too heavy then what about molecules, atoms, or any particle for that matter? The mass difference should be millions of millions times; isn't it enough for orbiting? The Moon is 1% of the Earth's mass, yet we don't see 1kg rocks orbiting 100kg ones.</p> | 6,078 | [
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<p>Is the Lorentz Force acting on a wire, that has current $I$ in a magnetic field $B$ <a href="http://en.wikipedia.org/wiki/Conservative_force" rel="nofollow">conservative</a>? Or non-conservative? I understand that all the fundamental forces are conservative, am I correct? </p> | 212 | [
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<p>This was just a random thought that crossed my mind, and I'm sure there is a simple physics explanation. Say we have a leaky faucet 20 feet above the ground that lets out the exact same amount of water every few seconds or so. Why is it that the splash pattern for every drop is never the same as any other drop? Now it's not like I've sat there and observed every splash, but sometimes its obvious, like a certain splash's water reached a farther distance than another splash. If we are letting the same amount of water come out everytime, and nothing else changes, why do the splashes vary? (pretend the water disappears off the ground after every splash, that way every splash has the same environment.)</p> | 6,079 | [
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<p>I'm trying to calculate two 2d disks' collision with rotational motion.
The collision is perfectly elastic: the sum of translational and rotational energy is conserved. In the instant of the collision the contact of disks' surface is without slipping. Could you give me hint for this calculation?</p>
<p>Thanks.</p> | 6,080 | [
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<p>I'm trying to relate them, I'm trying to find the key relation that would show how the <a href="http://en.wikipedia.org/wiki/Conservative_force" rel="nofollow">conservative forces</a> serve <a href="http://en.wikipedia.org/wiki/Conservation_of_energy" rel="nofollow">conservation of energy</a>.
How would they relate?</p>
<p>Also, how are non-conservative forces related to conservation of energy? </p> | 6,081 | [
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<p>I've read this question and answer: <a href="http://physics.stackexchange.com/questions/1493/how-efficient-is-an-electric-heater">How efficient is an electric heater?</a>
, but still don't understand.</p>
<p>If I have an electric radiator it heats the room with 1000 Watts of power. And I feel the room's getting warmer.
In contrast, if I turn on a vacuum cleaner which consumes 1000 Watts as well as the radiator, it doesn't seem to heat the room as well.</p>
<p>Why? Won't all kind of energy transform into heat ultimately?</p> | 6,082 | [
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<p>Metallic strings exist in different kinds but I would like to measure for a metallic string:</p>
<ul>
<li><p>the section/geometry along its length, to a precision of 1/100 mm</p></li>
<li><p>its elongation when a tension is exerted on it on a measuring system I would construct to apply forces of the order of 1 to 100 daN. I have no clear idea of the order of magnitude of the elongation, so it might be too small to measure with reasonable equipement.</p></li>
</ul>
<p>Any ideas, suggestions, theoretical and practical remarks welcome.</p> | 6,083 | [
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<p>What is the difference between conduction of electric wave in conductor and propagation of electromagnetic wave in dielectric?</p>
<p>Why propagation term is used for dielectric and conduction for conductor?.</p>
<p>Somehow why propagation of electromagnetic wave (is it energy wave) is not possible in conductor, but in dielectrics, and conduction (power signal) not possible in dielectric.</p> | 174 | [
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<p>In our undergraduate E&MWaves class, we learned about a rectangular waveguide, and how to calculate the electric and magnetic field distribution throughout the waveguide using finite element analysis. Out of curiosity, I am interested in how you would go about solving the boundary conditions of a funnel wave guide, as a series of rectangles that radially get larger and larger. Can each 'rectangle' be calculated separately, and 'stitched' together at the end? How do you figure out the boundary conditions of this waveguide?</p> | 6,084 | [
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<p>Can we make a jacket using an electronic circuit that uses electric force to cancel the effect of gravity so that we get lifted in air.</p> | 6,085 | [
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<p>I was doing <a href="http://www.ocr.org.uk/Images/62267-question-paper-unit-g485-fields-particles-and-frontiers-of-physics.pdf" rel="nofollow">this</a> past paper and am a little confused by question 5) part c)ii)</p>
<p><img src="http://i.stack.imgur.com/3MBvP.png" alt="Diagram from past paper"></p>
<p><img src="http://i.stack.imgur.com/3p3C8.png" alt="force on current carrying wire from magnetic field"></p>
<p><img src="http://i.stack.imgur.com/PzDp1.png" alt="how this force affects the balance"></p>
<p>I correctly calculated that the force acting on the rod due to the magnetic field is 0.016N but I can't see how this could affect the scales as they're not touching.
I had a look at the <a href="http://www.ocr.org.uk/Images/58522-mark-scheme-unit-g485-fields-particles-and-frontiers-of-physics-june.pdf" rel="nofollow">mark scheme</a> but I can't understand why Newton's 3rd law would apply when the two objects aren't touching? </p>
<p>Here is a print screen of the mark scheme for part c):</p>
<p><img src="http://i.stack.imgur.com/gfeWK.png" alt="mark scheme"></p>
<p>Could anyone explain this please?</p>
<p>Thank you :)</p> | 6,086 | [
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<p>If we put earth like planet made up of antimatter (same mass of earth, same diameter etc..) with same distance from moon as current distance between earth and moon then center of gravity of moon and earth will change or not? Why?</p> | 6,087 | [
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<p>Can anyone resolve this contradiction:</p>
<p>$$\vec{r}\cdot\dot{\vec{r}}=\frac{1}{2}\frac{d}{dt}\left(\vec{r}^2\right)=\frac{1}{2}\frac{d}{dt}\left(\left|\vec{r}\right|^2\right)\equiv\frac{1}{2}\frac{d}{dt}\left(r^2\right)=r\dot{r}, \qquad r=|\vec{r}|.$$</p>
<p>But the velocity $\vec{v}=\dot{\vec{r}}$ has not to be parallel to $\vec{r}$, so actually:</p>
<p>$$\vec{r}\cdot\dot{\vec{r}}=r \dot{r} \cos{\angle\left(\vec{r},\dot{\vec{r}}\right)}$$</p>
<p>What am I doing wrong? Has anyone an idea?</p>
<p>P.S. I have this problem from the book "Electromagnetic Theory" from Ferraro (p. 543).</p> | 6,088 | [
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<p>I've seen the following formula for the potential energy of a body in a gravitational field ($\rho$ is the density, $g$ is the gravitational acceleration):</p>
<p>$$ \rho g \int_E z dV $$</p>
<p>Can you please explain to me how this formula is deduced? Thank you.</p> | 6,089 | [
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<p>In physics, the word entropy has important physical implications as the amount of "disorder" of a system. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable $X$ is defined as</p>
<p>$$H(X)=-\Sigma_x P(x)log_2 [P(x)]$$</p>
<p>bits, where $P(x)$ is the probability that $X$ is in the state $x$ , and is defined as $0$ if $P=0$. </p>
<p><strong>Question:</strong> Can anyone explain to me why "disorderness" of a system defined as this?
especially, where has "$log_2$" come from in this formula?</p> | 6,090 | [
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<p>Considering all of the appliances that the average home uses--microwaves, light bulbs, dishwashers, refrigerators--is it safe to say that all of the electrical energy in a home will be converted to thermal energy inside the home?</p>
<p>If you think about the resistance going through wires, that is converted to heat. The photons from the light will eventually be converted to heat. The refrigerator makes excess heat. Is there anything that doesn't end up as thermal energy?</p> | 6,091 | [
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<p>Suppose we have 2 fixed end connected with a wire and now we insert a vibrator <strong>in the middle</strong> of the wire, and resonance occur. How would the fundamental frequency looks like?</p>
<p>I know the case when the vibrator is at one ends and another ends are fixed while in this case, there are 2 fixed point and the vibrator is at the middle.</p>
<p><img src="http://i.stack.imgur.com/WMmgR.png" alt=""></p>
<p>Is the fundamental frequency like this? I imagine half of the original wire acts like a wire with a vibrator at one end and get the result.</p>
<p>Would the fundamental frequency be different if the vibrator which is on the string is vibrating with a very large amplitude?</p> | 6,092 | [
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<p>As I know, the working fuel of choice is Hydrogen because of its low molecular mass. When it comes to escape velocity, the estimate vary too much, from $8$km/s to $50$km/s (gas core reactor).</p>
<p>Wouldn't it be better to use water as a fuel since it's so much easier to store? </p>
<p>If we used water as a fuel, what would the escape velocity be (since it's molecular mass is higher than hydrogen)? </p>
<p>Is there a formula? What's the relation between the fuel's molecular mass, temperature and escape velocity?</p> | 6,093 | [
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<p>If you take the homogenous wave equation:</p>
<p>$$-\Delta_x u(x,t) + \frac{1}{c^2} \frac{\partial^2 u}{\partial^2 t} (x,t) \ = \ 0 \ \ \mathrm{in} \ \Omega \times (0, \infty),$$</p>
<p>with some proper initial- and boundary conditions and make the ansatz:</p>
<p>$$u(x,t):= e^{-i \omega t} v(x),$$</p>
<p>i.e. we seek a time-harmonic wave to the angular frequency $\omega \geq 0 $. <br>
Inserting this ansatz into the equation, we get the eigenvalue problem:</p>
<p>$$-\Delta v(x) \ = \ \underbrace{\frac{1}{c^2} \omega^2}_{:= \ \lambda} v(x) \ \ \mathrm{in} \ \Omega.$$</p>
<p>Now, if i solve this eigenvalue problem and got a eigenvalue $\lambda$, i can retrieve the angular frequency by $\omega = \sqrt{\lambda \ c^2}$.<br>
Here, i'm trying to model the 2D case of acoustic waves ($\Omega \subset \mathbb{R}^2$), but the 3D-case could be also interesting. Doing some research, i found out (<a href="http://en.wikipedia.org/wiki/Acoustic_wave_equation" rel="nofollow">for example here</a>), that $c$ is in my case the speed of sound, i.e. for example $c$ = 343 m/s in air at 20°C. I assume, since $c$ is dealing with meters, my domain $\Omega$ has also the dimension m$^2$ (or m$^3$ in 3D-case). <br>
But, by the upper formula, $\omega = \sqrt{\lambda \ c^2}$, the angular frequency would have dimension "m/s" (meters per second) - that doesn't sound right, shouldn't this be of dimension s$^{-1}$? <br></p>
<p>I would be glad if anyone, who's more experienced in physics, could explain this. Thanks in advance!</p> | 6,094 | [
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<p>I fear that I have a fundamental misconception about the "wave particle duality" of light, but in a related question, the answerer said, in some sense, that a light wave propagates until it hits something, at which point in time it (can) act(s) like a photon. Which is fine to me, but there are a finite number of photons in a wave front, so what happens if you "run out" of photons in a wave front? Certainly the wave needs to interact with everything it touches, so if you have a wave that only effectively has one photon, and it "hits" two electrons, how does it interact with both? Say you have two electrons both a distance $R$ from a photon emitter, emitting circular waves. Or something like that. </p> | 6,095 | [
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<p>My textbook presents an idealization of a conductor as made up of infinitesimal units of charge and derives results. I was not convinced, so I started thinking of how electric fields are in real metals. Here is what I think now:</p>
<p>i) My description of a "static" situation - There are electric fields inside a conductor, but no net electric field that does anything meaningful. Electrons are flying around left, right, up and down, but no net current is present. If you take a reasonably macroscopic chunk of metal, no net current will flow.</p>
<p>ii) The entire metal is at a lower potential energy than let's say, air, that surrounds it. Therefore, there exists a work function to pull out electrons. It's like a well. You need to pull hard enough to yank an electron out. Precisely, you have to do more work than the force that holds the electron in.</p>
<p>iii) The interior of a metal cannot have excess charge in a static condition - If the interior does have charge, the charges will repel until they reach a point of equilibrium, which means the metal isn't static. This is almost like a definition.</p>
<p>iv) Excess charge lies at the surface of a conductor, in static situation - Since the metal is static, all the inside of the metal should behave like a regular neutral metal. For a sphere for example, this means that excess charges must form a ring of equal charge density. This is similar to saying that electric field is zero inside a ring. The excess charges will not contribute any field</p>
<p>v) The excess charges exert a field on each other. For a sphere for example, every charge exerts charge on every other charge, so that net electric field points outside. The existence of the work function means that any attempts to pull out charges will be countered. This countering force (essentially due to protons in atoms) will counter outside field. An equilibrium is reached and therefore, those charges are more or less still.</p>
<p>vi) The electric field inside is more or less zero, and the excess charge on the outside experiences zero field too, so the metal is an equipotential surface. </p>
<p>vii) Any external electric field is countered using by an arrangement of charge that yields zero field inside, and these charges do not fly out because they are held in.</p>
<p>I know there are many approximations in this model. But I think it beats the unrealistic idealization given by my textbook isn't really connected to real life. Are there any flaws in it? I will learn better models when I do solid state physics, right?</p> | 6,096 | [
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<p>During a very short time after the big bang, the universe must have had an edge of space-time which is very close to all the matter in the universe. The particles which are close to or on the edge must also have gravity and other forces. How did those forces interact with the space-time boundary? And, if the range of gravity is infinite, then, does that mean that the space-time of the universe became infinite right after big bang?</p> | 6,097 | [
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<p>In an effort to destabilize hurricanes by burning the necessary cold zones it has been proposed that planes drop various <a href="http://www.dailygalaxy.com/my_weblog/2007/10/can-man-control.html" rel="nofollow">carbon trash</a> on them. That seems like a minor amount of heat change over a large area. What not just bomb the hurricane with remote detonation? How powerful would a series of rail guns or bombs need to be to destabilize a hurricane? </p> | 6,098 | [
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<p>The philosopher of physics Laura Reutsche argues in her book Interpreting Quantum Theories (review/summary here: <a href="http://philsci-archive.pitt.edu/9493/1/ruetsche-review.pdf" rel="nofollow">http://philsci-archive.pitt.edu/9493/1/ruetsche-review.pdf</a> ) that a "pristine" interpretation (1-1 correspondence between theory and reality) is impossible in quantum field theory, and all that exist are "adulterated" interpretations dependent on particular applications and contexts. Can anyone who thinks this is wrong explain why?</p>
<p>Thanks.</p>
<p>(I realize this is more a a philosophy of physics question, but I'm asking it here b/c Reutsche's book (with C*-Algebras, even a tiny bit of non-commutative geometry, and Hans Halvorson's comment that it is "perhaps the most sophisticated engagement that with mathematical physics that we have ever seen in a "philosophical" monograph") is way too mathematically advanced for most philosophers to understand.)</p> | 6,099 | [
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<p>Volcanic eruptions sometimes cool the earth. How many eruptions of what magnitude, locality and type would be required per year to neutralize Global Warming?</p> | 6,100 | [
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<p>How to find the intrinsic covariant derivative component?</p>
<p>In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant derivative with respect to proper time.
where the covariant derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
$\frac {DU^{\mu}}{d\tau}=\frac {dU^{\mu}}{d\tau}+\delta A^{\mu}$.
infact:
$A^{\mu}_{GR}=A^{\mu}_{SR}+\delta A^{\mu}$.
(GR represent General Relativity and SR represent Special relativity)</p>
<p>I don't know how the $\delta A^{\mu}$ becomes $\Gamma^{\mu}_{\alpha\beta}U^{\alpha} U^{\beta}$.</p> | 6,101 | [
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<blockquote>
<p>A nonuniform horizontal bar of mass $m$ is supported by two massless wires against gravity. The left wire makes an angle $ϕ_1$ with the horizontal, and the right wire makes an angle $ϕ_2$. The bar has length $L$.</p>
</blockquote>
<p><img src="http://i.stack.imgur.com/LB8VB.png" alt="enter image description here"></p>
<blockquote>
<p>What is the position of the center of mass of the bar, measured as distance $x$ from the bar's left end?
Find x in terms of $ϕ_1, ϕ_2,$ and $L$</p>
</blockquote>
<hr>
<p>I've found that
$$x={T_2sin(ϕ_2) \over T_1sin(ϕ_1)+T_2sin(ϕ_2)}$$
and
$$0=T_2cos(ϕ_2)-T_1cos(ϕ_1)$$
The hint it's giving me is</p>
<blockquote>
<p>Use what you know about the x components of the forces acting on the bar.</p>
</blockquote>
<p>I'm just not sure where to proceed.</p> | 6,102 | [
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] |
<p><a href="http://i.stack.imgur.com/ph7zc.png" rel="nofollow">click to view the image</a> <br/></p>
<p>Before I start, I want to say that this is not a duplicate of "Is it possible for information to be transmitted faster than light by using a rigid pole?", Since point A is not a real object, it is possible for A to exceed the speed of light. <strong><em>Notice that I have already taken the bending effect into account and decided that if the widths of the bars are small enough, then it's safe to treat the scissor as if it weren't bending</em></strong>. In other words, when they are thin, then the motion will obey the equation in a small region around point O. Please read to the end and have a look on my analysis in the last part before you explain.</p>
<p><br/>Suppose we have a "scissor", which is composed by two bars with a same width of length "l". Bring them closer to each other, during this process, we have the equation(#):$$v=-\frac l4\frac{cos\frac{\theta}2}{sin^2\frac{\theta}2}\omega$$, where v is the velocity of A, \theta is the angle between BC and BA as shown in the picture.It's also an arbitrary function of time t, or $\theta=\theta(t)$. $\omega$ is the angular velocity of $\theta$, thus $\omega=\frac d{dt}\theta(t)$.Here is the derivation:<br/>
1. Draw a line vertical to L from A, intersecting with L at C. <br/></p>
<ol>
<li>Since $sin\theta=\frac{AC}{AB}$,
$$AB=\frac{AC}{sin{\theta}}$$.</li>
<li>Since $cos\frac{\theta}2=\frac{AO}{AB}$,
$$AO=AB*cos\frac{{{\theta}}}2$$</li>
<li>from 2 and 3 we have:
$$AO=\frac{cos\frac{\theta}2}{sin{\theta}}*AC$$</li>
<li>Since $AC=l$, the equation in 4 becomes:
$$AO=\frac{cos\frac{\theta}2}{sin\theta}*l$$or$$AO=\frac l{2sin\frac {\theta}2}$$</li>
<li>Differentiate each side with respect to t, and denote $\frac d{dt}AO$ as "v", getting the equation(#):
$$v=-\frac l4\frac{cos\frac{\theta}2}{sin^2\frac{\theta}2}\omega$$
<br/>Again, we just look at a small area around point O where our equation is obeyed. Now, at any time $t_0$, $\theta$ and $\omega$ are independent of each other, thus $\omega$ is a free variable. At t= 0, let's set the value of $\omega$ really large and constant, so that the instantaneous speed of A is larger than even the speed of light. Later, since the factor $\frac{cos \frac{\theta}2}{sin^2 \frac {\theta}2}$ in equation(#) is quite large when $\theta$ is close to 0, and since large $\omega$ leads to small $\theta$ during a short period of time, the velocity of A will be increased by this factor. Thus overall, $\omega$ is set to be a large constant, l is a constant, the only changing factor is increasing. Therefore, the velocity of A will increase. Meanwhile, we emit a beam of light. A is in a superluminal motion. And A will arrive at a detector first, causing the bars to touch the detector, and sending the command to the machine. Then the light beam comes later, thus being detected later. Notice that here again, we place the detector close enough to point O, so that our equation is valid.</li>
</ol>
<p>Remark: Someone may argue that we are not sure if A could attain a speed higher to that of light. So suppose at t=0, the force has already affected the area around point O, and is now traveling further.Meanwhile, ${\theta}_0=\frac{\pi}3$ A is speeded up to 0.999c, then since our formula is valid at t=0 and later period of time, let's apply it and set $l=0.001m, c=3*10^8m/s,{\theta}_0=\frac{\pi}3$. Then plug them in, we get $$\omega=-3.461*10^{11}rad/s $$. Now we want to know its velocity at $t=1*10^{-12}s$, and we get $v=6.89062*10^8m/s$, faster than light.</p>
<p><br/>According to relativity, this shall violate casualty, and therefore could never happen. With which kind of mechanism is this paradox solved? I have taken the bending effect into account, and decided that as long as the widths of the bars are small enough, then it's safe to treat the scissor as if it weren't bending.</p> | 4 | [
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