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<ol>
<li><p><strong>Free field theory:</strong> Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other?</p></li>
<li><p><strong>Interacting field theory:</strong> How exactly does the addition of non-linear term in the Lagrangian make the Fourier modes couple to each other?</p></li>
</ol>
| 2,438 |
<p>Why is <a href="http://en.wikipedia.org/wiki/Earnshaw%27s_theorem" rel="nofollow">Earnshaw's Theorem</a> inapplicable for moving ferromagnets?</p>
<p>Can I get a mathematical classical proof for this?</p>
| 2,439 |
<p>I am wondering if I mix up the notion of <em>proper distance</em> and <em>proper length</em>.</p>
<p>I have two cuves in Schwarzschild space-time describing the flight of two photons (think of it as photons guided in by optical fiber, not geodesics).</p>
<p>$\gamma_1(\lambda) = \big(t_1(\lambda),r_1(\lambda),\frac \pi 2, 0\big )$ and $\gamma_2(\lambda) = \big(t_2(\lambda),R,\frac \pi 2, \varphi_2(\lambda)\big )$ where $r_1(0)=R$ and $\varphi_2(0)=0$. </p>
<p>I now want both photons to fly the same length. With "same length" I mean, that I trimmed the two optical fibers in my lab so that they have the same lenght, measured by a tape.</p>
<p>For this purpose I arbitrarily define the end of the first curve at $\lambda=\Lambda$ and define the corresponding radial coordinate by $r_1(\Lambda)=R+\Delta R$.
Now comes the crucial point: Is it correct to say, that the length of the first optical fiber is</p>
<p>$\displaystyle l = \int_0^\Lambda \frac{1}{\sqrt{1-\frac{r_s}{r_1(\lambda)}}} r_1'(\lambda) \; d\lambda = \int_R^{R+\Delta R} \frac{1}{\sqrt{1-\frac{r_s}{r}}} \; dr$ ?</p>
<p>If this is true this would also define a $\Delta \varphi$ for the second optical fiber by</p>
<p>$\displaystyle l \overset{!}{=}\int_0^{\Delta \varphi} R \, d \varphi = R\Delta\varphi$</p>
<p>Is that correct?
My problem is, that <a href="http://en.wikipedia.org/wiki/Proper_length" rel="nofollow">wikipedia states</a> that I am dealing with <em>proper distance</em> instead of <em>proper length</em>. This confuses me since I want to be sure to have the same length for both optical fibers (measured by a tape).</p>
| 2,440 |
<p>I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in <a href="http://scitation.aip.org/content/aip/journal/pop/13/11/10.1063/1.2388265" rel="nofollow">this paper</a>. This is the step I can't follow:</p>
<blockquote>
<p>$$\epsilon_o\frac{d}{d\varphi}\biggl(\frac{E^2}{2}\biggr) = \sqrt{\frac{m_e}{2e}}\frac{j_{eo}}{\sqrt{\varphi}} - \sqrt{\frac{m_i}{2e}}\frac{j_{eo}}{\lambda_I}\int_\varphi^{\varphi_w}\frac{\varphi'/\varphi_I - 1}{\sqrt{\varphi' - \varphi}}\frac{d\varphi'}{E'},\tag{4}$$</p>
<p>where $\lambda_I =1/\sigma_o n_a$ is the ionization mean free path for the electron energy $E_I$. The integration of Eq. (4) over $\varphi$ leads to an integral equation for the electric field $E(\varphi)$,</p>
<p>$$\begin{multline}\frac{\epsilon_o}{4j_{eo}}\sqrt{\frac{e}{2m_e}}(E_w^2 - E^2) = \bigl(\sqrt{\varphi_w} - \sqrt{\varphi}\bigr)\\- \frac{1}{\lambda_I}\sqrt{\frac{m_i}{m_e}}\int_\varphi^{\varphi_w}\biggl(\frac{\varphi'}{\varphi_I} - 1\biggr)\sqrt{\varphi(z') - \varphi(z)}\frac{d\varphi'}{E(\varphi')},\tag{5}\end{multline}$$</p>
</blockquote>
<p>The paper claims that instead of solving the integro-differential equation numerically from the form above, both sides of the equation can be integrated with respect to $\varphi$. I am not sure if it is valid to integrate both sides of this equation with respect to this variable since it appears in the lower limit of the integral. </p>
<p>Can someone explain how you would handle integrating both sides when $\varphi$ is in the the limit of the integral? Thanks!</p>
<p>Or if anyone has an argument for why a mistake might have been made in this step that would be helpful too. </p>
| 2,441 |
<p>I have $N$ distinguishable particles in a 1D harmonic oscillator potential with 'proper' frequency $\omega$. The particles also have internal spin-$\frac12$ degrees of freedom in a magnetic field $B$ with magnetic dipole $\mu$. The spin induced energy level splitting is
$$\varepsilon=2\mu B=0.1\hbar\omega$$</p>
<blockquote>
<p>Question: Show that the partition function is a product of the oscillator ($Z_1$) and spin ($Z_2$) partition functions.</p>
</blockquote>
<p>I have a response but I'm pretty dissatisfied with it. </p>
<p>I argued that
$$Z=\sum_n\exp(\beta\varepsilon_n)$$
so for $\varepsilon_i$ (oscillation energy) and $\varepsilon_j$ (spin energy) we have:</p>
<p>$$Z=\sum_n\exp(\beta\varepsilon_n)=\sum_{i,j}\exp\bigl(\beta(\varepsilon_{i}+\varepsilon_{j})\bigr)=\sum_i\sum_j\exp(\beta\varepsilon_i)\exp(\beta\varepsilon_j)=Z_1Z_2$$</p>
<p>(Edit: I actually don't think $\sum_{i,j}\leftrightarrow \sum_i \sum_j$ is valid)</p>
<p>This is where I left things for the time being but I wonder if I should elaborate.</p>
<p>For example, I considered... $$\varepsilon=2\mu B$$ ...for the 3 level spin and argued that...
$$\begin{align}
Z_2
&=\sum \exp(\beta\varepsilon_2)=\exp(-0.5\beta\varepsilon_2)+1+\exp(0.5\beta\varepsilon_2) \\
&=\exp(-0.05\beta\hbar\omega)+1+\exp(0.05\beta\hbar\omega)
\end{align}$$ ...for 3 energy levels $-0.5\varepsilon$, $0$ and $0.5\varepsilon$</p>
<p>For the harmonic oscillator I argued that... $$\varepsilon_1=\biggl(i+\frac{1}{2}\biggr)\hbar\omega$$ ...such that...
$$\begin{align}
Z_1
&=\sum_i \exp(\beta\varepsilon_1)=\sum_i \exp\biggl[\beta \biggl(i+\frac{1}{2}\biggr)\hbar\omega\biggr] \\
&=\frac{1}{2}\hbar\omega \sum_i \exp(\beta i\hbar\omega)=\frac{1}{\exp(-0.5\beta\varepsilon)-\exp(0.5\beta\varepsilon)}
\end{align}$$</p>
<p>If the above is correct then I should be able to arrive at $Z=Z_1Z_2$ easily. However, the outcome of the multiplication is unrecognisable to me. This is one of my challenges with this topic - trying to reconcile the maths with the physics</p>
<p>Clarification: </p>
<p>The above is the homework part. I'm also experiencing difficulty grasping some of the ideas in statistical physics, in particular, the relationship between the partition function, entropy and specific heat capacity</p>
<p>My understanding is that the partition function shows the number of microstates that are accessible within thermal energy range. Hence, the number of microstates that are accessible within thermal energy range is the product of microstates in the 1d oscillator and the 3 level system. Would this be 4 degrees of freedom in total (1 from the oscillator and 3 from the splitting)?</p>
<p>I have to say I'm very uncomfortable with my lack of physical understanding for the material. The relationship between the partition function, entropy and specific heat capacity is still very fuzzy in my mind.</p>
| 2,442 |
<p>For example say the gravitation constant instead of equaling G, was actually a range bounded between 0 and infinity. </p>
<p>Our Universe would be at a point on this range (equal to our G value) where things could physically exist and produce stars, galaxies and life. Yet in the gravitation constant's dimension it is able to be lower and higher then our G value. </p>
<p>All other physical constants being equal, this would allow the sort of full universe we recognise to exist on only a very tiny part of the full range of the gravitation constant's dimension. The rest of the dimension would be full of non-universes that would not be able to support anything.</p>
<p>The other non-dependent physical constants perhaps also represent dimensions. And there might even be a set of useful life supporting universe's on a continuous 'line' inside this n-dimensional structure.</p>
| 2,443 |
<p><img src="http://i.stack.imgur.com/gQ3jg.gif" alt=""></p>
<p>I recently stumbled upon the above image describing partial transmittance, and was wondering what sort of equation would model such a wave propagating through varying mediums. Is there also an equation for continuous mediums, as opposed to the discontinuous transition in the picture?</p>
| 2,444 |
<p>In the book <a href="http://books.google.ca/books?id=UaY1MLmC780C&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false" rel="nofollow">Nonlinear Optics</a>, it is stated that the nonlinear effects start to become a problem in WDM systems (around 1550 nm) after about 1 mW of optical power. However, I measured the optical power at transmission of a transmitting laser of a 10GBASE-ZR on a short, 2 km link, and found that it was 2.5 mW. </p>
<p>What is the operating optical power of telecommunications lasers? Does it differ for inter-continental links compared to short links, i.e. between buildings on a campus? If they are above 1 mW, are nonlinear effects a concern?</p>
| 2,445 |
<p>Assume a balloon filled with Hydrogen, fitted with a perfect valve, and capable of enduring vacuum (that is to say, it would retain it's shape and so well insulated that the extremes of temperature at high altitudes and in space would have little effect) were to be launched. </p>
<p>As long as the balloon were in atmosphere it would ascend upwards (and also affected by various winds/currents, and gravity). As the balloon passed through increasingly rare atmosphere, would it rise faster? </p>
| 2,446 |
<p>I'm reading about how the soon-to-be-launched <a href="http://en.wikipedia.org/wiki/Nuclear_Spectroscopic_Telescope_Array" rel="nofollow">NuSTAR</a> is on the cutting edge of focusing x-rays, which captures 5 to 80 keV radiation by focusing them with optics that have a 10.15 meter focal length onto 2 sets of 4 32×32 pixel detector arrays. These are particular "hard" (high energy) x-rays, which is a part of what makes the task difficult and the NuSTAR telescope novel.</p>
<p>If I understand correctly, imaging gets particularly difficult with electromagnetic radiation beyond a certain energy, as true gamma rays (above 100 keV) are detected with a family of radiation detectors that sense the Compton scatter or photoelectric absorption with an electrical pulse that is (in a naive sense) insensitive to the originating direction or location within the detector. It should be obvious that imaging can still be done with the use of an array of detectors, each constituting a single pixel, and these capabilities may improve with time as semiconductor detector technology evolves.</p>
<p>So the critical distinction I'm trying to establish is between x-rays and gamma rays. It would seem that we focus x-rays and do not focus gamma rays. For a very good example of researchers <em>not</em> focusing gamma rays, consider Dr. Zhong He's <a href="http://czt-lab.engin.umich.edu/" rel="nofollow">Radiation Measurement Group</a> at UM, who do actual imaging of a gamma ray environment (the UM Polaris detector). They use a grid of room temperature semiconductors laid out bare in a room and use back-processing of the signals to triangulate a sequence of scatter-scatter-absorption reactions in 3D space. This is a lot of work that would be completely unnecessary if you could focus the gamma rays like we do for a large portion of the EM spectrum.</p>
<p>Both of the technologies I reference, the NuSTAR telescope and the UM Polaris detector, use CdZnTe detectors. Functionally they are very very different in that the telescope uses optics to capture light from just a few arc-seconds of the sky.</p>
<p>My question is what is the specific limitation that prevents us from focusing photons above a certain energy? It seems this cutoff point is also suspiciously close to the cutoff between the definition of x-rays and gamma rays. Was this intended? Could future technology start using optics to resolve low-energy gamma rays?</p>
| 2,447 |
<p>The text I am reading (Stars and Relativity by Ya. B. Zel'dovich) discusses the possible existence of a repulsive force proportional to total baryon number. At the time of the book's publication it was stated that if such a force existed, it woukd need to be on the order of $10^{46}$ times weaker than the Coulomb force between two protons. </p>
<p>It's also stated as a footnote that such a force would be incompatible with a homogeneous, isotropic universe. Why is this so? Has any of the above changed in the ~40 years since this text was published?</p>
| 2,448 |
<blockquote>
<p>What are some common applications, uses, exploitations of the properties of surface tension? </p>
</blockquote>
<p>Here is what I mean. A water strider can walk on water, that is a consequence of surface tension. This is a consequence, but it is not human made.</p>
<p>On the other hand, I heard that in the construction of some tents, the upper cover of the tent is the rain protector. It is not really impermeable, but if water is placed on it then the water surface tension does not let the water pass through the fine, small pores of the tent cover. However, if you touch the cover while water is on it, you break the surface tension and water passes through. </p>
<p>I would say that the above fact is a clever use of the effect of surface tension. Are there any other known applications, or interesting experiments regarding the surface or interfacial tension?</p>
| 2,449 |
<p>English Wikipedia in the <a href="http://en.wikipedia.org/wiki/Poisson%27s_ratio#Width_change" rel="nofollow">Poisson's ratio article</a> gives an equation for large deformation:
$$
\frac{\Delta d}{d}=-1+\frac{1}{\left(1+\dfrac{\Delta L}{L}\right)^\nu}
$$</p>
<p>I couldn't find any reference for this equation. Could someone help? I need it in my thesis, and I'd like to reference some other source than Wikipedia. Every other publication I've seen so far gives the approximate (linear) equation.</p>
| 2,450 |
<p><em>It's really shocking that the following question was voted down twice although the question is yet to be answered by anyone.</em></p>
<p>So please read the following and give an answer why physics community has taken a double standard in the following case.</p>
<p>We know about Michelson's experiment. We also know that despite of the null result how many times the experiment was revisited with the hope to get a positive result. </p>
<p>Unfortunately, this is not the case for Ehrenhaft's experiments (<a href="http://www.rexresearch.com/ehrenhaf/ehrenhaf.htm" rel="nofollow">http://www.rexresearch.com/ehrenhaf/ehrenhaf.htm</a>).</p>
<p>We have already performed a lot of experiments to find a real or artificial magnetic monopole but no monopole is found yet through any experiment.</p>
<p>Furthermore, while studying literature on magnetic monopole, I have found that very few researchers even recognized the work of <a href="http://en.wikipedia.org/wiki/Felix_Ehrenhaft" rel="nofollow">Ehrenhaft</a> who published more than 60 papers on this topic.</p>
<p>I don't understand why his work is so much ignored in the mainstream physics.</p>
<p><strong>Is there a large number of references</strong> (I don't expect this number to be comparable to the number of experimental attempts made in case of Michelson's experiment. I only expect a significant number of repetitions which can firmly conclude about the experimental outcome.) <strong>which show that his experiments were revisited thoroughly, carefully and honestly but his claimed outcome could not be reproduced?</strong></p>
<p>Please give your opinion about the following possible answers:</p>
<ol>
<li>Michelson's NULL result was favorable to Einstein's Relativity, while Ehrenhaft's result was not.</li>
<li>Ehrenhaft's magnetic monopole was not in agreement with Dirac's theory of magnetic monopole.</li>
</ol>
| 2,451 |
<p>I've been lightly studying GR lately. Something that has been bothering me has been the lack of (Ricci) curvature produced from the Schwarzschild metric in the few lectures I've watched, as well as the few snippets of text book I've been able to read. Why is there no (Ricci) curvature outside this spherically symmetric, non-rotating, uncharged body that still has mass? Shouldn't there always be curvature in the presence of mass or am I missing something? I've read a bit about certain information that is unobtainable when dealing with Schwarzschild coordinates, is the curvature outside the body one of the specific quantities that cannot be defined with these coordinates?</p>
| 2,452 |
<p>Now consider I went into a rocket which goes an infinite distance far from earth. By infinite I mean very far. The gravitational attraction between me and the earth will significantly decrease and after a certain distance it will cease to act as the inertia of my rocket would be bigger than the gravitational force between my rocket and the earth. So where will the gravitational potential energy be lost? It is no more stored in the rocket since it does not get affected by that force. I know to recover it I can go back closer to the earth and my rocket will automatically start getting affected by the gravitational force again. So my question is, where is the gravitational potential energy stored for an object or a massive body that goes very very far from earth or any massive object?</p>
<p>When the rocket is at earth the total energy is </p>
<p>$E = E_m = mc^2$</p>
<p>When it takes off, the mass of the rocket decreases due to the fuel consumption but this is compensated by the increase in kinetic energy and potential energy so the total energy becomes</p>
<p>$E = E_m + E_k + E_g$</p>
<p>But when it reaches a point where gravity no longer affects it then $E_g$ = 0 because there is no more potential energy.</p>
<p>So the energy of the system becomes</p>
<p>$E = E_m + E_k$</p>
<p>So where did $E_g$ vanish?</p>
| 2,453 |
<p>I have a strong background in Fourier analysis, and I'm looking for QM resources that can build on that. Is there a book around or a little above the level of Griffiths that has that kind of emphasis? Any other resources are good too.</p>
| 2,454 |
<p>The question basically amounts to whether I can construct the illusion of superposition with adjacent sine waves of varying frequency.</p>
<p><strong>Context</strong></p>
<p>I'm trying to play music on a Tesla Coil (like OneTesla and the works). If I want to play an A for instance, I'll modulate my input signal so that the Tesla coil sparks 110 times per second. Each time, it will turn air into plasma, a volume change whose pressure wave propagates through the air as sound.</p>
<p>However, as of now I can only play one note at a time. For example, I can play a C, followed by a F, followed by a G, but I can't play a CFG chord. In terms of sine waves (i.e. why I'm posting to the Physics StackExchange), I'm trying to model $\Sigma_{i=1}^{n} sin(\alpha_i x)$ from, say $0<x<\pi$ time units, with the approximation $sin(\alpha_i x)$ for $\frac{(i-1)\pi}{n}<x<\frac{i\pi}{n}$ for i from 1 to n, and x from 0 to $\pi$.</p>
<p>At least with light, I think that humans have a visual memory - e.g. if I show you red, and then blue, and then red, and alternate really, really, fast, you see purple (I think...). Is the same possible for sound? If so, how fast do I need to alternate notes?</p>
<p>I'm sure the same question has been asked somewhere else, but I couldn't find any answers that apply. Also, if this is in the wrong forum, feel free to move it.</p>
| 2,455 |
<p>I'm programming a 1 dimensional physics simulator in python. How do I calculate the amount of time until two particle points that are traveling towards eachother collide? I am using the equation: </p>
<pre><code>time = (m2.pos - m1.pos) / (m1.velocity - m2.velocity).
</code></pre>
<p>You can assume m1 is always on the left, and m2 is always on the right. This equation works fine in my program if both points are traveling in the same direction at slightly different speeds, but when m1 has a positive velocity and m2 has a negative velocity, that gives the denominator of my equation a negative number which produces a negative amount of time until the next collision.</p>
| 2,456 |
<p>I'm working my way through Griffith's "Introduction to Electrodynamics". In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric potentials $V_0$, we can create a new set of equivalent magnetic and electric potentials given by:</p>
<p>$$
\textbf{A} = \textbf{A}_0 + \nabla\lambda \\
V = V_0 - \frac{\partial \lambda}{\partial t}
$$</p>
<p>These transformations are <strong>defined</strong> as a "gauge transformation". The author then introduces two of these transformations, the coloumb and lorentz gauge, defined respectively as:</p>
<p>$$
\nabla \cdot \textbf{A} = 0 \\
\nabla \cdot \textbf{A}= -\mu_0\epsilon_0\frac{\partial V}{\partial t}
$$</p>
<p>This is where I am confused. I do not understand how picking the divergence of $\textbf{A}$ to be either of these two values actually constitutes a gauge transformation, as in it meets the conditions of the top two equations. How do we know that such a $\lambda$ even exists for setting the divergence of $\textbf{A}$ to either of these values. Can someone convince me that such a function exists for either transformation, or somehow show me that these transformations are indeed "gauge transformations" as they are defined above.</p>
| 2,457 |
<p>There are known formulae relating a capacitor's voltage and current in its classic form (battery, wires, a capacitor circuit), but what if we had a charged plate (-ve), then we put on one of its surfaces a dielectric, then we put another plate (neutral, connected to the ground)? </p>
<p>Technically, this is a capacitor (2 differently charged plates separated by a dielectric). The other plate will be +ve charged (like charging by induction) and a current will flow in the wire connecting the second plate to the ground. If I wanted to calculate this current, should I use the same formulae or will they change because the voltage is changing only on one plate and the second plate will reach a equal and opposite voltage pretty quickly?</p>
| 2,458 |
<p>I'm trying to calculate the induced $-V$ in a DC circuit when current starts to flow, and reaches it's maximum value.
The inductance of the conductor is 1000nH it carries large amounts of current (1000A), what formula is best to know the $-V$ due to self inductance?</p>
| 2,459 |
<p>I want to understand the derivation of the partition function for two distinguishable non-interacting particles. </p>
<p>Let the energy of particles $1$ and $2$ be $E_1$ and $E_2$ respectively. Setting $\beta = 1/kT$, the partition function becomes:
$$
Z = \sum_{s} e^{-\beta[E_1(s)+E_2(s)]}
$$
Where $s$ represents the state of the whole system. But then the author says: the set of state of the system is equivalent to the set of all possible pairs of states, $(s_1,s_2)$, for the two particles individually. And he states:
$$
Z_{total} = \sum_{s_1} \sum_{s_2} e^{-\beta E(s_1)} e^{-\beta E(s_2)}
=\sum_{s_1}e^{-\beta E(s_1)}\sum_{s_2}e^{-\beta E(s_2)} = Z_1 Z_2
$$
My question is: how can you formally go from the summation over $s$ to the double sum over $s_1$ and $s_2$? And why is one allowed to do so? I believe this has to do with the fact the since $s_1$ and $s_2$ do not intersect, then $P(s) = P(s_1)P(s_2) \sim e^{-\beta E(s_1)} e^{-\beta E(s_2)}$. But that's still not enough to justify the double sum...</p>
<p>(If I have we written too much, could someone cut the unnecessary parts? I don't know how much information I should give to make myself understood.)</p>
| 2,460 |
<p>What does the following mean with respect to <a href="http://en.wikipedia.org/wiki/Special_relativity">special relativity</a>?</p>
<p>$$\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$</p>
| 2,461 |
<p>If one is able to travel into the past but at a spatial distance that puts him outside of his own past light cone would this be considered a causality violating trip? Looking at a Minkoski diagram, it would seem that one ought to be able to travel to a spatially displaced past without producing causality violations. In fact, I'm not sure you could even say whether this was the past of not for sure. <img src="http://i.stack.imgur.com/U34I5.png" alt="Generic Minkowski diagram"></p>
<p>For instance: If you found a one way wormhole, that took you outside of your own lightcone but into the past, could you say for sure that you traveled through time? Would this be a form of timetravel that did not violate causality? Note; I'm not necessarily suggesting that it is an FTL trip, it may be considered to be instantaneous teleportation, but in the reverse direction of x'. </p>
| 2,462 |
<p>I was reading up on the history of the solar neutrino problem, and as far as I can understand it, neutrinos supposedly oscillate from one form to another, thus explaining why there were only one-third the number of neutrinos detected than were expected, when they began neutrino observations in the 1960's.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Solar_neutrino_problem">Wikipedia article on the topic</a> ends with this statement:</p>
<blockquote>
<p>The convincing evidence for solar neutrino oscillation came in 2001
from the Sudbury Neutrino Observatory (SNO) in Canada. It detected all
types of neutrinos coming from the Sun, and was able to distinguish
between electron-neutrinos and the other two flavors (but could not
distinguish the muon and tau flavours), by uniquely using heavy water
as the detection medium. After extensive statistical analysis, it was
found that about 35% of the arriving solar neutrinos are
electron-neutrinos, with the others being muon- or tau-neutrinos. The
total number of detected neutrinos agrees quite well with the earlier
predictions from nuclear physics, based on the fusion reactions inside
the Sun.</p>
</blockquote>
<p>But as far as I can see, none of this or anything else I've read seems to give any <em>proof</em> that solar neutrinos change type while <em>en route</em> to the Earth. It seems that the sun just emits about 1/3 of each of the three types. </p>
<p>Or is it that at the temperature of the solar core only electron neutrinos are emitted, and then they oscillate (randomly?) from that type to the others and back again? I'd welcome a little clarity about this.</p>
| 2,463 |
<p>It is claimed that <a href="http://www.google.com/#q=de+Sitter+temperature" rel="nofollow">de Sitter temperature</a> is $$T=\frac{1}{2\pi}H,$$</p>
<p>where $H$ is the Hubble constant. I presume it is expressed in natural units with which I am not familiar. So what it will be in Kelvins? Is it higher than the temperature of CMB? If so, is there any heat exchange between the both?</p>
| 2,464 |
<p>We know that level-k Abelian 2+1D Chern-Simons theory on the $T^2$ spatial torus gives ground state degeneracy($GSD$):
$$GSD=k$$</p>
<p><strong>How about $GSD$ on $T^2$ spatial torus of</strong>: </p>
<p><strong>SU(N)$_k$ level-k Chern-Simons theory?</strong></p>
<p><strong>SO(N)$_k$ level-k Chern-Simons theory?</strong></p>
<p><strong>Sp(N)$_k$ level-k Chern-Simons theory?</strong></p>
<p>What are the available methods to compute them? such (i) algebraic geometry; (ii) Lie algebra; (iii) topological theory or (iv) quantum hall fluids parton construction?</p>
<p>an example of SU(2)$_k$ using <a href="http://physics.stackexchange.com/questions/80177/follow-up-question-on-wilson-loops-as-raising-operators">this approach</a> shows $GSD$ on $T^2$ is
$$GSD=(k+1)$$</p>
<p>SU(3)$_k$ using <a href="http://physics.stackexchange.com/questions/80177/follow-up-question-on-wilson-loops-as-raising-operators">this approach</a> shows $GSD$ on $T^2$ is
$$GSD=(k+1)(k+2)/2$$.</p>
<p>If there are examples of $G_2,F_4,E_6,E_7,E_8$ (level-k?) Chern-Simons theory and its $GSD$ on $T^2$, it will be even nicer. Reference are welcome.</p>
| 2,465 |
<p>I know classical physics, quantum mechanics, special relativity, basic nuclear physics.
I would like to get into some particle physics. </p>
<p>I want to get into that higgs boson, lepton, quark things :D </p>
<p>Where to start? Any books, websites, video lectures? </p>
<p>Mathematical rigor required! </p>
<p>Thanks </p>
| 154 |
<p>Why is it that when we cut bread or anything else with a knife, the less effective way to cut it is just by pressing on it from above? And is it true that we can cut things with knife because of the thin edge that makes an interaction with the other material (like bread)?</p>
| 2,466 |
<p>Quantum Mechanics is very successful in determining the overall statistical distribution of many measurements of the same process.</p>
<p>On the other hand, it is completely clueless in determining the outcome of a single measurement. It can only describe it as having a "random" outcome within the predicted distribution.</p>
<p>Where does this randomness come from? Has physics "given up" on the existence of microscopic physical laws by saying that single measurements are not bound to a physical law?</p>
<p>As a side note: repeating the same measurement over and over with the same apparatus makes the successive measurements non-independent, statistically speaking. There could be a hidden "stateful" mechanism influencing the results. Has any study of fundamental QM features been performed taking this into account? What was the outcome?</p>
<hr>
<p>Edit: since 2 out of 3 questions seem to me not to answer my original question, maybe a clarification on the question itself will improve the quality of the page :-)</p>
<p>The question is about why single measurements have the values they have. Out of the, say, 1000 measure that make a successful QM experiment, why do the single measurements happen in that particular order?
Why does the wave function collapse to a specific eigenvalue and not another? It's undeniable that this collapse (or projection) happens. Is this random? What is the source of this randomness?</p>
<p>In other words: what is the mechanism of choice?</p>
<hr>
<p>Edit 2: More in particular you can refer to chapter 29 of "The road to reality" by Penrose, and with special interest page 809 where the Everett interpretation is discussed - including why it is, if not wrong, quite incomplete.</p>
| 2,467 |
<p>$g\phi^3$ , $d=4$ , 3 point One loop diagram (three external legs) Divergence</p>
<p>I am trying to find where the divergence factor/pole is on the following diagram in 4 dimensions so that I can use minimal subtraction...</p>
<p><img src="http://i.stack.imgur.com/lV1cy.png" alt="enter image description here"></p>
<p>I know it is proportional to...</p>
<p>$$
\int d^dy_1 \ d^dy_2 \ d^dy_3 \ D_f(x_1,y_1)D_f(x_2,y_2)D_f(x_3,y_3)D_f(y_1,y_2)D_f(y_2,y_3)
D_f(y_3,y_1)$$</p>
<p>where $D_f$ are Feynmann propagators. Which I can manipulate to get...
$$
\int d^dp_1 \cdots d^dp_3 \left(\frac{\textrm{Exp}[ip_1x_1]}{p_1^2+m^2}\right) \cdots \left(\frac{\textrm{Exp}[ip_3x_3]}{p_3^2+m^2}\right) \delta[p_1+p_2+p_3] \bullet \int d^dp_4 \left(\frac{1}{p_4^2+m^2}\right) \left(\frac{1}{(p_2+p_4)^2+m^2}\right) \left(\frac{1}{(p_2+p_3+p_4)^2+m^2}\right)
$$</p>
<p>Which looks like a renormalization to the coupling constant $g$. So I only mess around with the $p_4$ integral, however it seems to be finite (no divergences) when I use Feynmanns trick...
$$
\frac{1}{abc}=2 \int_0^1 dx \int^{1-x}_0 dy \ (ax+by+c(1-x-y))^{-3}
$$
and the following integral...
$$
\int \frac{d^dp}{(p^2+2p \cdot q +m^2)^n} \textrm{(Minkowski)} = i \pi^{d/2}(m^2-q^2)^{d/2-n} \frac{\Gamma[n-d/2] \Gamma[d/2]}{\Gamma[n]}
$$</p>
<p>Was just wondering if I am going in the right direction or if I did something wrong.</p>
| 2,468 |
<p>Given reflectivity $R = 0.75$, an etalon is used as an interference filter, transmitting light of wavelength $4.3\mu m$ at normal incidence. The full width half maximum is about $\Delta \lambda = 0.2 \mu m$. Find the initial spacing of plate.</p>
<p>(a) Find the spacing of the plate.</p>
<p>(b) Find the change in mirror spacing to shift the centre wavelength to $4.5\mu m$.</p>
<p>(c) The spacing is now fixed at a value that gives maximum intensity at $4.3\mu m$ at normal incidence. The angle of incidence is changed to $17.5^o$, find the wavelength of light transmitted.</p>
<p>(d) Find a wavelength shorter and longer than $4.3\mu m$ that gives maximum intensity at normal incidence.</p>
<p>I'm quite confused by this question, as usually either the spacing or wavelength is fixed...now you can even change the wavelength of light merely by changing the spacing?!</p>
<p><strong>Attempt</strong></p>
<p>(a) Intensity is given by: $I = I_0 \frac{1}{1 + \frac{4R}{(1-R)^2}sin^2(\frac{\delta}{2})}$. At $I = \frac{I_0}{2}$,</p>
<p>$$ \delta = \frac{1-R}{\sqrt R} $$</p>
<p>Thus FWHM $\Delta \delta$ = $2 \times \delta = \frac{2(1-R)}{\sqrt R} $</p>
<p>Using $\delta = 2nd = \frac{4\pi d}{\lambda}$, $\Delta \lambda = \frac{4\pi d}{\lambda^2}\Delta \lambda$.</p>
<p>Using n = 1, $d = \frac{\Delta \delta}{\Delta \lambda} \frac{\lambda^2}{4\pi} = 4.25 \mu m$.</p>
<p>(b) Condition of maxima is $2d cos\theta = m\lambda$. Central maxima is given by $m_0 = \frac{2d}{\lambda} \approx 2$. (Quite a lousy etalon right? Only able to see 2 fringes)</p>
<p>Now for the same value of $m_0$, we want to increase $\lambda$, so naturally the LHS term $d$ must increase as well.
$$\Delta d = \frac{m_0}{2}\Delta \lambda = \frac{d}{\lambda}\Delta \lambda = +0.2 \mu m $$</p>
<p>(c) Again fixing $m_0$, and now the spacing is the original $4.25\mu m$.
$$2d cos (17.5^o) = m_0 \lambda$$</p>
<p>$$\lambda = 4.05 \mu m$$</p>
<p>The wavlength is now shorter! At this point I have no idea what I'm doing anymore, how can the wavelength be changed?!</p>
<p>(d) Since it's either m = 1 or m = 2,</p>
<p>$$m = \frac{2d}{\lambda} $$</p>
<p>For $m=1$, $\lambda = 2d = 8.5 \mu m$</p>
<p>For $m=2$, $\lambda = d = 4.25 \mu m$</p>
<p>I would really appreciate if anyone could explain what's going on, as I'm pretty much confused..</p>
| 2,469 |
<p>I've had it explained to me in a <a href="http://physics.stackexchange.com/questions/117013/is-there-a-difference-between-photons-that-act-as-virtual-particles-and-photon">separate post</a> that charged particles are constantly exchanging virtual particles with other charged particles and their energy is a steady state. How it is a surety that all of the virtual photons sent out by the charged particle will be exchanged with other charged particles? Are the particles sent in a directed manner toward other charged particles? How much do we know about this interaction/mechanism? Are some virtual particles just radiated away, leaving the charged particle with less energy?</p>
| 2,470 |
<p>I always thought that $force$ is $mass * acceleration$. Well, that's what I learnt at school a while back. Now, I have been enlightened that force is in fact the rate of change of momentum. </p>
<p>What makes this interesting to me, is that this takes into account a change in mass with respect to time as also playing a role in the force. A nice example of this is that of a rocket taking off. But clearly, this was not an example that Newton had in mind. </p>
<p>So my question: What did Newton have in mind of something where the mass changes with time?</p>
<p>I appreciate that my question is similar to say: <a href="http://physics.stackexchange.com/questions/2644/how-did-newton-discover-his-second-law">How did Newton discover his second law?</a> but I didn't read anywhere there about the concept of change of mass with time.</p>
| 47 |
<p>This question is related to the Big Bang Nucleosynthesis of light elements, more concretely I do not understand some features of the graph</p>
<p><img src="http://i.stack.imgur.com/GSfcM.png" alt="BBabundances"></p>
<ul>
<li><p>Why do the $^3$He and D abundances go down with increasing baryon to photon fraction while the Li abundance goes up?</p></li>
<li><p>And what is that strange dip in the Li abundance? </p></li>
</ul>
| 2,471 |
<p>How does sub-ambient cooling work?</p>
<p>There are water cooling systems for computers that can cool components to below room temperature. The problem I see here is that the water is cooled using room temperature air. How can the cooling system keep a 150 watt computer component at a temperature below room temperature? The only powered device on the water cooling loop is the pump and since that isn't being cooled by anything else than the water it cannot contribute to the lower temperature.</p>
<p>I'm guessing this has something in common with a Geothermal heat pump, but reversed.</p>
| 2,472 |
<p>I have a model that represents a bicycle (a wood block with wheels), and I'm balancing the center of gravity so it's the same as a real bike. However, when the center of mass is kept constant, does the weight of it affect the effect torque has on it when it hits a wall?</p>
<p>I'm planning to measure the angle the back wheel bounces up.</p>
| 2,473 |
<p>Order parameter is used to describe second order <a href="http://en.wikipedia.org/wiki/Phase_transition" rel="nofollow">phase transition</a>. It seems that in some papers it is used in the first order phase transitions. Can first order phase transition have an order parameter? If so, how can we define the order parameter in liquid-gas transition (first order)?</p>
| 2,474 |
<p>Dirac equation for the massless fermions in curved spase time is $γ^ae^μ_aD_μΨ=0$, where $e^μ_a$ are the tetrads. I have to show that Dirac spinors obey the following equation:
$$(−D_μD^μ+\frac{1}{4}R)Ψ=0\qquad(1)$$</p>
<p>where $R$ is the Ricci scalar.</p>
<p>I already know that $[D_\mu,D_\nu]A^\rho={{R_{\mu\nu}}^\rho}_\sigma A^\sigma$, but a key point is to know what $[D_\mu,D_\nu]\Psi$ is.</p>
<p>($D_μΨ=∂_μΨ+A^{ab}_μΣ_{ab}$ is the covariant derivative of the spinor field and $Σ_{ab}$ the Lorentz generators involving gamma matrices).</p>
| 2,475 |
<p>It is well-known that Hooke's Law is only approximately true and thus that linear relation
is merely an idealization not strictly corresponding to the reality. Wouldn't it be necessary/appropriate that all linear relations decribing physical phenomena be reformulated to contain non-linear terms for use in cases where higher accuracy is desired? In particular, what would oppose Maxwell's Equations eventually being modified to have non-linear terms? (Or put in other words, why these equations "must" be linear?)</p>
| 2,476 |
<p>I spent some time studying about temperatures and color of objects. It turns out that as we heat something it turns to red, then yellowish white and if we heat it more it turns to bluish-white.</p>
<p>Like we can say a blue star is hotter than a red star.
But why isn't it the same with flames?</p>
<p>Blue flame isn't always hotter than a red flame. It's just the chemistry of it all. </p>
<p>I mean I know the electrons jump from one orbit to another and gets into excited state when heated.</p>
<p>But the questions in my mind are:</p>
<ol>
<li><p>If one object appears blue and another red, that doesn't always mean that the blue object is hotter than red. Is that correct? If yes then how is it so? I am sorry but I'm bit confused over it.</p></li>
<li><p>With the "Color Temperature" concept on Wikipedia. They say 1,700 K to match a flame. But 15,000–27,000 K for a clear blue poleward sky.</p></li>
</ol>
<p>This confuses me. The sky appears blue, does that mean its hot? But it isn't right? It's colder. I am not sure if I'm able to frame it properly. It's a bit confusing to me.</p>
<p>Is this something like a glowing object vs reflecting object?
The sun is white but Earth is blue. But sun is hotter than the earth. It's the sun radiating light but earth is just scattering it.</p>
<p>But the surface temperature of sun is 5778K while the Wikipedia states our sky is about 15000K-27000... I know there's something I'm missing here. I'm hoping someone can tell me what is it.</p>
| 2,477 |
<p>According to <a href="https://en.wikipedia.org/wiki/Mohs_scale_of_mineral_hardness" rel="nofollow">Wikipedia's article on the Mohs scale of mineral hardness</a>, materials / minerals with a higher rating cannot be "visibly" scratched by materials with a lower rating.</p>
<p>It goes on to admit that microscopic dislocations on the harder material do emerge though by trying to scratch it.</p>
<p>I always thought that physical damage mainly depends on the force being exerted to the material, however the explanation on Mohs hardness suggests that force and work are negligible because there is no mention on <em>how</em> exactly the scratching is to be performed, at which velocity, for how long, at which temperature etc.</p>
<p>This implies that sliding and rubbing two materials against one another will basically have the same outcome regardless of whether the sliding happens through manual movement or at cosmic scales like satellite movement.</p>
<p>Also, would a material be considered "harder" on the Mohs scale if we were to scratch it near its melting point so that it sort of "self-repairs" scratches when being scratched due to friction-induced melting ?</p>
<p>Sorry if these questions sound silly to physics experts and mineralogists.</p>
| 2,478 |
<p>I can unterstand why because the integration over Grassman variables has to be translational invariant too, one has </p>
<p>$$
\int d\theta = 0
$$</p>
<p>and </p>
<p>$$
\int d\theta \theta = 1
$$</p>
<p>but I dont see where the rule for this double integration</p>
<p>$$
\int d^2 \theta \bar{\theta}\theta = -2i
$$</p>
<p>comes from.</p>
<p>So can somebody explain to me how this is motivated and/or derived?</p>
| 2,479 |
<p>I consider an electron (charge $-e$) in $x=0$ and a constant electric field $E(x) \equiv E $. If the electron has initial velocity $v_0$ with the same direction of $E$, then its potential energy is
$$ U(x) = -eV(x) = -e E x $$
The total energy for $x=0$ is
$$ K(0) + U(0) = \dfrac{1}{2} m v_0^2 $$
Now if I try to obtain the point where the electron will eventually stop and begin moving in the opposite direction I obtain from the energy conservation
$$ \dfrac{1}{2} m v_0^2 = K(0) + U(0) = K(z) + U(z) = -e E z $$
$$ z = -\dfrac{m v_0^2}{2eE} $$
But if $E > 0$ I obtain $z<0$, while $z$ should be $z>0$ since $v_0$ has the same direction as $E$!</p>
| 2,480 |
<p>I've recently seen that space is or could be a quantum vacuum full of particles like matter and anti matter appearing and possibly colliding causing in theory the same effects that dark energy has. My question is could Dark energy or dark matter be a left over waste product of matter/anti matter collisions? </p>
| 2,481 |
<p>Basically, what are all the parameters that completely describe an electron in quantum theory.</p>
<p>In classical physics a complete and fundamental description of an electron is given by its mass, charge, and position. So $(m, q, \vec{x}(t))$ gives you complete information about the electron.</p>
<p>I would like the analogous description of the quantum electron. Typically I only see electrons described by a state vector over position or a state vector over spin. But <a href="http://www.eng.fsu.edu/~dommelen/quantum/style_a/complexs.html#SECTION07451000000000000000" rel="nofollow">this page</a> explains how to represent both . I am guessing the quantum picture of an electron is something like $(m, q, \phi(\vec{x},s;t))$ where $\left| \phi\right\rangle$ is the state vector and $(\vec{x},s)$ is its "index". $s\in\{-{1\over2}\hbar,{1\over2}\hbar\}$ is the spin measured along some axis in space. Am I missing anything? Like for instance, lepton number? Does this picture change when relativity is introduced? I don't care about how the state evolves for this question, just what mathematical objects are present in the computations.</p>
| 2,482 |
<p>The basic Randall-Sundrum model is given by the metric,</p>
<p>$$\mathrm{d}s^2 = e^{-2|\sigma|}\left[ \mathrm{d}t^2 -\mathrm{d}x^2-\mathrm{d}y^2 - \mathrm{d}z^2 \right]-\mathrm{d}\sigma^2$$</p>
<p>where $\sigma$ denotes the additional fifth dimension. Notice the brane is localized at $\sigma=0$; this 'slice' is precisely Minkowski spacetime. To compute the stress-energy tensor, I define a vielbien,</p>
<p>$$\omega^\mu = e^{-|\sigma|}\mathrm{d}x^\mu \qquad \omega^\sigma = \mathrm{d}\sigma$$</p>
<p>where $\mu=0,..,3.$ Taking exterior derivatives and expressing in the orthonormal basis yields,</p>
<p>$$\mathrm{d}\omega^\mu = \epsilon(\sigma) \, \omega^\mu \wedge \omega^\sigma$$</p>
<p>where we have defined,</p>
<p>$$\epsilon(\sigma)=\theta(\sigma)-\theta(-\sigma)$$</p>
<p>which arises because of the absolute value function in the exponent, and $\theta(\sigma)$ is the Heaviside step function. By Cartan's first equation, the non-vanishing spin connections $\gamma^a_b$ are,</p>
<p>$$\gamma^{\mu}_{\sigma}= \epsilon(\sigma)e^{-|\sigma|} \mathrm{d}\omega^\mu$$</p>
<p>Taking exterior derivatives once again, and expressing in terms of the basis yields,</p>
<p>$$\mathrm{d}\gamma^\mu_\sigma = \left[ \epsilon^2 (\sigma)-2\delta(\sigma)\right]\, \omega^\mu \wedge \omega^\sigma$$</p>
<p>which arises by applying the product rule, and noting that,</p>
<p>$$\frac{\mathrm{d}\epsilon(\sigma)}{\mathrm{d}\sigma} = 2 \delta(\sigma)$$</p>
<p>because the delta function is the first derivative of the step function. From Cartan's second equation,</p>
<p>$$R^a_b=\mathrm{d}\gamma^a_b + \gamma^a_c \wedge \gamma^c_b$$</p>
<p>the components of the Ricci tensor are,</p>
<p>$$R^\mu_\sigma = \left[ \epsilon^2 (\sigma)-2\delta(\sigma)\right]\, \omega^\mu \wedge \omega^\sigma$$
as the second term vanishes. By the relation,</p>
<p>$$R^a_b = \frac{1}{2}R^a_{bcd} \omega^c \wedge \omega^d$$</p>
<p>we may deduce the Riemann tensor components,</p>
<p>$$R^\mu_{\sigma \mu \sigma} = 2\epsilon^2 (\sigma)-4\delta(\sigma)$$</p>
<p>I believe, in this case, both tensors in the coordinate basis and orthonormal basis are identical. Therefore we obtain the rank $(0,2)$ Ricci tensor,</p>
<p>$$R_{\sigma \sigma}=8\epsilon^2 (\sigma)-16\delta(\sigma)$$</p>
<p>As the only diagonal component, the Ricci scalar is identical to the Ricci tensor at $(\sigma,\sigma)$. Using the Einstein field equations, the stress-energy tensor is given by,</p>
<p>$$T_{55} = \frac{1}{8\pi G_5}\left[ 4\epsilon^2 (\sigma)-8\delta(\sigma) + \Lambda\right]$$</p>
<p>where $\Lambda$ is the cosmological constant, and $G_5$ is the five-dimensional gravitational constant. The function $\epsilon^2(\sigma)$ is given by,</p>
<p>$$\epsilon^2(\sigma)=\theta^2(\sigma)+\theta^2(-\sigma)-2\theta(\sigma)\theta(-\sigma)$$</p>
<p>The last term appears to be the delta function, as it is zero everywhere, but singular at zero. The first terms are unity everywhere, but undefined at zero, therefore,</p>
<p>$$T_{55}= \frac{1}{8\pi G_5}\left[ 4\theta^2(\sigma)+4\theta^2(-\sigma) -16\delta(\sigma) + \Lambda\right]$$</p>
<p>However, this disagrees with Mannheim's <em>Brane-Localized Gravity</em> which states,</p>
<p>$$T_{ab}=-\lambda \delta^\mu_a \delta^\nu_b \eta_{\mu\nu}\delta(\sigma)$$</p>
<p>where $\lambda = 12/\kappa^2_5$. In his text, $T_{\mu\nu}\propto -\eta_{\mu\nu}$, but in my calculation the entire purely 4D stress-energy vanishes. I can only assume I've done something wrong.</p>
| 2,483 |
<p>They say <a href="http://en.wikipedia.org/wiki/Inflation_%28cosmology%29" rel="nofollow">inflation</a> must have occured because the universe is very homogeneous. Otherwise, how could one part of the universe reach the same temperature as another when the distance between the parts is more than light could have traveled in the given time?</p>
<p>Why can't this problem be solved without inflation? If each part started with the same temperature to begin with, then they can have the same temperature irrespective of the distance between them. Am I missing something here?</p>
| 2,484 |
<p>I have an 80 mm <a href="http://en.wikipedia.org/wiki/Refracting_telescope" rel="nofollow">refractor</a> telescope on a tripod, but it shakes on every touch. It's very hard to see via 6 mm (x120) ocular. Even a little wind causes the image to become too unsteady. </p>
<p>How can I make my tripod more steady?</p>
| 2,485 |
<p>I'm not an expert but I've come to understand that the universe is expanding at enormous speed. That means that all of the visible galaxies are moving away from us at great speed. </p>
<p>I also came to understand that, eventually (in many many years), all the galaxies and all of the rest of the objects outside our own galaxy will move so far away from us that it will be impossible for us to look (or measure) them.</p>
<p>This means that eventually our entire observable universe will be our own galaxy and it will be impossible for us to measure anything else, and scientific measurements at that point in time will not correspond to the actual reality... because data will show that other galaxies don't exist.</p>
<p>But I have two questions regarding this phenomena:</p>
<ol>
<li><p>How is it that the light won't reach us anymore? Is the expansion happening at greater speed than light itself, making it impossible for it to ever reach anything?</p></li>
<li><p>If scientific measurements, at that point in time, will prove to be wrong, because they will show that galaxies don't exist (while they actually do exist), doesn't that mean that something similar could be happening right now as well? We could be measuring something about the universe that we're <em>dead</em> sure about, but it won't be the actual reality.</p></li>
</ol>
| 2,486 |
<p>If general relativity is the newest model of Gravity which is so far been proven. Does it still have any anomalies such as the <a href="http://physics.stackexchange.com/q/26408/2451">problem</a> of Mercury's orbit during Newtonian gravity period?</p>
<p>If so are there other types of physics to be discovered?</p>
| 2,487 |
<p>We can see that when a charge sphere is at rest & we are to put it in motion with any desired velocity than we will have to apply the same force for a longer time as if it were applied to the identical uncharged sphere to put it in motion with the same desired velocity.Here the due to electromagnetic fields associated with the charged sphere are responsible for this fact & by methods of integration we find that the additional inertial property gained by the body due to its charge can be given by a simple function of universal constants charge of the body and its radii.I have varified that in linear as well as planar or 3d motion we can apply normal mechanics by adding this factor of additional inertial property in its actual mass.so in one way we can say that the particle show more inertia without any external interference to it so its inertial mass is increased.And I have read that all experiments till today show that inertial mass and gravitational mass are completely identical.so due to charge on any body its gravitational mass is also changed affected or not ?</p>
| 2,488 |
<p>So it's Thanksgiving here in the states and an odd combination of things are on my mind. In the past day, I've</p>
<ul>
<li>Brined a turkey whole, skin on</li>
<li>Taken a long epsom salt bath</li>
</ul>
<p>(Same thing, right? What a turkey! Haha.)</p>
<p>Anyways, it got me wondering, can a long salt water bath increase water retention? It's basically the same process as <a href="http://en.wikipedia.org/wiki/Brining" rel="nofollow">brining meat</a>:</p>
<blockquote>
<p>Salt is added to cold water in a container, where the meat is soaked usually six to twelve hours [making] cooked meat moister by hydrating the cells of its muscle tissue before cooking, via the process of osmosis</p>
</blockquote>
<p>I'm sure we've all experienced dry, shriveled or even cracking skin (called <a href="http://en.wikipedia.org/wiki/Xeroderma" rel="nofollow">xeroderma</a> when chronic), which can <a href="http://www.medicinenet.com/dry_skin/page6.htm#what_are_some_home_remedies_for_dry_skin" rel="nofollow">apparently be improved by warm baths or showers</a>. And here is where I'm in danger of leaving the bounds of rigorous science: can you cause or worsen <a href="http://en.wikipedia.org/wiki/Water_retention_%28medicine%29" rel="nofollow">water retention</a> by soaking in a salt water bath? How much water can the skin take on? If you weighed yourself regularly and rigorously, would you be able to perceive a difference in weight before and after such a bath?</p>
<p>If it doesn't occur, what makes human skin different in this regard from poultry skin? Would brining not be effective if the body cavity of the bird weren't opened and vacated? (Brining and marinating of course works perfectly well on skinless cuts of meat.)</p>
| 2,489 |
<p><strong>Question:</strong></p>
<p>Will this work as I've shown it here, more or less, or will the iron cylinder being hollow produce results not shown? </p>
<p><strong>Topic:</strong></p>
<p>I'm experimenting with magnetic fields and have been testing electromagnets. For a specific device that I'm interested in engineering, I'll need to look at the effects of a hollow iron tube wrapped by a standard charged wire coil. The reason that it needs to be hollow is that other machinery will need to fit down the center of the device.</p>
<p>This is what I'm expecting from this:</p>
<p><strong>Model 1:</strong></p>
<p><img src="http://i.stack.imgur.com/Nobbo.png" alt="enter image description here"></p>
<p>In this model, the poles originate from the edges of the cylinder. </p>
<p>Another possibility that I thought might occur would be this model:</p>
<p>Model 2:</p>
<p><img src="http://i.stack.imgur.com/DGdo6.png" alt="enter image description here"></p>
<p>Here, the poles originate from the object's true center. </p>
<p><strong>Are either of these models accurate? If not, what's the answer?</strong> </p>
| 2,490 |
<p>The pressure cooker is heated at the base to cook the food. The pressure cooker releases steam when the force exerted by pressure exceeds the counterweight, at which point the weight rises up allowing steam to escape. The time to cook is measured by the number of times steam is released.</p>
<p>When food(dry lentils) is kept in a container inside the pressure cooker it does not cook as well as if it were dumped in there directly. The same amount of water is added to the food in both cases, though in the former case there is also some water in the base of the cooker outside the container. In both cases the cooker was allowed to release steam 3 times.</p>
<p>I thought that based on Pascal's law the same pressure would be exerted on the water in the container as it would if kept there directly. Since the temperature is the same, in both cases the food should be completely cooked. However practical results differ. Please suggest a theory which explains this.</p>
| 2,491 |
<p>Hope this isn't something trivially wrong, I am a beginner in classical mechanics. But anyways, say there is a cube, of side length $a \ \text{cm}$. The volume of the cube is $a^3 \ \text{cm}^3$. If the total mass of the cube is $m \text{ kg}$, the <a href="http://en.wikipedia.org/wiki/Density" rel="nofollow">density</a> of the cube will be $\frac{m}{a^3} \text{kg/cm}^3$. Again, consider the cube as the combination of many square plates of negligible thickness. If the mass of each is $m_2 \ \text{kg}$, their density will be $\frac{m_2}{a^2} \ \text{kg/cm}^2$. Here's the question: we found the density in two ways, and their dimensions don't match. I think I know why this happens- we are finding the density of the cube by integrating the densities of each plate. But how would you answer the following question?</p>
<p>The density of a cube is $X \text{ kg/cm}^3$. Find the density of each square plate.</p>
<p>Again, I am sorry if this is something trivial, I am a novice. :)</p>
| 2,492 |
<p>I had read somewhere that a star, whose light passes very close to the sun and reaches the earth produces 4 images of the same star (left, right, top and bottom) in a telescope due to gravitational lensing. However, this cannot be possible. Since light from the star passes on all sides of the sun (continuous 360 degrees) and bends to reach the earth, the star should be visible as a continuous ring around the sun. Is there any such star that appears as a continuous ring around the sun due to gravitational lensing? What is the reason for the 4 separate images and not the continuous ring?</p>
| 48 |
<p>As far as I understand a new pattern of crystal growth has been found experimentally. How does it relate to the known 2D and 3D nucleation and growth of crystals? The dominating theory of crystal growth nowadays is that initiated by Cabrera and Frank. How is the new finding related to what we know so far or is it at odds with what we know and how?</p>
| 2,493 |
<p>When judging if relativity is important in a given phenomenon, we might examine the number $v/c$, with $v$ a typical velocity of the object. If this number is near one, relativity is important. In optics, we examine $\lambda/d$, with $\lambda$ the wavelength of light involved, and $d$ a typical size-scale for the object interacting with light. If this number is near one, physical optics becomes important. In fluid mechanics we have the Reynolds number, etc.</p>
<p>Can we compute a number to tell roughly when a gas will deviate significantly from the ideal gas law?</p>
<p>For simplicity, you may want to deal only with a monatomic gas, but a more general approach would be welcome.</p>
| 2,494 |
<p>Suppose I do two experiments to find the triple point of water, one in zero-g and one on Earth. On Earth, water in the liquid or solid phase has less gravitational potential per unit mass than water in the gas phase. Therefore, the solid and liquid phases should be favored slightly more on Earth than in zero-g.</p>
<p>In a back-of-the-envelope calculation, how does the temperature of the triple-point of water depend on the gravitational acceleration and, if necessary, on the mass of water and volume and shape of container?</p>
<p><b>Edit</b>
Let's say I have a box in zero-g. The box is one meter on a side. It has nothing in it but water. Its temperature and pressure are just right so that it's at the triple point. All the water and ice and steam are floating around the box because it's zero-g.</p>
<p>Now I turn on gravity. The liquid water and ice fall to the bottom of the box, but the average height of the steam remains almost half a meter above the bottom of the box. So when gravity got turned on, the potential energy of the ice and liquid water went down significantly, but the potential energy of the steam didn't. Doesn't this mean that once gravity is turned on, water molecules would rather be part of the ice or liquid phase so that they can have lower energy? Wouldn't we no longer be at the triple point?</p>
<p>Several people have posted saying the answer is "no". I don't disbelieve that. Maybe the answer is just "no". I don't understand why the answer is no. Answers such as "No, because gravity doesn't affect the triple point," or "No, because the triple point only depends on pressure and temperature" simply restate the answer "no" with more words.</p>
| 2,495 |
<p>I have a horizontal bar on vertical rails and we can consider that does not exist any friction. </p>
<p>Perpendicular to the plane of the rails it exists a magnetic field $B$. </p>
<p>I am asked to find the movement equations and the bar position depending on time. </p>
<p>And I have made some considerations: </p>
<p>I should separate the phenomenon in two parts: first one until the forces are equal and one after. </p>
<p>I'm stuck on finding the equations before the forces get equals. This is what I have done so far: </p>
<p>$$mg-F_{m}=m·a$$ with $F_m$ the magnetic force, with the Ohm Law: $$\epsilon=\frac{I}{R}$$ and with: $$\epsilon=-\frac{d \phi }{dt}=-Bl\frac{dx}{dt}$$
and all together:</p>
<blockquote>
<p>$$mg-\frac{B^2l^2}{R}·\frac{dx}{dt}=m·\frac{dx^2}{dt^2} $$</p>
</blockquote>
<p>So, what now? I am in my first physics course so I am not supposed to be solving this by two order differential equations tools. </p>
<p>I know that I can find the $v_{limit}$ if I make the sum of forces equal to $0$, but that is not enough. I'm really stuck, maybe I should be making some assumptions, but I don't know what! </p>
<p>(I have tried my best with my English language knowledge)</p>
<p>Thank you in advance. </p>
| 2,496 |
<p>While reading about the <a href="http://en.wikipedia.org/wiki/Clausius-Clapeyron_equation" rel="nofollow">Clausius Clapeyron equation</a> from the Feynman lectures on Physics, I couldn't understand a few things from its derivation:</p>
<p>Although the argument was pretty clear, when the system consists both gas(vapour) and liquid why would it have constant pressure on heating and increasing volume?? </p>
<p>Second and more fundamental, how can we assign a single pressure value to this composite system in spite of it having two states of matter; wouldnt the pressure in the gas part (on top part of container) be different from that at the bottom or side in the liquid part? On top of this how can this be the vapour pressure (which I think is the pressure of the vapours on the surface of liquid)? </p>
| 2,497 |
<p>In <a href="http://arxiv.org/abs/hep-th/0702178" rel="nofollow">this 2007 paper by Alan Guth</a> discussing eternal cosmic inflation, he start's using a value $\phi$ on page 8. My understanding is that $\phi$ is the scalar field representing the dark energy of a "false vacuum" - but then why does $\phi$ appear as the dependent variable of the energy density function $V$?</p>
| 2,498 |
<p>I have some question about interpreting PDG particle cross section data sets' metadata fields. The data sets I'm having questions for are <a href="http://pdg.lbl.gov/2011/hadronic-xsections/hadron.html" rel="nofollow">http://pdg.lbl.gov/2011/hadronic-xsections/hadron.html</a></p>
<p>For example, the data set for p-bar p collision <a href="http://pdg.lbl.gov/2011/hadronic-xsections/pbarp_total.dat" rel="nofollow">http://pdg.lbl.gov/2011/hadronic-xsections/pbarp_total.dat</a></p>
<p>The data set has meta data fields as shown at the top of the .dat file. I have questions about the following metadata fields. Apologies if the questions are silly.</p>
<ul>
<li><p><code>BEAM_MASS</code></p>
<p>Is this the energy of the beam particle w.r.t. lab coordinates?</p></li>
<li><p><code>TARGET_MASS</code></p>
<p>Again, w.r.t. lab coordinates?</p></li>
<li><p><code>THRESHOLD</code></p>
<p>Not exactly sure what this is, but most files has 0 for this value, what's it?</p></li>
<li><p><code>FINAL_STATE_MULTIPLICITY</code></p>
<p>Is this multiplicity from phase space configuration? Could you given a reference or an example so I may check details of the derivation?</p></li>
<li><p><code>PLAB(GEV/C)</code></p>
<p>Momentum of the beam particle w.r.t. lab reference coordinate?</p></li>
<li><p><code>LAB_MIN</code>, <code>PLAB_MAX</code></p>
<p>Minimum and maximum of PLAB? So a data point in the plot will have an x coordinate with uncertainty? how should it be plotted then? It seems in the plot in the PDF the x-coordinate doesn't have "bars" but only the y-coordinate has it. How is the uncertainty of x taken into account?</p></li>
<li><p><code>SY_ER+(PCT)</code>, <code>SY_ER-(PCT)</code>:</p>
<p>What's this? is it visible in the plot at all?</p></li>
<li><p><code>REFERENCE FLAG</code></p>
<p>For example, "ELIOFF 62 PR 128, 869", but what is the last number here? My guess is .</p></li>
</ul>
| 2,499 |
<p>I'm trying to write N=4 SYM in terms of N=1 superfields. I have the lagrangian</p>
<p>$$\mathcal{L}=\frac{1}{16 k} \int d^2 \sigma \text{Tr} \big[W^a W_a\big]+c.c+\int d^4\theta \text{Tr}\big[\bar{\Phi}^i e^V \Phi^i e^{-V}\big]+\frac{\sqrt{2}}{3}\int d^2\theta \text{Tr}\big[\phi^i [\phi^j,\phi^k]\big]\epsilon_{ijk}+c.c $$</p>
<p>Where the $\Phi^i$ are chiral superfields and V is a vector superfield. In components, this is
all fine except for the Yukawa terms
$$\mathcal{L} \supset i\sqrt{2} f^{ABC} Z^{i\dagger}_A \psi^i_B \lambda_C - \sqrt{2}\epsilon_{ijk} Z^i_A \psi^j_B \psi^k_C+c.c $$</p>
<p>Where $A,B,C$, are $SU(N)$ gauge group indices, $i,j,k$ number my 3 chiral superfields, which have an explicit $SU(3)$ symmetry, the $Z^i_A$ are the complex scalars from my chiral superfields, $\psi^i_A$ are the fermions from my chiral superfields, and the $\lambda_A$ is the fermion from my vector superfield. </p>
<p>The fermions combine into an fundamental $SU(4)$ multiplet $\chi^I=(\psi^i, \lambda)$, and I decompose my complex scalars into real ones in a fundamental $SO(6)$ (isomorphic to $SU(4)$) multiplet, $Z^i=X^a+iX^{a+3}$. I should be able to write the Yukawa terms as</p>
<p>$$\mathcal{L} \supset f^{ABC} X^a_A C^a_{IJ}\chi^I \chi^J +c.c $$</p>
<p>essentially putting the scalars into the antisymmetric matrix representation of $SU(4)$, $X_{IJ}=X_{[IJ]}=X^a_{IJ}X^a$.</p>
<p>So I need to show that the $C^a_{IJ}$ that I have are an invariant symbol of $SO(6)=SU(4)$, and thus my Lagrangian has that R symmetry. Not sure how to do that... a reference I found says they should be related to the $SO(6)$ gamma matrices (<a href="http://arxiv.org/abs/hep-th/0201253">http://arxiv.org/abs/hep-th/0201253</a>, below equation 3.1), but that hasn't been very helpful.</p>
| 2,500 |
<p>Charged pions $\pi^\pm$ decay via an intermediate $W$ to (e.g.) a lepton-neutrino pair. </p>
<p>The pions being scalar (spin-0) particles and the intermediate $W$ having spin 1, how is spin conserved in this interaction?</p>
| 2,501 |
<p>Are there any books or articles that describe models for transport in a metal/semiconductor junction where the thickness of the semiconductor is less than the thickness of the depletion/accumulation layer?</p>
| 2,502 |
<p>Calculate $r$ vector and $v$ vector in AU and AU/TU respectively for the Halley's comet on September 12, 2014 given:
= 17.9 AU
= 0.967
= 2.8274 rad
Ω = 1.0210 rad
= 1.9513 rad
= 0 rad on February 9, 1986</p>
| 2,503 |
<p>In relation to my question <a href="http://biology.stackexchange.com/questions/10945/isotropicity-of-sarcomere-bands-in-skeletal-muscle-cells">here</a> I wanted to make sure that my physical argument was not flawed. </p>
<p>Anisotropic properties, (especially refractive index) is characteristic of a well-ordered solid (usually crystalline). On the other hand Isotropic properties is indicative of an amorphous solid, i.e lack of well-ordered arrangement (a random arrangement). The question is:- </p>
<blockquote>
<p><strong>Can any composite arrangement of isotropic elements give an
anisotropic system? Can any arrangement of anisotropic elements give an isotropic system?</strong> </p>
</blockquote>
<p>Intuitively, I think, it should not be possible, if isotropicity implies random arrangement (of individually anisotropic components) since any arrangement of random elements should not give a well-ordered system capable of exhibiting anisotropic refractive index. </p>
<p>As for the other part of my question, intuitively, it must be possible, since a random arrangement of anisotropic elements will impart an overall randomness and hence isotropic refractive index. (If I am not mistaken, this is what happens in short-range order amorphous solids, whre individually anisotropic grains are randomly arranged to produce an isotropic solid).</p>
| 2,504 |
<p>What is the transfer matrix size for a strip lattice of width $n$ vertices, with arbitrary $q$??<br>
I am not sure if it is $q^n$ x $q^n$ or something else.</p>
<p>Any reference is also welcome.</p>
| 2,505 |
<p>My textbook says talking about Cygnus X-1 (the first black hole to be identified as such) has variations in brightens of the order of 0.01 seconds and that this means that it's dimeter must be on the order of 3000km (=speed of light times 0.01). Why must this be so? The reason I asked is could (if it was not a black hole) it just not have vary active outer layers and be much larger. What the above seems to imply is that the light must be able to travel from one side of the source to the other in the time between differences in intensity. But I can not see why this should be the case. </p>
| 2,506 |
<p><strong>Context</strong>: Solid state physics. Monoatomic linear chain.</p>
<p><strong>Question</strong>: To prove that the total momentum of the chain is zero.</p>
<p><strong>Attempted solution</strong>: I consider the sum:</p>
<p>\begin{align*}
p = \sum_{n=1}^{N} m \dot{u_n}
\end{align*}</p>
<p>where $u_n$ is the displacement from the equilibrium position of the $n$-th atom. The displacement is given by the formula:</p>
<p>\begin{align*}
u_n = u_0 \exp\left[-i\left(\omega t \pm k n a\right)\right]
\end{align*}</p>
<p>where $k$ is the wavevector, $n$ is the $n$-th atom and $a$ is the distance between atoms. If I substitute the above formula to the first sum, I get a result of the form:</p>
<p>\begin{align*}
p \sim \sum_{n=1}^{N} \exp(i k n a)
\end{align*}</p>
<p>I wonder, how could I prove that this is always zero ? If I treat it as the sum of a geometric series with $\alpha_1=\exp(ika)$ and $\lambda=\exp(ika)$, I still get a result that isn't necessarily zero.</p>
<p>\begin{align*}
S_{1\to N} = \alpha_1 \frac{\lambda^N-1}{\lambda - 1} = e^{ika} \frac{e^{ikNa}-1}{e^{ika}-1}
\end{align*}</p>
<p>If I further require that the first and last atoms are fixed, then $\exp(ikNa) = \exp(ika)$ and $S_{1\to N} = \exp(ika)$. Then</p>
<p>\begin{align*}
p = -i\omega u_0 m \exp(-i\omega t) \exp(ika) = -i\omega m \underbrace{u_0 \exp\left[-i(\omega t - k a)\right]}_{u_1} = -i \omega m u_{1} = 0
\end{align*}</p>
<p>since we assumed that the 1st atom is fixed. Does this sound correct ?</p>
| 2,507 |
<p>In the solution of Schrödinger Equation for harmonic oscillator why the distance between peaks of the probability distribution function decreases when n increases?</p>
<p>Is there a good reason for it or is it just the consequence of solving the Schrödinger Equation?</p>
| 2,508 |
<p>Given complete knowledge of the precise state of every property of every particle and energy phenomenon existing in our universe for a given infinitely small frame of time, is it possible to completely determine the status of every property of every particle and energy phenomenon existing in the succeeding time frame?</p>
<p>This situation assumes that there are no restrictions towards the access of any information, particularly particle states, existing in the universe.</p>
<p>How can this situation be mathematically modeled?</p>
| 49 |
<p>If right in the centre of the Earth would be like a hole and I could teleport right there, what would happen to me? Would I explode, implode? My own theory is that maybe I would levitate because since I'm right in the middle, the gravity wouldn't affect me...</p>
<p>What do you guys think? </p>
| 50 |
<p>I am appreciating particle physics and I read about mesons. In quark's model, mesons are pairs of quark-antiquark. Now I think that in general matter-antimatter annihilate and so I don't understand how meson could be possible. Why don't a couple up-antiup annihilates? </p>
<p>Thank you!</p>
| 2,509 |
<p>Linear antenna directed along z, photons (EM waves) propagate along x. Momentum of photons have only x component. Why electrons in antenna have z component of momentum?</p>
| 2,510 |
<p>The rate at which time passes is relative depending on speed and the gravity as predicted in general relativity. This theory has been tested by scientists by comparing two identical atomic clocks, one on Earth the other on a rocket speeding at escape velocity. The initially synchronised clocks measured different amounts of time when the rocket returned. </p>
<p>Given the current scientific definition of time rate, </p>
<p>"The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom",</p>
<p>does this mean that 1) the above definition is only true on the surface of an object with Earths mass and 2) the fundamental properties of sub-atomic particles change on the speeding rocket so the rate at which electrons of the caesium 133 atom oscillate between the two energy levels is different?</p>
| 2,511 |
<p>I'm working on a research project involving absorption spectra of particulates in solution. I was curious if someone could clarify or direct me to a resource that explains broadening mechanisms <em>specifically</em> for absorption spectra. For example, the oft-cited Heisenberg and Doppler broadening effects both make sense to me in the context of emission spectra, but I don't see the mechanism by which they would be influential in absorption spectra. But I also can't find a source to verify this, since emission and absorption spectra are often lumped together in the familiar statement "the absorption spectrum is simply the inverse of the emission spectrum"</p>
<p>Thanks very much for any advice.</p>
| 2,512 |
<p>When a light beam reaches a dielectric surface, the incident and reflected beams have different intensities depending on polarization. For the so-called Brewster's angle, the reflected light is lineary polarized.</p>
<p>My question is: how does this law work in case of mirror-like surface, when (ideally) all the light is reflected?</p>
| 2,513 |
<p>The question reads as such: "What is the mass of a 1.05 µCi carbon-14 source?"</p>
<p>First I convert to decays/s: $R = 1.05 µCi=3.885 \times 10^4 decays/s$.</p>
<p>The half-life for carbon-14 that we've been using is 5730 years or $T_{1/2}=1.808 \times 10^{11} s$, so $\lambda=\frac{ln(2)}{T_{1/2}}=3.834 \times 10^{-12}s^{-1}$</p>
<p>$R=\lambda N$, so $N=\frac{R}{\lambda}=1.0133 \times 10^{16}nuclei$</p>
<p>Now just to convert to grams: $1.0133 \times 10^{16}nuclei \times \frac{1mol}{N_A} \times \frac{12.011g}{1mol}=2.02 \times 10^{-7}g$. Which unfortunately is the wrong answer. Maybe someone sees a conversion error I made somewhere?</p>
| 2,514 |
<p>The magnetic monopoles does not exist which can be shown by $ \int {\vec{B} \cdot d\vec{A}} = 0 $.</p>
<p>But in Faraday's Law of electromagnetic induction, we clearly show the EMF induced is the <em>time rate of change of the magnetic flux</em>, which is $ E = -\frac{d\Phi_B}{dt} = -\frac{d\int{\vec{B}\cdot d\vec{A}}} {dt}$.</p>
<p>Now if $ \int {\vec{B} \cdot d\vec{A}} = 0 $ then shouldn't the induced emf be zero?</p>
| 2,515 |
<p>The energy of a single planet in a gravitational potential is $$E=\frac{m\dot{r}^2}{2}+\frac{L^2}{2mr^2}-\frac{GMm}{r}$$
And the effective potential energy is defined as the last two terms. Note this satisfies $$F=-\nabla U$$
$$mr^2 \dot{\theta}-\frac{MmG}{r^2}=-\frac{d}{dr}(\frac{L^2}{2mr^2}-\frac{GMm}{r})$$
$$=\frac{L^2}{mr^3}-\frac{GMm}{r^2}$$</p>
<p>Similarly, in quantum mechanics I have encountered the energy of an electron in an electric potential as being
$$E=\frac{\hbar^2}{2mr^2}-\frac{Zq_e^2}{4 \pi \epsilon_0r}$$</p>
<p>With the first term arising from energy due to confinement in a small volume (uncertainty principle).</p>
<p>I'm aware that this question is probably site-pollutingly basic, but:</p>
<ul>
<li>I assume, by analogy, $\hbar$ is the electron's angular momentum (or integer multiples of $\hbar$: it's probably quantised). Does this form for $L_e$hold firm in post-1920/30's quantum mechanics, or is this result as spurious an idea as the Bohr atom model?</li>
<li>If I'm not incorrect, why aren't I? The energy-angular momentum equation is wholly classical, surely electrons behave very differently from this? </li>
<li>To what extent does the electron energy equation hold true, or is it again spurious?</li>
<li>Are there any other analogies exist between the equations that I've missed?</li>
</ul>
| 2,516 |
<p>I have the following question to answer:
a force of160 N stretches a spring 0.050m from its equilibrium position.
A. what is the spring constant of the spring?</p>
<p>The equation for Hooke's Law in my text is F=-kx, where k is the spring constant. So with some simple algebra we can find an equation for k: k = -(f/x). After plugging all my values in I get -3200. </p>
<p>My question: Am I supposed to disregard the negative sign when plugging in value? </p>
| 2,517 |
<p>A proton p collides with a neutron(at rest) n at relatively low-energies and creates a 'deuterium-core' d:</p>
<p>$$p+n->d+\gamma$$</p>
<p>Find the wavelength for the photon as a function of the proton's momentum and the angle that the photon creates with the proton.Do this relativistic.</p>
<hr>
<p>So seems I'm supposed to set up two equations: energy- and momentum-balance.
I tried to set up the relativistic energy equation as function of momentum, but became messy to solve.
Any hint on what this energy equation could look like?</p>
| 2,518 |
<p>In Misner, Thorne, Wheeler (henceforth written as "MTW"), "Gravitation", Box 16.4, there's an experimental setup construction (or method) presented by which</p>
<blockquote>
<p>"<i>Each geodesic clock is constructed and calibrated as follows: [...]</i>", where "<i>Those that pass through [event] $\mathcal A$ are calibrated [...] against the [specific timelike] standard interval $\mathcal A B$</i>" and "<i>Any interval $\mathcal P Q$ along the world line of a geodesic clock can be measured by the same method as was used in calibration</i>".</p>
</blockquote>
<p>Regarding importance and motivation of this method it is stated that</p>
<blockquote>
<p>"<i>The Einstein principle that spacetime is described by Riemannian geometry exposes itself to destruction by a "thousand" tests. Thus, from the fiducial interval, $\mathcal A B$, to the interval under measurement, $\mathcal P Q$, there are a "score" of routes of intercomparison, all of which must give the same value for the ratio $\mathcal P Q / A B$.</i>". </p>
</blockquote>
<p>Box 16.4 contains two separate sketches of the setup construction; one concerned with interval $\mathcal A B$, the other equivalently with interval $\mathcal P Q$. Explicitly evident in either sketch are two separate participants ("<i>particles</i>", "<i>timelike world lines</i>") who </p>
<ul>
<li><p>observe pings between each other, and</p></li>
<li><p>both observe one ping to one particular event (event $\mathcal{B}$ in the first setup sketch; or event $\mathcal{Q}$ in the second setup sketch), where</p></li>
<li><p>each participant finds some (integer) number of successive pings to the other as same as the one ping to the "external" event ($\mathcal{B}$, or $\mathcal{Q}$). (The "<i>intercomparison</i>" to be achieved then involves counting those successive pings by the two participants.)</p></li>
</ul>
<p>There are additional "<i>demands</i>" stated (which the two sketches alone don't make explicit); among them that the participants who took part in event $\mathcal A$ (referring to the first sketch), or in event $\mathcal Q$ (referring to the second sketch) are required to have been "<i>freely falling</i>"; and their world lines correspondingly "<i>paths of freely falling particles</i>" and "<i>geodesics</i>".</p>
<p><strong>Question 1</strong>:<br>
Has there at least one such test been carried out already, presumably by "<i>intercomparison</i>" along at least two distinct "<i>routes</i>", by explicitly using the method indicated by MTW ? (i.e. not involving any further "model dependent assumptions" such as in the arguments presented at the end of box 16.4)</p>
<p><strong>Question 2</strong>:<br>
Instead of demanding of the setup(s) that certain participants were "<i>freely falling</i>" and obtaining "<i>a "thousand" tests of the Einstein principle that spacetime is described by Riemannian geometry</i>",<br>
could the described method be explicitly adapted in turn, by assuming "<i>the Einstein principle</i>" as outright valid and by suitably combining sufficiently many (if not "a thousand") of the described setups, to obtain a test of whether (or to which accuracy) a given particle had been "<i>freely falling</i>" (or correspondingly, whether, or to which accuracy, a given timelike world line was "<i>geodesic</i>") ?</p>
| 2,519 |
<p>I'm well aware of transmutation as a way to effectively make radioactive material decay faster, however that isn't really what I mean.</p>
<p>Doing a quick Google search I found references to several theoretical treatises on the subject, and a few references to experiments with, at best, inconclusive results. Among them, there were a few on the quantum anti-zeno effect, which I already knew about, as well as a couple on something called "field enhanced beta decay" and various methods utilizing lasers. Nevertheless, much to my frustration, there was essentially no example I could find with any sort of empirical testing of these effects. So, I suppose I have two questions: </p>
<ol>
<li>How "good" is the physics behind these claims?</li>
<li>Have any of these effects ever been <em>unequivocally</em> observed experimentally?</li>
</ol>
| 2,520 |
<p>I have read that quantum mechanics says that the amount of possible particle configurations is $10^{10^{122}}$ to be exact in the universe. Do we know this figure to be exactly true to the exact figure? Wouldn't we need to know a true theory of quantum gravity to know the exact answer? Is the amount exactly that figure or just an estimate?</p>
| 2,521 |
<p>My dad and I have tried to calculate the strength of the explosion if the lid was suddenly freed. We took some measures:</p>
<ul>
<li>Lid mass: $0.7 \textrm{kg}$</li>
<li>Lid surface: $0.415 \textrm{m}^2$</li>
<li>Internal pressure (above external): $ 1\textrm{atm}\approx 100 \textrm{kPa} $</li>
</ul>
<p>Now we are kind of struggling about what to do with these. We'd like to know how to calculate the initial speed, for instance, or any other value that could get us started.</p>
<p>I hope I was clear enough. :)</p>
| 2,522 |
<p>I'm trying to find the electric field due to an electric dipole $\mathbf{d}$. There are plenty of approaches to doing this online, but I want to do it "my way," which doesn't seem to be working (and I have yet to find this approach in textbooks/online). I start with the potential:</p>
<p>$$ \phi(\mathbf{r}) = \frac{\mathbf{d}\cdot \mathbf{r}}{4\pi\epsilon_0 r^3} = \frac{d\cos\theta}{4\pi\epsilon_0 r^2} $$</p>
<p>And then take the negative gradient to find:</p>
<p>$$ \mathbf{E} = -\nabla \phi = \frac{2d\cos\theta}{4\pi \epsilon_0 r^3} \hat{r} + \frac{d\sin\theta}{4\pi\epsilon_0 r^3}\hat{\theta} $$</p>
<p>But I don't see how to manipulate this into the form that I expect:</p>
<p>$$ \mathbf{E} = \frac{1}{4\pi\epsilon_0}\left[ \frac{3(\mathbf{d}\cdot\hat{r})\hat{r}-\mathbf{d}}{r^3} \right] - \frac{1}{3\epsilon_0}\mathbf{d} \delta^3(\mathbf{r}) $$</p>
<p>Since I haven't done anything mathematically illegal, I don't see why my approach shouldn't get me to the correct answer - but I can't figure out what to do (and I've also been unable to find this derivation online).</p>
| 2,523 |
<p>So, let's say we have a spacecraft in deep space. It activates its rocket engines, to keep it simple. The engine reaction channels exhaust through the aft nozzle, right? How does that chemical reaction actually induce a change in the velocity of the spacecraft? The particles <em>have</em> to push against the geometry of the spacecraft to provide a force. Do they do that and how when their direction vector clearly goes away from the spacecraft?</p>
<p>And here's something else. A pressurized spacecraft opens up an airlock which decompresses the entire deck, would that create a small thrust for the spacecraft and how? Is it the contact hull - air as it is accelerating outwards?</p>
| 2,524 |
<p>I accept the Big Bang theory. What I can't understand is how there can be a where or when to the Big Bang if space time did not exist prior to it. Did space and time exist prior to the Big Bang? </p>
| 51 |
<p>How strong would have to be nuclear explosion on exo-planet that orbits some other star for it to be detectable outside of that system.<br>
Or it would be impossible due to amount <a href="http://www.neatorama.com/2008/03/22/trivia-solar-energy-1-trillion-1-megaton-atom-bombs-per-second/#!8m5ue" rel="nofollow">of radiation coming from that star</a>?<br>
Could the angle at which radiation would go out of that system be indicator, or light interference would make it impossible? </p>
<p><strong>Update:</strong>
Atmospheric nuclear explosions produce a unique signature, often called a "double-humped curve": a short and intense flash lasting around 1 millisecond, followed by a second much more prolonged and less intense emission of light taking a fraction of a second to several seconds to build up. The effect occurs because the surface of the early fireball is quickly overtaken by the expanding atmospheric shock wave composed of ionised gas. Although it emits a considerable amount of light itself it is opaque and prevents the far brighter fireball from shining through. As the shock wave expands, it cools down becoming more transparent allowing the much hotter and brighter fireball to become visible again.</p>
<p><a href="http://en.wikipedia.org/wiki/Vela_%28satellite%29" rel="nofollow">No single natural phenomenon is known to produce this signature.</a> </p>
<p>Could this be used to identify or "double-humped curve" would be of no help in space due to radio pollution, of many double humped curves produced by stars?</p>
| 2,525 |
<p>I am presently in my senior year and I am considering fluid mechanics for my thesis.
What area of research of fluid mechanics which is purely analytical and very mathematical since I am an applied mathematician can I look into.
NOTE 1: I am from a 3rd world country and the academic standard is low.
NOTE 2: Please provide references that I can consider.</p>
| 2,526 |
<p>Here is a drawing of the circuit that is confusing me:
<img src="http://i.stack.imgur.com/Kag3r.png" alt="enter image description here"></p>
<p>I don't quite understand how batteries work in this diagram. If a battery has a negative and positive terminal, there must be a barrier preventing them from neutralizing one another, so how can the potential from either negative terminal ever make it through the top half of the circuit without passing through a battery?</p>
| 2,527 |
<p>This is about a step in a derivation of the expression for the relativistic Doppler effect.</p>
<p>Consider a source receding from an observer at a velocity $v$ along the line joining the two. Light is emitted at frequency $f_s$ and wavelength $\lambda_s$. The frequency $f_0$ and wavelength $\lambda_0$ received by the observer will be different.</p>
<p>Some textbooks now argue that the wavelength received by the observer is given by $\lambda_0=(c+u)T_0$, where $T_0$ is the time period of the wave in the observers frame. The argument given is that the successive "crests" will be an extra distance $uT_0$ apart due to movement of the source. Upon relativistically transforming the time period in the source frame to the observer frame, the correct result $f_0=f_s\sqrt{\frac{c-u}{c+u}}$ is obtained.</p>
<p>But the argument for the wavelength seems reminiscent of the classical Doppler effect for sound. Is it really applicable here through this argument? And is there a way to show the same result for $\lambda_0$ mathematically?</p>
| 2,528 |
<p>I am reading Wald for the interior solutions of a static spherical metric. Assume it to be of the form
$$ds^2 = -f(r)dt^2 + h(r)dr^2 + r^2 ( d{\theta^2} \sin^2{\theta}d{\phi^2})$$</p>
<p>Wald states: For a perfect fluid tensor $T_{ab}= \rho u_a u_b + P ( g_{ab}+ u_{ab})$</p>
<blockquote>
<p>In order to be compatible with the static symmetry of space time, the
four velocity of the fluid should point in the direction of the static
killing vector $\xi^a$</p>
</blockquote>
<p>i.e. $u^a=-(e_0)^a=-f^{\frac{1}{2}}(dt)^a$</p>
<p><strong>EDIT: It also seems $(e_0)_a=f^{\frac{1}{2}}(dt)_a=f^{-\frac{1}{2}}(\frac{\partial}{\partial t})_a$. Please could someone tell, why this is so?</strong></p>
<ol>
<li><p>First, why is the the static killing vector $\frac{\partial}{\partial t}$ equal to $-f^{\frac{1}{2}} dt$?</p></li>
<li><p>Second, why is the velocity, along the killing time vector? What would happen if there is a component perpendicular to it? Does this mean, the fluid doesn't move through space?</p></li>
</ol>
| 2,529 |
<p>I'm reading the QFT textbook by Weinberg. In volume one chapter 10 page 451, at the lower part of the page he says,</p>
<blockquote>
<p>Now, because $\Pi^*_{\mu\nu}(q)$ receives contributions only from one-photon-irreducible graphs, it is expected not to have any pole at $q^2=0$.</p>
</blockquote>
<p>$\Pi^*_{\mu\nu}(q)$ is the sum of all one-photon-irreducible graphs, with the two external photon propagators omitted.</p>
<p>Weinberg states it within one sentence as if it's self-explanatory, but I cannot understand why it is true. Is there something simple I missed? </p>
<p><strong>Update:</strong> I think what Weinberg had in mind was Luboš Motl's answer, that why he's so brief. In addition Peskin & Schroeder used the same reasoning in page 245:</p>
<blockquote>
<p>...the only obvious source of such a pole would be a single-massless-particle intermediate state, which cannot occur in any 1PI diagram</p>
</blockquote>
<p>However P&S also put a footnote immediately after:</p>
<blockquote>
<p>One can prove that there is no such pole, but the proof is nontrivial. Schwinger has shown that, in two spacetime dimensions, the singularity in $\Pi$ due to a pair of massless fermion is a pole rather than a cut; this is a famous counterexample to our argument. There is no such problem in four dimensions.</p>
</blockquote>
<p>Thus my original question stands justified. I'd be grateful if one can give a reference that elaborates P&S's footnote. Of course explanations by any SE user himself/herself are even more welcomed. </p>
| 2,530 |
<p>A cylinder of mass M and radius R is in static equilibrium as shown in the diagram. The cylinder
rests on an inclined plane making an angle with the horizontal and is held by a horizontal string
attached to the top of the cylinder and to the inclined plane. There is friction between the cylinder
and the plane. What is the tension in the string T?</p>
<p><img src="http://i.stack.imgur.com/aHsY6.png" alt="Sketch of the problem"></p>
<p>Ans:
$$T =Mg\sin(\theta)/(1 + \cos(\theta))$$</p>
<p>I'm having trouble arriving at this solution.
I first look at the forces.
I set my $x$-axis such that the Force of friction is parallel with the x-axis.</p>
<p>So my forces are:</p>
<p>$F_T$ = Force of Tension<br>
$F_f$ = Force of friction<br>
$F_N$ = Normal Force<br>
$F_g$ = Force of Gravity</p>
<p>Since the cylinder is not moving. The forces and torques equal zero and balance.</p>
<p>Forces in the X direction: </p>
<p>$$0 = F_f + F_g\cos(\theta) - F_T*\cos(\theta)$$</p>
<p>Forces in the $y$ direction:</p>
<p>$$0 = F_N - F_g\sin(\theta) + F_T\sin(\theta) $$
$$F_T\sin(\theta) = F_g\sin(\theta) - F_N$$</p>
<p>$$F_T = (F_g\sin(\theta) - F_N)/\sin(\theta)$$</p>
<p>I'm not sure if I'm setting up my forces correctly. Can someone help? </p>
| 2,531 |
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