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<p>Consider this figure
<img src="http://i.stack.imgur.com/epLm6.jpg" alt="A charge $e$ moves with certain velocity $\mathbf v_e$"></p>
<p>Now, when I measure a field produced by the charge $e$ at the point $\mathbf r$, at the time $t=t_1$, it means that the charge sent the signal field at the time $t=t_r$, where $t_1$ and $t_r$ are related by
$$t_{r}=t_1-\frac{||\mathbf{r}-\mathbf{r}_{e}(t)|{}_{t=t_{r}}||}{c} $$
Now, my question is, how is it possible that we can take $t_r$ like a time variable? I mean, when we want to measure the velocity of the charge $\mathbf v_e$, I must derive $\mathbf r_e$ respect to $t_r$:
$$\mathbf{v}_{e}=\frac{d\mathbf{r}_{e}}{dt_{r}}$$
but why? I mean, why not to derive respect to $t$? So, what is the physical meaning of $t_r$? Or, in other words, how can we interpret the time $t_r$? Are there actually two time-axes?</p> | g10838 | [
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<p>I am trying to find a book on electromagnetism for mathematician (so it has to be rigorous).
Preferably a book that extensively uses Stoke's theorem for Maxwell's equations
(unlike other books that on point source charge, they take Stoke's theorem on
$B-\{0\}$ with $B$ being closed ball of radius 1, but this does not work, as Stoke's theorem only works for things in compact support)
Preferably if it mentiones dirac delta function, hopefully it explains it as a distribution (or a measure...)</p>
<p>P.S. This question is posted because there is no questions about electromagnetism books for mathematician. I have background in mathematics as <a href="http://www.math.washington.edu/~lee/" rel="nofollow">John Lee</a> <em>Smooth Manifolds.</em></p> | g10839 | [
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<p>The traditional physics career is an academic job at some university, with the eventual goal of becoming a tenured professor. Is it possible for a mostly self-educated outsider working outside academia to come up with significant results in physics? Let's take significant to mean accepted in a high ranking peer review journal with a high citation count. Let's just say due to external life circumstances, the academic path is unfeasible.</p>
<p>Are there any examples of notable physics results coming from outsiders? e.g. a third class patent clerk coming up with light quanta, Brownian motion and relativity.</p> | g10840 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/44967/mass-points-of-a-mass-spring-model">Mass points of a Mass-spring model</a> </p>
</blockquote>
<p>Say I have a spring like the one in the picture below:</p>
<p><img src="http://i.stack.imgur.com/8kdlv.png" alt="enter image description here"></p>
<p>The point at the top is fixed to a ceiling.</p>
<p>The red coloured arrow is the direction in which I pull the spring to, and its vector is $(3, 6)$. Let's say the triangluar spring is equilateral and gravitational force is considered to be as $mg$.</p>
<p>Now, I want to find its restorative force in the <em>y-axis direction</em>. Based on Hooke's Law,</p>
<p><img src="http://i.stack.imgur.com/sn3Nz.png" alt="enter image description here"></p>
<p>where $k$ is stiffness.</p>
<p>But I am not sure if I did this correctly. </p>
<p>Do I have to include the force of $-6k$, which is the change in length from the pulled direction vector in the y-axis?</p> | g350 | [
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<p>$\alpha$ radiation consist of positive charged helium nuclei, $\beta$ radiation of negative charged electrons. So why don't the $\alpha$ particles take those electrons to get neutral?</p> | g10841 | [
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<p>I am spending some time reading about Bose-Einstein condensation. I want to know if it is possible to use atom lasers to realize the kind of holography traditionally associated with nano-fabrication.</p>
<p>Many papers say that the motivation for BEC is the aforementioned holography, but I can't find if they were able to actually realize it. I am very much a novice at searching through academic literature.</p>
<p>The closest I have found is the work of some Japanese scientists in the 90s who were able to make 2D hologram, without a laser style beam.</p>
<hr>
<p>Does anybody know if it was done, or possible can comment on some of the challenges.</p> | g10842 | [
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<p>How would you explain string theory to non physicists such as myself? I'm specially interested on how plausible is it and what is needed to successfully prove it?</p> | g10843 | [
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<p>I've been looking for a textbook in classical mechanics that's readily available (like can be found in the library of James Cook University of Townsville, Australia) and full of fully-answered questions in the Lagrangian/Hamiltonian formalisms yet I can't find any. My first port of call was the Schaum's Outlines and Demystified series but the only member of these series I could find that was relevant was <a href="http://rads.stackoverflow.com/amzn/click/0070692580" rel="nofollow">Lagrangian Dynamics</a> which is difficult for me to track-down in real-life. </p>
<p>It would be particularly helpful if one could point me to a free eBook with this material. </p> | g12 | [
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<p>I'm trying to understand how to correctly think about a change of reference frames for a conducting rod moving sideways through a uniform magnetic field.</p>
<p>Thinking of the rod in motion through a uniform B field is easy. The Lorentz force explains the charge separation in the rod and I'm fine with that.</p>
<p>But if you change your point of view to where the rod is stationary, I immediately have an issue with the idea of the magnetic field "moving." Sure, if you picture magnetic field lines drifting one way or the other there's an appearance of movement, but isn't that a wrong way to look at it? Field lines aren't real. It seems to me that 'moving' a uniform B field changes nothing at all--it just stays the same as it always was. I'm tempted to say that a constant B field just cannot be thought of as moving at all. The field is the same as if it was not moving.</p>
<p>Yet clearly there has to be the same charge separation in that rod whether we think of it being in motion or not. Relative motion of the rod and field has to lead to the same effect no matter how you look at it. So where am I going wrong?</p>
<p>I have a vague recollection of something in Griffiths' Electrodynamics book that may have talked about this and a problem with the idea of a uniform B field throughout space violating necessary boundary conditions or something like that. Is that where the trouble is?</p>
<p>Is there a simple way out of my dilemma?</p>
<p><i>Added:</i> I should have pointed out that my issue was motivated by reading an introductory section from a textbook on Faraday's Law and induced E fields. There the discussion is all about "flux change" as usual. But this argument doesn't work for the rod. Is there some other intuitive way to see the origin of the E field?</p>
<p><i>More added:</i> I came across this <a href="http://en.wikipedia.org/wiki/Faraday_paradox" rel="nofollow">Faraday Paradox</a> today and perhaps it addresses some of my uneasiness. I still would like an intuitive way to see how it all works out though.</p> | g10844 | [
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<p>I read in <a href="http://rads.stackoverflow.com/amzn/click/0030839939" rel="nofollow">Ashcroft & Mermin's Solid State text</a> that for the <a href="http://en.wikipedia.org/wiki/Thomas%E2%80%93Fermi_model" rel="nofollow">Thomas-Fermi approximation</a> to be applicable, the external potential needs to be "slowly varying," What does it mean for a function (in this case potential) to be slowly varying? </p>
<p>And also I would like to ask what is the difference between <a href="http://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method" rel="nofollow">Hartree-Fock method</a> and the method of Thomas-Fermi?</p> | g10845 | [
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<p>Firstly I would like to know if this is valid for every kind of wave, or are there any conditions/exceptions where this is not valid.</p>
<p>But the main question is, is it possible to prove this fact for a general wave intuitively, rather than taking specific waves and deriving equations and showing it?</p> | g351 | [
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<p>In the textbooks that I read (namely Ashcroft/Mermin , Marder, etc.) it seems that a distinction is made between the correlations in electron gas and a Couloumb interaction between the electrons. What is exactly meant by the concept of correlations? How is that connected to the interactions in electron gas, and how does the screening enters the picture?</p> | g10846 | [
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<p>Consider a body released from a height $h$ and assume a drag force is linearly proportional to the velocity. Then by Newton's Second Law, $$m\mathbf{\dot{v}} = \mathbf{F_g} + \mathbf{F_{drag}} = m\mathbf{g} -\mu \mathbf{v} $$ Take the positive $x$ axis upwards, then we can write $$m \mathbf{\dot{v}} = -mg \hat{x} - \mu v (-\hat{x})\,\,\,\,\,\,(1)$$ where $\mathbf{v} = v(-\hat{x})$ The decomposition is therefore $$m\frac{dv}{dt} = -mg + \mu v,$$ which when integrated yields an exponentially growing velocity, and hence there is a problem with the above.</p>
<p>I do not see where the problem is. The starting point is that $\mathbf{F_{drag}} = -\mu \mathbf{v}$ and since $\mathbf{v} = v (-\hat{x})$ in the coordinate system, we have the above. In $(1)$, it makes sense that the drag force and gravitational force are in opposite directions when the body falls down.</p>
<p>I looked in some books (in particular D. Morin P.63) and he has a sign reversal in the $\mu v$ term (i.e both terms on the right hand side of the equation are negative) However, in this link, they have alternating signs (see Example 1,$\,$ $\approx$ half way down - just reversed coordinate system) <a href="http://www.math24.net/newtons-second-law-of-motion.html" rel="nofollow">http://www.math24.net/newtons-second-law-of-motion.html</a></p>
<p>Many thanks for any input.</p> | g10847 | [
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<p>Does <a href="http://en.wikipedia.org/wiki/String_theory">string theory</a> have a notion of vacuum? If yes, what is known about it?</p> | g10848 | [
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<p>Consider the space-time domain Klein-Gordon propagator:</p>
<p>$$G_F(x)=\int\frac{d^4p}{(2\pi)^4}e^{ipx}\frac{1}{p^2-m^2+i\epsilon}$$</p>
<p>I understand this as the amplitude at location $x$ created by a source located at spacetime event $(0,0)$. I also see it as plane waves propagating with momentum $p$ weighted by $\frac{1}{p^2-m^2}$ : the density if sharply peaked at on-shell particles by the pole in the denominator.</p>
<p>Now consider the same propagator after integration on $p^0$ :
$$G_F(x)=\frac{-i}{2}\int\frac{d^3p}{(2\pi)^3}e^{i\bf{p.x}}\frac{e^{-iEt}}{E}$$</p>
<p>What is the physical interpretation of this? What happened to the off-shell modes?</p> | g10849 | [
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<p><strong>Review and hystorical background:</strong></p>
<p><a href="http://en.wikipedia.org/wiki/Gravitomagnetism" rel="nofollow">Gravitomagnetism</a> (GM), refers to a set of formal analogies between <strong><em>Maxwell's field equations</em></strong> and an approximation, valid under certain conditions, to the <strong><em>Einstein field equations</em></strong> for general relativity. The most common version of GM is valid only far from isolated sources, and for slowly moving test particles. The GM equations coincide with equations which were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law:</p>
<p>$\nabla \cdot \vec G = -4\pi\gamma\rho$</p>
<p>$\nabla \cdot \vec \Omega = 0$</p>
<p>$\nabla \times \vec G = - \dfrac{\partial \vec \Omega}{\partial t}$</p>
<p>$\nabla \times \vec \Omega = -\dfrac{4\pi\gamma}{c^2} \vec J + \dfrac{1}{c^2} \dfrac{\partial \vec G}{\partial t}$</p>
<p>$\vec G$ is gravitational field strength or gravitational acceleration, also called gravielectric for the sake of analogy; $\vec\Omega$ is intensity of torsion field or simply torsion, also called gravitomagnetic field; $\vec J $ is mass current density; $\gamma$ is gravitational constant.</p>
<p><strong>Magnetic monopole and Maxwell's field equations:</strong></p>
<p>It is known that the <strong><em>Maxwell's field equations</em></strong> have some asymmetry, in the absence of a magnetic monopole, although formally we can say that the problem can be solved theoretically (PAM Dirac and other works).</p>
<p>$\nabla \cdot \vec E = \dfrac{1}{\epsilon_0}\rho_e$</p>
<p>$\nabla \cdot \vec B = \mu_0 c \cdot g_m$, $g_m$ - magnetic monopole charge dencity.</p>
<p>$\nabla \times \vec E = \mu_0 J_{mag} - \dfrac{\partial \vec B}{\partial t}$, $J_{mag}$ - magnetic charge current </p>
<p>$\nabla \times \vec B = -\dfrac{1}{c^2 \epsilon_0} \vec J_{el} + \dfrac{1}{c^2} \dfrac{\partial \vec E}{\partial t}$, $J_{el}$ - electric charge current </p>
<p><strong>General relativity and gravitomagnetic monopole:</strong></p>
<p>Formally, a massive body in the linearized general relativity, is the gravielectric charge.</p>
<p>Now there is another interesting issue associated with the hypothesis of the existence gravimagnetic charge.</p>
<p>If we suppose its existence, what <em>changes</em> should be made <em>to the equations of general relativity</em> $G_{ik}= \kappa T_{ik}$ ($G_{ik}$ - Einstein tensor, $T_{ik}$ stress-energy tensor)? And what are the properties of such a charge?</p> | g10850 | [
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<p>I realise the question of why this sky is blue is considered reasonably often here, one way or another. You can take that knowledge as given. What I'm wondering is, given that the spectrum of Rayleigh scattering goes like $\omega^4$, <em>why is the sky not purple, rather than blue</em>? </p>
<p>I think this is a reasonable question because <a href="http://physics.stackexchange.com/questions/20114/purple-doesnt-occur-in-rainbow-or-does-it">we do see purple</a> (or, strictly, violet or indigo) in rainbows, so why not across the whole sky if that's the strongest part of the spectrum?</p>
<p>There are two possible lines of argument I've <a href="http://www.physicsforums.com/showthread.php?t=318518">seen elsewhere</a> and I'm not sure which (if not both) is correct. Firstly, the Sun's thermal emission peaks in the visible range, so we do actually receive less purple than blue. Secondly, the receptor's in our eye are balanced so that we are most sensitive to (roughly) <a href="http://physics.stackexchange.com/questions/3145/sensitivity-of-eye">the middle of the visible spectrum</a>. Our eyes are simply less sensitive to the purple light than to the blue.</p> | g601 | [
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<p>I have had a question since childhood. Why do we always get circular waves (ripples) in water even when we throw irregularly shaped object in it? </p> | g10851 | [
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<p>I am thinking of a (greatly simplified) computer simulation of a universe that followed something like Newtonian rules. Inside the simulation are A.I.s that are made from those same rules, and can only use those rules interact the world around them. Would there be some fundamental limits on what those A.I.s could work out about their universe, like their own version of an uncertainty principle?</p>
<p>Sorry for phrasing this question in such a convoluted way. If anyone recognises what I am asking, and can point me in the right direction that would be appreciated.</p> | g10852 | [
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<p><strong>Context:</strong> There have been a few papers out recently which mention how photosynthesis in plants might have connections to entanglement, or even perhaps that entanglement is causing photosynthetic complexes to capture the sunlight and go through the conversion stages. Here are two papers worth mentioning: </p>
<p><a href="http://mukamel.ps.uci.edu/publications/pdfs/676.pdf" rel="nofollow">Quantum oscillatory exciton migration in photosynthetic reaction centers.</a></p>
<p><a href="http://arxiv.org/abs/1006.4053" rel="nofollow">Quantum entanglement between the electron clouds of nucleic acids in DNA</a></p>
<p><strong>Question:</strong> In QFT, the Reeh–Schlieder theorem is thought to be an analogue of some sense to quantum entanglement. My question is that can one use the Reeh–Schlieder theorem instead of entanglement and try to do the work the other papers mentioned above have done but in a QFT context? To what extent does that analogue relationship hold? In that sense it is usable as a dual description under certain constrains? </p> | g10853 | [
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<p>This was explained to me many years ago, by a physics teacher, with the following analogy:</p>
<p>"If someone on the beach wants to reach someone else that is in the water, they will try to travel as much as they can on the beach and as little as possible on the water, because this way they will get there faster."</p>
<p>I'm paraphrasing of course, but this is as accurate as I recall it.<br>
This explanation makes no sense to me. Was he telling me the light knows where it is going? It wants to get there faster? It chooses a different direction?<br>
(No need to answer these questions, this was just me trying to understand the analogy.)</p>
<p>My attempts to clarify the issue were without success and many years later I still don't know. </p>
<p>Why does light change direction when it travels through glass?</p> | g10854 | [
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0.021309... |
<p>In odd number of space-time dimensions, the Fermions are not reducible (<em>i.e.</em> do not have left-chiral and right-chiral counterparts).</p>
<p>Does this mean that there is no such thing as 'chiral' anomalies in odd number of space-time dimensions, when these fermions are coupled to gauge fields?</p> | g10855 | [
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<p>A recent joke on the comedy panel show <em>8 out of 10 cats</em> prompted this question. I'm pretty sure the answer's no, but hopefully someone can surprise me.</p>
<p>If you put a person in a balloon, such that the balloon ascended to the upper levels of the atmosphere, is it theoretically possible that an orbiting satellite's (i.e. a moon's) gravity would become strong enough to start pulling you towards it, taking over as the lifting force from your buoyancy?</p>
<p>Clearly this wouldn't work on Earth, as there's no atmosphere between the Earth and the moon, but would it be possible to have a satellite share an atmosphere with its planet such that this would be a possibility, or would any shared atmosphere cause too much drag to allow for the existence of any satellite?</p>
<p>If it were possible, would it also be possible to take a balloon up to the satellite's surface, or would the moon's gravity ensure that its atmosphere was too dense near the surface for a landing to be possible thus leaving the balloonist suspended in equilibrium?
Could you jump up from the balloon towards the moon (i.e. jumping away from the balloon in order to loose the buoyancy it provided).</p>
<p><a href="http://www.channel4.com/programmes/8-out-of-10-cats/4od#3430968">http://www.channel4.com/programmes/8-out-of-10-cats/4od#3430968</a></p> | g10856 | [
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<p>In most nuclear fission examples and exercises, the products of a nuclear fission of Uranium-235 are two light nuclei of Krypton and Barium:</p>
<p>$$\mathrm{ _0^1n + U \longmapsto Kr + Ba + energy }$$</p>
<p>Is there some fission reaction that produces more stable nuclei instead of Krypton and Barium?</p>
<p>Please mention all possible fission equations of Uranium-235.
There's another question, is Uranium-235 the only fissile nucleus?</p> | g10857 | [
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<p>This happens in Peskin and Schroeder, <em>An Introduction to QFT,</em> on page 285. They set out to calculate correlation functions for the free <strong>real</strong>-valued Klein-Gordon field $\phi(x)\in \mathbb{R}$. They define a 4D space-time lattice with lattice spacing $\epsilon$, and define</p>
<p>$$\mathcal{D}\phi~=~\prod_id\phi(x_i).$$</p>
<p>Next they write the Fourier series of $\phi(x_i)$ as</p>
<p>$$\phi(x_i)~=~\frac{1}{V}\sum_n e^{-i k_n\cdot x_i}\phi(k_n).$$</p>
<p>My questions are:</p>
<ol>
<li><p>Why do they treat the real and imaginary parts of $\phi(k_n)$ as independent variables?</p></li>
<li><p>Why does the fact that the change of variables is unitary let them write the measure as
$$\mathcal{D}\phi(x)=\prod_{k_n^0>0}dRe \phi(k_n) dIm \phi(k_n)?$$</p></li>
</ol>
<p>I've never seen an integration measure split into its real and imaginary part, so maybe I'm missing something obvious.</p> | g10858 | [
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<p>I am on a boat docked at Cape Charles, VA, about 30 or 40 miles from the center of Hurricane Irene. This understandably got me thinking about the force of wind on the boat. Since air friction is proportional to the speed squared (except for some friction types I'm sure someone will be kind enough to remind me of), is the wind's force on the boat also proportional to the wind's speed squared? In other words, will a 70 knot wind produce almost twice the force on the stationary boat as a 50 knot wind, all other factors being equal?</p> | g10859 | [
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<p>Assume you start with a car at rest. Then you accelerate the car to 100 kilometers per hour. In terms of the energy needed, does it matter how fast you accelerated the car? </p>
<p>In other words, do you need to provide the same amount of energy to accelerate a car from 0 to 100 kilometers per hour whether you do it in 4 seconds or 20 seconds? </p> | g10860 | [
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<p>I know the flavor and mass eigenstates are different, but are they related? What I mean is, in a process like fusion where electron neutrinos are created, do they start in the 1 mass eigenstate? My knowledge of QFT is nonexistent, so I've never really seen a Neutrino field written out mathematically, so I don't really know what's going on. I just see the mass eigenkets and I am comfortable with that, but I wasn't clear on this question. The mass and flavor eigenstates are different, but in what ways are they related?</p> | g10861 | [
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<p>My question arises from two ideas that seem to be contradictory. </p>
<p>Idea One: Wheeler's Delayed Choice experiment is an interesting variation of the double slit experiment.</p>
<p>Idea Two: In the "reference frame" for a photon, it is emitted and absorbed in the same instant; it 'experiences' no time.</p>
<p>If we accept the premise and conclusions of idea two, then how can idea one be at all interesting? It seems that it simply demonstrates that and instant for a photon may be several nanoseconds for an experimenter.</p>
<p>Which of these concepts am I not grokking correctly?</p> | g10862 | [
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<p>If we pass current through a gas, like in the discharge tube, the electrons will accelerate in the electric field. The accelerated electrons will collide with gas molecules, and transfer some of their energy to them. Thus, producing gas molecules in the excited state. The excited molecule can come to ground state by emitting its excess of energy in the form of light. Thus, we can see gasses emitting light when current is passed through them. </p>
<p>Will the current carrying solid conductors (example,copper wire) emit light similar to that of gases? </p>
<hr>
<p>$\large{\color{red}{\checkmark}}$This particular question is about the current carrying conductors which are not insulated (seen in everyday life) and so as to why they don't emit light. If we consider other cases, there will be radiation emitted from the conductor as discussed in this <a href="http://physics.stackexchange.com/questions/102334/why-doesnt-alternating-current-produce-light-while-a-vibrating-single-particle/102335#102335">particular question</a>. </p> | g10863 | [
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<p>When physicists say energy is conserved, do they mean that energy satisfies the continuity equation:</p>
<p>$$\triangledown \cdot j+\dot{\rho}=0$$</p>
<p>On the internet there is plenty of talk of how the continuity equation applies to conservation of charge, fluid dynamics, and so forth, but I can't find any mention of how it applies to the conservation of energy. Why? Is it because it is problematic to talk about energy current density ($j$)?</p> | g10864 | [
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<p>The inverted pyramid puzzle has the following premises:</p>
<blockquote>
<p>A giant inverted steel pyramid is perfectly balanced on its point.
Any movement of the pyramid will cause it to topple over. Underneath
the pyramid is a $100 bill. How do you remove the bill without
disturbing the pyramid?</p>
</blockquote>
<p>The answer to puzzle is <a href="http://www.questionotd.com/2010/04/are-you-feeling-greedy.html" rel="nofollow">here</a>.</p>
<p>Physics Questions:</p>
<ol>
<li>A friend, who did not provide the standard answer, claimed that the bill can be removed easily by pulling it out from underneath. I do not think this is correct if there is non-zero friction, as the mass of the pyramid would make the frictional force too great. The force of friction is independent of the area of contact, so it would be just as difficult to remove as if the pyramid was not inverted. Please confirm.</li>
<li>If we assume that the pyramid is solid steel and the size of an Egyptian pyramid, <strong>does a human being have the strength to even move such a massive pyramid intentionally?</strong> (I understand that <em>if</em> the pyramid was moved, it would topple over, because of the location of the center of gravity. However, my question is whether such a pyramid can be moved by a human in the first place. A very tall styrofoam column can be toppled easily, but an iron column of the same dimensions may be unmovable, due to the difference in mass.)</li>
</ol>
<p><em>Note: Your answers should not try to solve the puzzle. The physics question is whether the premise of the puzzle is physically plausible.</em></p> | g10865 | [
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<p>My new question here: has string theory been analyzed somewhere in the context of various quantization prescriptions formulated in a mathematically sound way? I mean something like geometric quantization, Klauder Quantization, Brownian quantization etc. It is important to know if string theory ever passed through all the mathematical requirements that define a quantum theory; and by the way, are all the mathematical prescriptions defined consistently? Do we have a mathematically sound (consistent) prescription for quantization that is free of ambiguities?
People generally boast with the fact that string theory is "mathematically consistent" as a quantization of 1 dimensional objects. Is it so indeed? Is the "quantum" part in string theory really exactly the way it should be? and is the "way it should be" really known? </p> | g10866 | [
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<p>This is a question on gauge invariance in quantum mechanics. I do some simple math on a 1D wave-function with periodic boundary conditions, and get that gauge invariance is violated. What am I doing wrong?</p>
<p>Consider one coordinate dimension configured as a ring. The gauge dependent momentum operator can be written: </p>
<p>$p_{op}=-i \frac{\partial}{\partial x} - k$</p>
<p>Units have been chosen so that $\hbar = 1$, $k$ is an arbitrary real constant different for each gauge and $x$ represents the coordinate.</p>
<p>The gauge dependent eigenfunction can be written</p>
<p>$\psi(x)= Ae^{i(n+k)x}$</p>
<p>where A is a constant determined by normalization. As is well known in quantum mechanics, an operator applied to one of its eigenfunctions should yield a real constant eigenvalue multiplying the same eigenfunction: Thus</p>
<p>$[-i \frac{\partial}{\partial x} - k] Ae^{i(n+k)x}= nAe^{i(n+k)x}$</p>
<p>so that the real number n is the eigenvalue, which must be determined by the boundary conditions. </p>
<p>The boundary condition for this periodic system must be that the wave function should join onto itself smoothly everywhere. Thus, if the coordinate is chosen such that x extends from –$\pi$ around the ring to $\pi$ then the eigenfunction in equation 3 must have (n + k) = m, where m is an integer. </p>
<p>Under these conditions, the eigenvalue n in equation 3 will be n = m – k. This eigenvalue depends explicitly on k, and so is not gauge invariant. </p>
<p>I'm assuming this simple situation should be gauge invariant, but I don't see where I goofed. I'd appreciate any help. </p> | g10867 | [
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<p>Please please help me out with this. I am trying to find a software/program that I could use the simulate the scattering effect of light when it strikes gas particles moving at supersonic speeds? Please help me as I have no idea what to use. I have googled several searches but I could find nothing. Please help. thanks a lot. </p> | g10868 | [
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<p>This is a thought I asked myself often, but never did real efforts
to get an answer. Barsmonsters question about number of fans of a
wind turbine made me think of it again</p>
<p>Why do blades of aircraft propellers or wind turbines cover only a small fraction of the area they circle? Propellers of ships or Kaplan turbine propellers cover almost the entire circle. (Independent of number of blades). Same for steam turbines and jet turbines. </p>
<p>I am shure that the forms of those propellers are very close to optimum, due to decades of
experience. </p> | g10869 | [
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<p>I would like to understand what is meant when one introduces a generator $G(z)$ as the superpartner of the energy-momentum tensor $T(z)$.</p>
<ul>
<li><p>How does one decide that this $G(z)$ should have a "conformal weight" of $\frac{3}{2}$ ?</p></li>
<li><p>How are the following OPEs for it derived ?</p></li>
</ul>
<p>$T(z)G(w) = \frac{\frac{3}{2}}{(z-w)^2}G(w) + \frac{ \partial_w G(w) } {z-w} + ... $</p>
<p>$G(z)G(w) = \frac{ \frac{2c}{3} }{(z-w)^3} + \frac{2T(w)}{z-w} + ... $</p>
<ul>
<li>If one wants to here create a $\cal{N}=2$ superconformal algebra then one apparently needs to introduce two such conformal weight $\frac{3}{2}$ supercurrents say $G^1(z)$ and $G^2(z)$ (with OPEs as above) and another "$U(1)$ current" $J(z)$ such that,</li>
</ul>
<p>$ G^1(z) G^2(z) = \frac{ \frac{2c}{3}}{(z-w)^3} + \frac{2T(w)}{(z-w)} + i( \frac{2J(w)}{(z-w)^2} + \frac{\partial _w J(w)} {(z-w)} ) +... $</p>
<p>$T(z)J(w) = \frac{J(w)}{(z-w)^2} + \frac{ \partial _w J(w)} {(z-w)} + ...$</p>
<p>$ J(z)G^{1/2}(w) = \pm \frac{ iG^{2/1}(w)}{(z-w)} + ... $</p>
<p>$J(z) J(w) = \frac{ \frac{c}{3} }{ (z-w)^3 } + ... $</p>
<p>I would be grateful if someone can give some explanations about how the above construction is made.</p>
<p>Especially regarding the need and motivation to introduce the field $J$ (..and its OPEs..)</p> | g10870 | [
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<p>I have been looking around to figure out how superconductors are made. What ways are there to create a superconductor that don't involve a coolant like liquid nitrogen? Is it possible to cause a material to become a superconductor by running electric current through it? How practical is it to create a superconductor using any of these methods?</p> | g10871 | [
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<p>Electronic aiming systems use automated rangefinders that can sometimes be based on lasers. Obviously a projectile itself will be subject to the Coriolis Force (among others) and deviate from its course slightly and this can be accounted for/calculated. Was doing some work on the accuracy of these when I reached this dilemma.</p>
<p>The scenario I have in mind is (for the sake of argument) a gun firing a projectile at a target to the <em>north</em> which should get deflected slightly east at latitude of around 51 degrees north. The laser would be shone in the same direction. </p>
<p>So, will the laser beam be subject to the Coriolis Force in the same manner as the projectile?</p>
<p>My initial reaction was that the equation for the Coriolis Force:</p>
<p>$$F_{Coriolis} = -2m \overrightarrow{\omega} \times \overrightarrow{\dot{x}}$$</p>
<p>Depends on the mass of the object, and therefore if I take photons to be massless they shouldn't be affected, but this doesn't seem fulfilling. With the huge velocity of the beam I suppose that the effects will be extremely small, but it is more the principle that's bugging me. </p>
<p><strong>Extra edit:</strong> In fact would the beam necessarily follow the curvature of the earth (at extremely large distances)? </p> | g10872 | [
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<p>In the derivation of the Boltzmann distribution they consider a system $A$, enclosed by a diathermal wall in a heat reservoir $R$. Then they calculate the probability that the system $A$ is in an energy state $E_r$, given that the reservoir has a temperature $T$ and energy $E_0$ and the two systems are in thermal equilibrium. </p>
<p>I don't understand though why we can speak about probabilities. I would rather expect that we can calculate the energy of $A$ exactly if indeed the systems are in thermal equilibrium. Because the temperature of $R$ is then exactly $T$, so we can calculate its energy exactly (this last step is what I think we can do, though I'm not completely sure). So I wonder: is the derivation indeed for thermal equilibrium, like they state explicitly in my book. Or is that just a confusing mistake, and does the Boltzmann distribution give the probability if the system $A$ is in contact with $R$ but not necessarily in equilibrium with it? In the last case the derivation seems actually to make sense, but I'm worrying about the fact that they explicitly say the it is for thermal equilibrium. The fluctuations in energy cause then also fluctuations in temperature, so according to this, there would also never be exactly thermal equilibrium (so I'm not exactly sure the last case should be the right one either).</p> | g10873 | [
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<p>In Quantum Optics and Quantum Mechanics, the time evolution operator </p>
<p>$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$ </p>
<p>is used quite a lot. </p>
<p>Suppose $t_i =0$ for simplicity, and say the eigenvalue and eigenvectors of the hamiltionian are $\lambda_i, \left|\lambda_i\right>$.
Now, nearly every book i have read and in my lecture courses the following result is given with very little or no explanation:</p>
<p>$$U(t,0) = \sum\limits_i \exp\left[-\frac{i}{\hbar}\lambda_it\right]\left|\lambda_i\right>\left<\lambda_i\right|$$</p>
<p>This is quite a logical jump and I can't see where it comes from, could anyone enlighten me?</p> | g10874 | [
0.01150329876691103,
0.09125225245952606,
-0.021602123975753784,
-0.04021681845188141,
0.05527792125940323,
0.019652724266052246,
0.07104070484638214,
0.031660810112953186,
0.014974339865148067,
0.015573146753013134,
-0.059174008667469025,
-0.019460337236523628,
0.0332733578979969,
0.05541... |
<p>I'm trying to determine if going through the trouble of ingesting ice is worth the hassle versus ingesting ice-cold <em>water</em>, but my physics skills are rusty.</p>
<p>If I drink a gram of ice <em>water</em> at ~0C, my body has to heat the water to 37C.</p>
<p><a href="http://en.wikipedia.org/wiki/Heat_capacity#Table_of_specific_heat_capacities" rel="nofollow">Per wikipedia</a>, the water will be heated by</p>
<p>$$1\text{ g} \times \underbrace{37\text{ K}}_\text{Temperature difference} \times \underbrace{ 4.1813 \frac{\text{J}}{\text{g K}}}_\text{Specific heat of water} = 155\text{ J}$$</p>
<p>Whereas if I ingest a gram of <em>ice</em>, in addition to those 155 Joules, it will need to heat by <a href="http://en.wikipedia.org/wiki/Latent_heat#Specific_latent_heat" rel="nofollow">334 Joules</a> to do the transformation from ice to liquid. So the payoff to ingesting <em>ice</em> instead of water is about 3 times more calories burned.</p>
<p>Is my reasoning sound?</p>
<p><em>Note: You could turn my question into a meta-question by attacking the logic of ingesting cold water for burning calories, but please be so kind as to keep your answers to the physics question I've stated.</em></p> | g10875 | [
0.04442311450839043,
0.0944264754652977,
-0.003568680491298437,
0.04199013113975525,
-0.010670697316527367,
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0.0158580020070076,
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-0.002183958189561963,
0.016682449728250504,
0.013397393748164177,
0.013665714301168919,
-0.03... |
<p>I have a problem set due tomorrow, and the last problem is driving me nuts. Been combing through griffiths trying to find similar examples to no avail, so it'd be greatly appreciated if stackexchange could help me out =)</p>
<p>Suppose that a quantum system has 4 independent states. The unperturbed energy of state $\lvert n_1\rangle$ is $E_1$ while the states $\lvert n_2\rangle$, $\lvert n_3\rangle$, and $\lvert n_4\rangle$ have energy $E_2$. Now a perturbation $H'$ acts on the system such that </p>
<p>$$\begin{align}
H'\lvert n_1\rangle &= -k\lvert n_1\rangle \\
H'\lvert n_2\rangle &= k\lvert n_2\rangle \\
H'\lvert n_3\rangle &= k\sqrt{2}\lvert n_4\rangle \\
H'\lvert n_4\rangle &= k\sqrt{2}\lvert n_3\rangle + k\lvert n_4\rangle
\end{align}$$</p>
<p>So the way I understand it, I have two unperturbed energy levels here, $E_2$ and $E_1$, the former of which is triply degenerate. When the system is perturbed, $E_2$ splits off into 3 new distinct energy levels. The first order correction would typically be given by</p>
<p>$$E_n = E_0 + \delta E_n,$$</p>
<p>where $E_n$ is the unperturbed Hamiltonian and $\delta E_n$ is the correction due to the perturbation. What I would have to do is add the perturbed states to the unperturbed states, but how do I work with the notation to express that mathematically? My base intuition is that I can treat the kets as eigenfunctions of $H^0$ and treat the $k$ coefficients as eigenvalues, but I'm not really sure if that would be going on the right track. </p> | g10876 | [
-0.009530561044812202,
-0.061021048575639725,
-0.01904447376728058,
-0.06377695500850677,
0.020134974271059036,
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0.06703019142150879,
-0.01913851499557495,
0.00635530287399888,
0.014738155528903008,
-0.01318453624844551,
0.027646809816360474,
0.... |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/27909/why-is-exhaling-more-forceful-than-inhaling">Why is exhaling more forceful than inhaling?</a> </p>
</blockquote>
<p>If I put my hand behind my computer's system block, I can feel a very strong flux of air coming out, as it is being pushed by the fans. However, if I put my hand in front of the box, where the air is coming in, I can barely feel the air flow. Why is that?</p>
<p>I understand that this is probably because the air is coming into the box from a greater solid angle than the air coming out, so the incoming air has a lower velocity. But what is the reason for such asymmetry? Can it be explained in simple terms, without solving the Navier-Stokes equations?</p>
<p>Just to clarify: it is definitely not due to the difference in the area through which the air can come in and out: my computer has holes all over the back side, so no nozzles there :) In addition, all regular floor fans has the same property: the air flow coming out feels stronger than the air flow coming in.</p> | g659 | [
0.03425988927483559,
0.05788678303360939,
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0.07370400428771973,
0.04258304089307785,
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0.09268610924482346,
0.05635349079966545,
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-0.042120303958654404,
0.020876824855804443,
-0.006383759435266256,
-0.0019314925884827971,
0.057... |
<p><a href="http://www.usatoday.com/tech/science/columnists/vergano/story/2012-01-02/international-linear-collider/52324768/1" rel="nofollow">USA Today</a> has an article on Japan's interest as the site for the $10 billion future International Linear Collider. This accelerator will utilize electron/positron collisions (like CERN's former LEP collider) at energies of 500 GeV and will be 30 km long. The <a href="http://www.linearcollider.org/" rel="nofollow">ILC</a> site claims precision capabilities over and above those of the 14 TeV LHC for Higgs and other "new physics" areas.</p>
<blockquote>
<p>What are the specific advantages of the ILC over the LHC and what new particle physics insites are more likely to be discovered?</p>
</blockquote> | g10877 | [
0.028789442032575607,
0.07874190807342529,
0.007326985243707895,
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0.04163678362965584,
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0.027955995872616768,
0.0036194915883243084,
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-0.03949007764458656,
0.03047998994588852,
0.025636885315179825,
-0.016520533710718155,
-0.... |
<p>What happens when two D-branes annihilate? Do we get a radiation of strings?</p>
<p>Thanks in advance</p> | g10878 | [
0.008072602562606335,
0.03732500597834587,
0.02775796316564083,
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0.060393329709768295,
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0.031022053211927414,
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0.015169902704656124,
-0.0647401288151741,
0.04403001815080643,
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<p>In Physics ordinary terms often acquire a strange meaning, action is one of them. Most people I talk to about the term action just respond with "its dimension is energy*time". But what is its historial origin?</p>
<p><a href="http://en.wikipedia.org/wiki/Action_(physics)#History">http://en.wikipedia.org/wiki/Action_(physics)#History</a> doesn't really give much insight, as it lacks citations and depth.</p>
<p>So, how did "action" become to mean what it means now?</p>
<p>Cheers</p> | g10879 | [
0.07359018921852112,
0.05282559245824814,
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0.04486182704567909,
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-0.015261676162481308,
-0.014872646890580654,
0.019663533195853233,
0.01893587037920952,
-0.013... |
<p>Has a system where conducting sites can percolate by hopping over/tunnelling through a non-conducting site been described? If so what are the characteristics, and where can I find more details (such as a paper on the subject)?</p>
<p>In the image below, if the edge of a black square touches the edge of another black it 'conducts' across. That could be described as singularly percolated.</p>
<p>I'm trying to describe a system whereby the 'conduction' can hop over a white square.</p>
<p>Does this have a name? Is it formally described in a paper anywhere?</p>
<p><img src="http://i.stack.imgur.com/RoIo1.gif" alt="Percolation grid"></p> | g10880 | [
0.004865430295467377,
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0.021593645215034485,
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0... |
<p>In a lot of laymen explanations of general relativity it is implied that the four dimensions of the space-time are equivalent, and we perceive time as different only because it is embedded in our human perception to do so.</p>
<p>My question is: is that really how general relativity treats the 4 dimensions?</p>
<p>If so - what are the implications (if any) this has on causality?</p>
<p>If no - can the theory support more than one time dimension?</p> | g10881 | [
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0.0020192868541926146,
0.07121320813894272,
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0.02584335394203663,
0.022894... |
<p>What direction is the angular momentum of right hand polarized light points to? Is it vertical to its propagating direction? I want to recognize this in quantum theory.</p> | g10882 | [
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0.04207296296954155,
0.012590132653713226,
0.003382330760359764,
0.004342189524322748,
-0.0... |
<p>Light hits a charge coupled element. The wavelength of the light somehow is translated into a color picture. Where can I learn about methods (algorithms) to decompose light hitting a CCD into frequency spectrum? </p> | g10883 | [
0.08011383563280106,
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0.0062665073201060295,
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-0.050128545612096786,
0.0018369992030784488,
0.014504684135317802,
0.06078331917524338,
0.04... |
<p>So from learning Band theory, and PN Junction and such, I've learned that photons are created when "holes" are filled in a band, and this is what can create light (Isn't this how LEDs work?)</p>
<p>Anyways, my question is - How come when Electrons move between conduction bands light isn't produced? Or is it and it's just so small we can't tell?</p>
<p>Because the conduction band is technically still an orbital, which means it can have "holes", right?</p>
<p>Secondary Question: Is there a "Point" in the orbitals in which the "Grip" a nucleus has on an electron becomes basically insignificant. Like I know it gets less and less as the Bands go outward, but is the conduction band basically a point where it's just "decided" that the electron at that band will probably not be attracted to the nucleus more than the thermal energy produced in a normal environment is much more?</p> | g10884 | [
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-0... |
<p>How does a state vector change under an exchange of a boson and a fermion ? That's how is $\Psi_{\alpha,\beta}$ related to $\Psi_{\beta,\alpha}$ where $\alpha$ and $\beta$ are a boson and a fermion respectively?</p> | g10885 | [
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... |
<p><img src="http://i.stack.imgur.com/RbNrA.png" alt="my poor attempt @ FBD..."><img src="http://i.stack.imgur.com/Pd2Sf.png" alt="enter image description here"></p>
<p>I did the FBD, and I found too many variables which are not eliminating...Moreover, I believe this question is based on kinetic and static friction. But, $\mu$ here is ambiguously defined...How Do I get the integral value?</p> | g10886 | [
0.0799911618232727,
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0.03395995497703552,
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0.... |
<p><strong>Motivation</strong>: The neutral pion decays to 2 photons ($\pi^0\to\gamma\gamma$) most of the time. For the decay of the neutral to 3 photons ($\pi^0\to 3\gamma$) we have an upper limit on the branching ratio of $3.1 \cdot 10^{-8}$ in the Particle Data Book (2012). The explanation is that this decay would violate charge conjugation. </p>
<p>I haven't found anything about angular momentum conservation in this decay: The pion has spin zero, the photon is a spin-one particle, but, being massless, a (free) photon cannot have 0 for the projection of the spin. </p>
<p>Now, my <strong>question</strong> is: Can I combine three photons to give a spin-zero state such that angular momentum in the $\pi^0\to 3\gamma$ decay is conserved?</p>
<p><strong>Thoughts</strong>: For "ordinary" spin-zero particles the spin addition is described by the Clebsch-Gordan coefficients. From the tabulated CG coefficients I see that I can combine $1\times1$ to give a $J=1$ state with zero contribution from $m_1=m_2=0$. So I could add two photons, giving $J=1$ and $M=0$ (the important point being that all other possibilities are ruled out by the requirement of the photon not having $m=0$??). I could then add the third photon on top, argueing along the same lines. </p>
<p>Does that make sense? Is the application of the formalism correct?</p>
<p>(Sorry for writing so much text about a simple question.)</p> | g10887 | [
0.003439271356910467,
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0.00908361840993166,
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0.082155242562294,
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0.0014807697152718902,
0.00710215512663126,
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-0.0441... |
<p>If <strong>the position of some charge Q is known</strong>, the boundary condition is <strong>u=0</strong> on some parabolic surface, and we <strong>know the image charge has its electric volume of Q'</strong>, then <strong>how can I determine the position of the image charge</strong>?</p>
<p><strong>Same questions goes for the hyperbolic curve boundary. How can I determine the position of Q'?</strong></p>
<p>I think may be there is a way to transform the coordinates to make everything into an easily-handled form, but I am not sure about it. Another solution I thought about is to put this question into a general question based on basic Poisson's equation and Laplace equation, but I do not have a specific idea on how to conduct this.</p> | g10888 | [
0.08068836480379105,
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0.006795772351324558,
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0.0... |
<p>My book "Concepts of Physics (Satish K. Gupta)" says: </p>
<blockquote>
<p>The electric field of a charge is the space property by virtue of which the charge modifies the space around itself. As a result, if any charge is brought in the space around the charge, it experiences electrostatic force. <em>It may be pointed out that the source charge does not experience any force due to electric field produced by it</em>. The electric field due to a source charge has its own existence and is present even if there is no test charge to experience the force. </p>
</blockquote>
<p>NEWTON'S LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. — Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. </p>
<p>By reading the statements given in my book and the newton's third law, I got a question. If we press a stone with our finger, the finger is also pressed by the stone. If a source charge exerts force on the test charge, it should also feel the same force in opposite direction. So, <em>won't it mean source charge is experiencing force due to its own electric field?</em> If it is true, it is in contradiction with my text book statement that the source charge experiences no force due to electric field produced by it. </p>
<p>I don't know whether I have misunderstood anywhere, or (if I am not misunerstood) is it that source charge experiences force due to it's own electric field? </p> | g10889 | [
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0.027779579162597656,
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0.0016705866437405348,
0.009128489531576633,
-0.01627... |
<p>Since gravity is three dimensional why planets are rotating only in one plane around sun.</p> | g231 | [
0.014675100333988667,
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0.013... |
<p>For a massless scalar field the equation of motion in a general curved Space time is $\frac{1}{\sqrt{g}}\partial_\mu(\sqrt{g}g^{\mu\nu}\partial_\nu\phi)=0$. Though, in the action, we can by hand include a term $\xi R\phi^2$ as the only possible local, dimensionally correct term which has the physical significance of coupling of the scalar field with the curvature of space time itself and it modifies the EOM as $\frac{1}{\sqrt{g}}\partial_\mu(\sqrt{g}g^{\mu\nu}\partial_\nu\phi)+\xi R\phi=0$.</p>
<p>Now for a massless spin-1 field, I can similarly first write down an EOM as $\frac{1}{\sqrt{g}}\partial_\mu(\sqrt{g}g^{\mu\nu}\partial_\nu A_\rho)=\Box A_\rho=0$ in Lorentz gauge. My idea was that now if I want, I can by hand put a coupling (with space time curvature) term which will modify the EOM likewise. Am I correct about that? For example Birrell-Davies book equation 3.184 says $F_{\mu\nu};\,^\nu+\zeta^{-1}(A^\nu_\,;\nu);\mu=0$ which goes with what I said (with gauge choice kept more general from what I said), but Eq. 3.185 says $A_{\mu;\nu}\,^\nu+R_\mu\,^\rho A_\rho-(1-\zeta^{-1})A_{\nu;}\,^\nu\,_\mu=0$.</p>
<p>Can we derive it from the previous equation? How? Sorry my references were not helpful. Synge's general relativity book might have the answer but I don't have it. So, I have now two questions. Whether it is just derivable from the previous equation 3.184 (in which case I will take 3.185 to be minimally coupled case EOM), or whether I need to introduce extra term (non minimal coupling term of gauge field with curvature just like scalar case) in the action to get 3.185. If so, then I guess it will be called the non-minimal contribution. I think its a non-minimal coupling term. And in this case I guess whether I introduce it or not is my choice (if I have an option to make this choice). Please verify this too. </p> | g10890 | [
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0.05153195187449455,
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0.024511191993951797,
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0.020782850682735443,
0.04356047511100769,
0.021... |
<p>My understanding is that QCD has three color charges that are conserved as a result of global SU(3) invariance. What about SU(2) weak? Does it have two types of charges? What I'm getting at is:</p>
<p>U(1) --> 1 type of charge</p>
<p>SU(2) --> ?</p>
<p>SU(3) --> 3 types of charge</p>
<p>Does SU(2) have two types? If not, what is the relation between SU(N) invariance and the number of charge types?</p>
<p><strong>Idea:</strong> Maybe both I and I_3 (weak isospin and its third component) are conserved before electroweak symmetry breaking? Is that true? If so, then that would answer my question.</p> | g10891 | [
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0.000530124525539577,
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... |
<p>I want to prove that the reflectancy of an Electromagnetic wave in a medium with $\sigma$ $\ne$ 0 is:</p>
<p>\begin{align*}
R_{\|}=R_{\perp}=\frac{(n-1)^2+k^2}{(n+1)^2+k^2}
\end{align*}</p>
<p>any advice, please?</p> | g10892 | [
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0.015428817830979824,
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0.06810903549194336,
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0.0... |
<p>I've got several questions regarding the so called second quantization of the Schroedinger equation.</p>
<p>My professor introduced the field operators for the Schroedinger field by simply stating them as follows:
$$
\hat\psi (\vec{r},\xi)=\sum\limits _i \psi_i(\vec{r},\xi) \hat a_i\\
\hat\psi^\dagger (\vec{r},\xi)=\sum\limits _i \psi_i^\star (\vec{r},\xi)\hat a^\dagger_i
$$
Where $\psi_i(\vec{r},\xi)$ are the time independent one particle wave functions and $\hat a_i,\, \hat a^\dagger_i$ the corresponding creation and annihilation operators.</p>
<p>Is there a way to explain, why one does this? If I understood correctly what I've been taught so far, in QFT one must find some way to quantize the fields obeying the field equation in question. I do, however, not quite understand why in this particular case it is done like this.</p>
<p>Shouldn't the $\psi_i(\vec{r},\xi)$ be the time dependent one particle wave functions? Because I thought the field operators for a system in a box look like this:
$$
\hat\psi (\vec{r},\xi)\sim\int\text{d}^3k\ \exp(i\omega_k t-i\vec k\vec x) \hat a_k
$$</p>
<p>My professor then proceeded to prove the (anti)commutation relations between the field operators, postulating the corresponding relations between the fermionic or bosonic creation and annihilation operators:
$$
\left[\hat\psi (\vec{r},\xi);\hat\psi^\dagger (\vec{r}',\xi')\right]_\pm=
\left[\sum\limits_i\psi_i (\vec{r},\xi)\hat a_i;\sum\limits_j\psi^\star_j (\vec{r}',\xi')\hat a^\dagger_j\right]_\pm
=\sum\limits_i\psi_i (\vec{r},\xi)\psi^\star_i (\vec{r}',\xi')\\
=\delta(\vec{r}-\vec{r}')\delta _{\xi,\xi'}
$$
Here I do not understand the last step. Is that conclusion possible? And shouldn't or couldn't one postulate the commutation relations between the field operators and arrive at the relations for the creation and annihilation operators?</p> | g10893 | [
0.009424874559044838,
-0.034471407532691956,
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-0.02351953834295273,
0.09393340349197388,
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0.014865976758301258,
0.0002195908600697294,
0.0009172825957648456,
0.010711822658777237,
-0.06131044775247574,
0.0480429083108902,
-0.025568928569555283,
0... |
<p>Note: My question is duplicate of <a href="http://physics.stackexchange.com/questions/16561/why-doesnt-water-come-out-of-tap-faucet-at-high-pressure-when-i-turn-it-on">Why doesn't water come out of tap/faucet at high pressure when I turn it on?</a>. None of the answers given there explains how the continuity equation fits properly. That's why I am asking this question. </p>
<hr>
<p>I do not understand the <a href="http://en.wikipedia.org/wiki/Ohm%27s_law#Hydraulic_analogy" rel="nofollow"><em>Hydraulic analogy</em></a> explained on wikipedea. </p>
<blockquote>
<p>I asked my <em>Network analysis</em> teacher for an explanation .<br>
He told me that cross-section area of a pipe represents its resistance and flow of water represents current passing through that pipe. Like in a tap when we change the area of aperture the flow rate of water changes, this is analogous to change in current in a wire due to the change cross-section area of the wire.</p>
<hr>
<p>There is a contradiction in my understanding. </p>
</blockquote>
<p><strong>Contradiction</strong>: When we decrease the area of the mouth of the tap by our thumb the amount of water flowing out remains same but if we decrease the area of aperture of the tap by turning the knob the amount of water flowing out decreases.<strong>why?</strong><br>
Is it due to the change in type of flow i.e the flow changes from <em>laminar</em> to <em>turbulent</em> or <em>choked</em>?<br>
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Tap.png/220px-Tap.png" alt="image">
I found a flaw by applying equation of continuity as: </p>
<p>Suppose two parallel resisters are connected to voltage source as shown:<br>
<img src="http://images.tutorvista.com/content/electricity/parallel-resistors-energised-battery.jpeg" alt="image 1"><br>
Let's name the three wires as <em>pipe-0</em>(having the battery) ,<em>pipe-1</em>(having resister $R_1$) and <em>pipe 2</em>(having resister $R_2$). </p>
<blockquote>
<p>All three pipes are of same length. Resistances of different wires are equivalent to the respective aperture radius of different pipes. </p>
</blockquote>
<p>Our analogy is to interchange the terms current $i=dq/dt$ (amount of charge crossed in a unit time) with flow rate of water i.e $i \equiv dm/dt$ (amount of mass of water crossed in unit time). </p>
<blockquote>
<p>Let's remove the wire having $R_2$. </p>
</blockquote>
<p>The current $I$ in <em>pipe 0</em> will decrease as the resistance of circuit increase.<br>
removing the resistance is analogous to change the area of pipe-2 to $0$.
On the other hand removing pipe 2 will not change water flow $dm/dt$ in <em>pipe-0</em> because of <a href="http://www.engineeringtoolbox.com/equation-continuity-d_180.html" rel="nofollow"><em>Equation of Continuity</em></a>. </p>
<hr>
<p>I wrote my understanding about the situation to tell the community that my question is not <em>Home-work like</em>
<strong>In essence my question is<br>
1.How the tap works? And how can we apply the equation of continuity to the water flow when we turn the knob and when we cover the tap with thumb?<br>
2. Where I'm getting wrong with my understanding of the Hydraulic analogy.</strong> </p> | g10894 | [
0.05473625659942627,
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0.03319062292575836,
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0.01531299203634262,
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0.0... |
<p>I read about the geometrical limit of wave theory. The source from where I read had a slightly different explanation to provide than <a href="http://physics.stackexchange.com/questions/65237/if-light-rays-obey-to-the-wave-equation-why-can-they-be-thought-as-straight-lin">here</a>(The more rigorous answer is too complicated for me to understand). Though I do not understand <em>completely</em> the easier method in the linked question either, I would like to understand what the source form where I read is trying to say:- </p>
<blockquote>
<p><strong>An Aperture of size $a$ illuminated by a parallel beam sends
diffracted beam (the central maxima) in angular width approximately
$\lambda/a$. Travelling a distance $z$, it aquires the width
$z\lambda/a$ due to diffraction. The distance at which this width
equals the size of the aperture is called the <em>fresnel distance</em>.
$z_F=a^2/\lambda$. It is the distance beyond which the divergence of
beam of width $a$ becomes significant. At distances smaller than
$z_F$, spreading due to diffraction is smaller than the width of the
beam, and at distances greater than $z_F$, spreading due to
diffraction dominates over that due to ray optics(width $a$ of the
aperture$.</strong></p>
</blockquote>
<p>I fail to see the meaning behind this line of argument. Is ray optics valid for distances less than the fresnel distance? Is it not that the validity holds when all objects are comfortably <em>larger</em>, and not smaller, than the wavelength of light?
How is the "divergence" due to diffraction, and its equivalence to the aperture width, related to the geometric limit of wave theory? </p> | g10895 | [
0.03457968309521675,
0.004574356134980917,
0.009087899699807167,
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-0.04453161358833313,
0.011927628889679909,
0.013331474736332893,
0.05330200865864754,
-0.0129... |
<p>If the application of the <a href="http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics" rel="nofollow">Bose-Einstein distribution</a> is in <a href="http://en.wikipedia.org/wiki/Black-body_radiation" rel="nofollow">blackbody radiation</a>, then what is <a href="http://en.wikipedia.org/wiki/Planck%27s_law" rel="nofollow">Planck's distribution</a>? Are they same? How did Planck know that he should use a Bose-Einstein distribution to model blackbody radiation?
Thanks</p> | g10896 | [
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0.007782059255987406,
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0.014688661321997643,
0.03565257787704468,
0.021491337567567825,
-0.0... |
<p>I work currently on the calculation of magnetic resistance for a air-gap of an electrical machine. For this I am using conformal transformations. I am doing this on a 2D slice. As the machine is obviously round I first use a logarithmic transformation to convert it to a polygonal shape. After that I use a Schwarz-Christoffel transformation to map the polygon to a rectangle.</p>
<p>For the said rectangle I assume a homogeneous field distribution. A fine mesh is set to represent the homogeneous field distribution. This mesh is maped back to the real(physical) geometry. What I get is a very nice representation of the field and equipotential lines. I approximate every of the small quadrilaterals as rectangles and calculate the resistance of them. So I get the resistance. For the Schwarz-Christoffel Transformation I use the <a href="http://www.math.udel.edu/~driscoll/SC/" rel="nofollow">SC-Toolbox</a> by Tobin A. Driscoll</p>
<p>Here you can see a simple example of what I'm trying to calculate:
<img src="http://i.stack.imgur.com/C0fdi.png" alt="enter image description here">
<a href="http://i.stack.imgur.com/C0fdi.png" rel="nofollow">Larger image</a></p>
<p>My question is if the there is a more elegant way to get the resistance directly from the transformed (canonical) space, meaning the rectangle. I asked this question already on the <a href="http://math.stackexchange.com/questions/754818/conformal-equivalence-of-resistance">math portal</a> in more general terms, but I wasn't able to get an adequate answer</p>
<p>I am currently in the phase of writing documentation for my project, so I wanted to see if there is some deeper physical/mathematical explanation for it.</p> | g10897 | [
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0.014352640137076378,
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0.015174604021012783,
0.023482652381062508,
0.0112... |
<p>To understand the phenomenon of diffraction as an interference effects of several dipole oscillators (like in case of several symmetrical, not sawtooth, scratches in a diffraction grating), we consider a linear array of $N$ particles, each of which acts as a source of EM wave, and their interference produces the diffraction pattern. This can also be used in case of width-zero slits, $N$ in number, arranged linearly.<br>
For a single slit diffraction pattern, we use an array of infinite sources (with separation $d\rightarrow 0$)along the width of the slit, a justification lies either in the Huygen's principle (sources of secondary wavelets) or as explained by assuming a plug at the slit in Feynman's lectures on physics. </p>
<p>How do we extend this methodology to study the diffraction pattern at the edge of an opaque object, producing a shadow? Do we consider infinite sources extending from the edge till infinity? Or we consider sources only upto some defined distance? </p>
<p><img src="http://i.stack.imgur.com/Av1KK.gif" alt="enter image description here">
<img src="http://i.stack.imgur.com/vzAfE.jpg" alt="enter image description here"></p> | g10898 | [
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0.04149268940091133,
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0.046178143471479416,
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-0.03509194403886795,
0.0074613806791603565,
0.001459451043047011,
-0.005265411455184221... |
<p>Currently I am stuck, trying to derive the self-inductance of a long wire. According to literature it should be </p>
<p>$$L=\frac{\mu_r\mu_0l}{8\pi}$$</p>
<p>and in literature its derived by looking at the energy of the magnetic field. I tried to derive this formula via the magnetic flux and I am getting $4\pi$ instead of $8\pi$. This are my considerations:</p>
<pre><code> _ _
/ |R \
| ' | a wire with radius R and length l
\ _ _ /
</code></pre>
<p>The magnetic flux density $B(r)$ is given according to Ampere's law:</p>
<p>$$B(r) 2\pi r=\mu_0\mu_r\frac{r^2}{R^2}I$$</p>
<p>$$B(r)=\mu_0\mu_r\frac{r}{2\pi R^2}I$$</p>
<p>where $r$ is the distance from the center of the wire, $R$ is its radius and $I$ is the total current through the wire. Now I know that the magnetic flux $\phi$ through the upper part of a longitudinal section is</p>
<p>$$\phi = \int_A B dA = \int_0^R B(r)l dr = \frac{\mu_0\mu_rIl}{4\pi}$$</p>
<p>where $l$ is the wire's length. No I use $\phi = LI$ and arrive at $4\pi$.</p>
<p>What am I doing wrong? Where is the mistake in my considerations?</p>
<p>Moreover I have the following problem. If I look at an entire longitudinal section of the wire and not only at its upper half the magnetic flux is zero:</p>
<pre><code> _ _
/ | \
| |2r | => Magnetic flux is zero (the magnetic field
\ _|_ / penetrating the upper half of the longitudinal cross section is
exactly opposite to the magnetic field penetrating the lower
half)
</code></pre>
<p>Hopefully I formulated my problem clear enough. If not please ask me for further details.</p> | g10899 | [
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0.058479975908994675,
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0.04049188271164894,
-0.07406876981258392,
0.00... |
<p>Here is the sketch:</p>
<p><img src="http://i.stack.imgur.com/0s1is.jpg" alt="Sketch for the problem"></p>
<p>The sketch is supposed to be side-view of the path of the object.</p>
<p>The following values are known:</p>
<ul>
<li>$r$ - radius of the circle that describes the path AB of the object</li>
<li>$a$ - angle that characterizes the part of a circle that describes the path AB of the point</li>
<li>$m$ - mass of the point</li>
<li>$V_0$ - velocity</li>
</ul>
<p>What I need to find out:</p>
<ol>
<li>Equation of motion for AB</li>
<li>Equation of motion for BC</li>
<li>velocity at B</li>
<li>The distance DC</li>
</ol>
<p>The dashed line is the object's trajectory after it leaves AB. $N$ is the normal force, $T$ is friction and $g$ is the gravitational acceleration.</p>
<p>I was able to solve this problem partially when AB is a straight line and $a$ represents the angle between AB and AD. So far I could come up with only this:</p>
<p>$m x'' = -T-mg \sin(?)$ <- in place of the question mark I would need the angle between AB and AD</p>
<p>$m y'' = N-mg \cos(?)$</p>
<p>$N = mg \cos(?)$</p>
<p>$T = \mu N = \mu mg \cos(?)$</p>
<p>$x'' = -g(\mu \cos(?) + \sin(?))$</p>
<p>$x' = -gt(\mu \cos(?) + \sin(?)) + c_1$</p>
<p>$x = g\frac{-t^2}{2}\left[ \mu \cos(?) + \sin(?) \right] + c_1 t + c_2$</p>
<p>where $\mu$ is the coefficient of friction. $x$ and $y$ are the coordinates with respect (both functions of time).</p>
<p>How do I deal with the fact the ramp is no longer a straight line but a curved line? Thank you very much for your help.</p> | g591 | [
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0.008744644932448864,
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0.04536915570497513,
-0... |
<p>As stated according to Newton laws of gravity, every object with mass attracts all other object with a force which produces acceleration. Basically there are several forces in the universe which affects our planet as well.</p>
<p>My question: can an inertial frame (which has net force zero) can exist in these condition, and does this frame plays any role in producing acceleration?
If two objects are many miles apart and one object has much greater mass then, will the heavier object still accelerate relative to lighter one?</p> | g10900 | [
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0.0198808666318655,
0.027345197275280952,
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0.013649377040565014,
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0.0082885... |
<p>Consider a metal surface which is being continuously irradiated with a light with frequency greater than the threshold frequency. After some time, all the electrons should have been emitted from the atoms of the metal on the surface. </p>
<p>Now since an electron has been removed from the valence shell, the ionisation potential should increase as the electrons are more tightly bound as compared to before. Thus the energy required to emit he electrons also increases. So does the threshold frequency change here?</p>
<p>Is there any other case in which the threshold freuqency changes</p> | g10901 | [
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0.0087498240172863,
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0.08127612620592117,
0.025837667286396027,
-0.0066... |
<p>What the theta in schwinger function and what is theta formula?</p>
<p>is theta formula general solution of klein gordon equation? if so, what is its coefficient of $\exp\left(-ipx\right)$?</p> | g10902 | [
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0.018295180052518845,
0.06612125784158707,
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0.04419485107064247,
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... |
<p>It is known that when a beam of lineary polarized light falls perpendicularly on a linear polarizer, the intensity of polarization changes according to Malus' law and the direction of polarization changes as cosine of angle between polarization vector and polarizer vector.</p>
<p>My question is: is anyone familiar with mathematical treatment of how the direction and intensity of polarization changes when the angle of incidence changes?</p> | g10903 | [
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-0.029002908617258072,
-0.003935144282877445,
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0.008057964034378529,
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0.019243936985731125,
0.03869149088859558,
... |
<p>I have stress tensors direct product of the form $T^{ab}(x)T^{cd}(y)$. I want to write this in terms of a tensor $I^{abcd}$ in the form.
$T^{ab}(x)T^{cd}(y)= I^{abcd}$.
This is like decomposing the direct product of the Rank 2 symmetric tensor into its irreps. How can I do a decomposition like this for $T\otimes T\otimes T...$??? </p> | g10904 | [
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0.03266393393278122,
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0.010144262574613094,
0.01278713345527649,
-0.0... |
<p>I am attempting to settle a friendly bet. Would a pendulum swing indefinitely in a hypothetical vacuum (i.e. no air resistance) having a hypothetical frictionless bearing (i.e. no energy lost due to friction) assuming the following</p>
<ol>
<li><p>The frictionless vacuum is on Earth (9.8 m/s^2).</p></li>
<li><p>The pendulum is already in motion and no other external forces other than gravity act on the pendulum.</p></li>
</ol> | g301 | [
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0.003... |
<p>I just have a question concerning the third law of thermodynamics. </p>
<p>The third law describes that the entropy should be a well defined constant if the system reaches the ground state which depends only on the temperature. Beside this fact we now that the temperature is independent on the measurement system we can assume that $S(T\rightarrow 0) = 0$.<br>
This is not difficult to understand. If you take a look on the definition of the entropy $S = k_B \log{\Omega}$ then this means that the number of microstates is equal to a constant or in other words, the system is in a well defined state. If you would measure $x$-times the system it will not change and you get $x$-times the same results. <em>I'm right so far?</em></p>
<p>Okay. Now we take a look on one example - the ideal gas. If we calculate the partition function we will get something like:
$$
Z(T,V,N) \propto T^{3N/2}
$$
And for the entropy we will get:
$$
S(T,V,N) \propto \ln{T^{3/2}}
$$</p>
<p>Both doesn't really fullfil the third law. Or is my assumption wrong? I'm mean that the entropy goes to zero?
Maybe the ideal gas doesn't fulfill the third law, but my concern is that the calculation for the partition function would be almost the same ($\propto T^\alpha$) if we calculated it for an other system, based on the definition. </p>
<p>Has anybody maybe something like a thumb rule for checking if the system fulfill the third law after calculating the entropy?</p>
<p>Thank you for your help. </p> | g10905 | [
0.003318848554044962,
0.013635657727718353,
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0.019111935049295425,
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-0.019039178267121315,
... |
<p>I have to develop a physical model for a certain type of biological oscillation that can be built upon periodic sequences.</p>
<p>From earlier <a href="http://en.wikipedia.org/wiki/Periodic_sequence" rel="nofollow">questions</a> I know that any periodic sequence (containing $0$s and $1$s) can be developed based on the form of such a Discrete Inverse Fourier:</p>
<p>$$\psi(n)=\sum_{k=1}^T\frac{1}{T} \cos \left( \frac{-2 \pi(k-1)n}{T} \right) \qquad; \;T,k,n \in \Bbb N \qquad(1)$$</p>
<p>Also I got an elegant <a href="http://math.stackexchange.com/questions/139881/how-to-construct-and-oscillation-with-exponentially-growing-period-times">answer</a> yesterday how to describe an oscillation which period exponentially increases along the abscissa by applying $\log x$ (in the example below visible on the zeros):</p>
<p>$$\varphi(x)=\sin(\pi\log_T x) \qquad;T \in \Bbb N \;;x \in \Bbb R \qquad (2)$$</p>
<p>What I am trying to do is to modify the equation $(1)$ in a way that it would describe a periodic sequence similar $\psi(x)$ (containing $0$s and $1$s) which period of occurance of $1$ (that is $T$) increases exponentially. That would help me to describe the behaviour of my biological oscillator.</p>
<p>What would such a function similar $\psi(x)$ be?</p>
<p>Many thanks in advance.</p> | g10906 | [
0.005034273955971003,
-0.0141774732619524,
0.013692156411707401,
-0.039897333830595016,
0.01325935311615467,
0.0018977936124429107,
0.003547289641574025,
-0.004676223266869783,
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-0.03006194345653057,
0.024719784036278725,
0.05328937992453575,
0.06632827967405319,
0.00... |
<p>Given a charge distribution $\rho(\vec{r})$ where $\vec{r}$ is the position vector and that $\rho$ is a function of only $|x|$, Why is it that the corresponding electric field $E$ is necessarily of the form $(E(x),0,0)$ and $E(x)$ is antisymmetric ? </p> | g10907 | [
0.05882726237177849,
-0.013225706294178963,
-0.038195572793483734,
0.038249630481004715,
0.10140299797058105,
0.02869037538766861,
0.02325899712741375,
0.010371768847107887,
-0.024407705292105675,
-0.008882083930075169,
-0.043461158871650696,
0.024519676342606544,
-0.004456077236682177,
0.... |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/25254/why-does-the-moon-sometimes-appear-giant-and-a-orange-red-color-near-the-horizon">Why does the moon sometimes appear giant and a orange red color near the horizon?</a> </p>
</blockquote>
<p>I made a little research about this and found this article <a href="http://en.wikipedia.org/wiki/Moon_illusion">http://en.wikipedia.org/wiki/Moon_illusion</a> that states the explanation of the moon being bigger at the horizon is still debated. </p>
<p>I found this which looks pretty 'big' <a href="http://www.psychohistorian.org/img/astronomy/deep-sky/photos/ayiomamitis/20090804-moonrise.jpg">http://www.psychohistorian.org/img/astronomy/deep-sky/photos/ayiomamitis/20090804-moonrise.jpg</a> but however this kind of repetitive shots <a href="http://apod.nasa.gov/apod/image/0706/UludagMoonrise_tezel.jpg">http://apod.nasa.gov/apod/image/0706/UludagMoonrise_tezel.jpg</a> show no decrease in the aparent moon radius.</p>
<p>What I am trying to understand is if it is a real illusion, if the atmosphere makes the image bigger or what other explanation could be possible and of course plausible.</p> | g106 | [
0.011503458023071289,
0.06647387892007828,
-0.02202748879790306,
0.013545488938689232,
-0.00428637582808733,
0.07853007316589355,
-0.0008651993703097105,
0.006374613847583532,
-0.024383289739489555,
0.020345939323306084,
0.06626460701227188,
0.012082046829164028,
0.020223919302225113,
0.03... |
<p>I understand that a 1.5 V cell will not deliver as much energy per coulomb as a 150 V power supply will.</p>
<p>What I do not understand is that why it is so.</p>
<p>I am digressing now. If we place two point charges at a distance r, then to increase the force between them, we need to increase their charges. This makes sense.</p>
<p>However, if we need to increase the Energy carried by 1 Coulomb of charge in a circuit, we need to increase the potential difference.</p>
<p>Here, the number of electrons remain the same but still, somehow, they are able to do more work.</p>
<p>The definition of electric potential is "electric potential at a point is the amount of electric potential energy that a unitary point charge would have when located at that point"</p>
<p>But why does electric potential energy act upon it at all if there is no charge to attract or repel it?</p>
<p>These seem to be different questions, but they are all linked to a fact that I cannot understand "The same amount of charge performs more work when a higher potential difference is applied"</p> | g10908 | [
0.041621990501880646,
0.034620050340890884,
-0.007342154625803232,
0.02488090470433235,
0.08063585311174393,
-0.01663268730044365,
0.026768838986754417,
0.038667261600494385,
-0.06421124935150146,
-0.0031400404404848814,
-0.02295399270951748,
0.02426828444004059,
-0.011282658204436302,
0.0... |
<p>Conventional physics as is usually presented in textbooks deals with the evolution of states in phase space parameterized by sharp instances in time, a real parameter. However, quantum fluctuations seem to suggest we have to smear a little over time to average over vacuum fluctuations and the like. What implications does this have over the meaning of 'now' and the nature of time?</p> | g10909 | [
-0.030080696567893028,
0.02927974984049797,
-0.0024268333800137043,
0.012126042507588863,
0.004394001793116331,
0.06044388934969902,
0.033424317836761475,
-0.010880490764975548,
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0.02544829435646534,
0.017582403495907784,
0.004551618359982967,
0.06404709070920944,
0.04... |
<p>For infinisesimal bispinor transformations we have
$$
\delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, \quad \eta_{\mu \nu} = -\frac{1}{4}(\gamma_{\mu}\gamma_{\nu} - \gamma_{\nu}\gamma_{\mu}). \qquad (.1)
$$
Then, by compairing $(.1)$ with transformation by the generators of the Lorentz group,
$$
\delta \Psi = \frac{i}{2}\omega^{\mu \nu}J_{\mu \nu}\Psi ,
$$
we can make the conclusion that in bispinor representation
$$
J_{\mu \nu} = -i\eta_{\mu \nu}. \qquad (.2)
$$
By the other way, from Noether theorem we can get spin tensor,
$$
S^{\mu, \alpha \beta} = \frac{\partial L}{\partial (\partial_{\mu}\Psi)}Y^{\alpha \beta} + \bar {Y}^{\alpha \beta}\frac{\partial L}{\partial (\partial_{\mu}\bar {\Psi})}.
$$
Then, by having $(.1)$ and Lagrangian
$$
L = \bar {\Psi}(i \gamma^{\mu}\partial_{\mu} - m)\Psi ,
$$
it's easy to show that
$$
S^{\mu, \alpha \beta} = i\bar {\Psi}\gamma^{\mu}\eta^{\alpha \beta}\Psi .
$$
It's clearly that I can get $(.2)$ by
$$
S^{\alpha \beta} = \int S^{\mu, \alpha \beta}dx_{\mu},
$$
but for me it's not obvious how to compute it. Can you help me? </p> | g10910 | [
-0.013228647410869598,
-0.021575966849923134,
-0.04372789338231087,
-0.03333457559347153,
0.05089052394032478,
-0.051594216376543045,
0.07860525697469711,
0.0437593087553978,
-0.04534745216369629,
0.019098233431577682,
-0.058994486927986145,
0.015876607969403267,
-0.027797041460871696,
-0.... |
<p>I'm quoting the definition of Resultant of two forces acting in the same line from the book "A FIRST COURSE IN PHYSICS" one of whose authors is Robert Andrews Millikan:</p>
<blockquote>
<p><em>The resultant of two forces is defined as that single force which will produce the same effect upon a body as is produced by the joint action of the two forces.</em></p>
</blockquote>
<p>I'm really confused as to whether the resultant of two forces say $A$ and $B$ is the force which is produced as a result of the two forces just mentioned or is it a completely separate force which is not caused by $A$ and $B$, but its effect is the same as that of the force produced as a result of $A$ and $B$? Even though, the force caused by $A$ and $B$, let's say it is $C$, is equal in magnitude to that of the resultant $R$ of $A$ and $B$ and the direction is also the same, but $C$ is caused by $A$ and $B$; however, $R$ has no primary causes as $C$ has. This is what I conclude from this definition; however, I'm not sure yet. </p> | g10911 | [
0.06322512775659561,
-0.021834881976246834,
0.01780220866203308,
-0.005619100760668516,
0.03484981879591942,
0.008390084840357304,
0.06426013261079788,
0.044526711106300354,
0.03512691333889961,
-0.07020505517721176,
0.012857276946306229,
0.012496343813836575,
-0.014859184622764587,
-0.022... |
<p>This is a question I would like to have an explanation with: It's in <a href="http://www.fisme.uu.nl/nno/opgaven/bestanden/Ronde1-Theorie-2013.pdf" rel="nofollow">this PDF</a>, question 9. </p>
<p><img src="http://i.stack.imgur.com/euGXK.png" alt="circuit"></p>
<blockquote>
<p>In that circuit, if switch 1 is closed, bulb A burns normally. If 2 is closed as well, do other lamps burn normally as well?</p>
</blockquote>
<p>The answer is D: yes, both B and C, but why? Could somebody explain the steps in order to think? This is not homework (even though in principal it is) but I did this voluntarily on my own interest in how more advanced electrical circuits work.</p> | g10912 | [
0.05075708404183388,
-0.00412454828619957,
0.00906070601195097,
0.017973551526665688,
0.03666854277253151,
0.00011061516124755144,
0.05334944650530815,
0.07385781407356262,
-0.005945079959928989,
-0.05094096064567566,
-0.030502745881676674,
0.05365322530269623,
0.016706787049770355,
0.0082... |
<p>How do you find a metric tensor given a coordinate transformation, $(t', x', y', z') \rightarrow (t, x, y, z)$? Our textbook gives a somewhat vague example as it skips some steps making it difficult to understand. What's the general definition for a metric tensor of a given transformation? The closest I could find was <a href="http://en.wikipedia.org/wiki/Metric_tensor#Coordinate_transformations" rel="nofollow">http://en.wikipedia.org/wiki/Metric_tensor#Coordinate_transformations</a>, but I'm having trouble understanding that.</p> | g10913 | [
0.022056184709072113,
0.04461749643087387,
-0.015527980402112007,
-0.02137821353971958,
0.03366491198539734,
-0.013142107985913754,
0.055880241096019745,
0.03820916637778282,
-0.04955548048019409,
-0.012952672317624092,
-0.04930765554308891,
-0.025172393769025803,
0.07353555411100388,
-0.0... |
<p>Up to my knowledge an electrified (charged) body can attract a non-electrified (neutral) body. I thought this because, when we bring a charged (suppose negatively charged) body near a neutral one. Electrified body can attract a non-electrified body by the opposite charge induced on neutral body due to electrostatic induction, then the answer to the above question would be likely to say that electrified body exerts attractive force on non-electrified body.<br>
But going through the Wikipedia's encyclopedia of <a href="http://en.wikipedia.org/wiki/Electric_charge" rel="nofollow">electric charge</a>, I found the following line:</p>
<blockquote>
<p><em>No force, either of attraction or of repulsion, can be observed between an electrified body and a body not electrified.[3]</em> </p>
</blockquote>
<p>My view on the concept is contradictory to the statement.</p>
<p><strong>EDIT:</strong> I have seen a article supporting my view.You can see at the bottom of <a href="http://www.physicsclassroom.com/class/estatics/u8l1c.cfm" rel="nofollow">this</a> link page about interaction between charged(electrified) and neutral(non-electrified)body. </p> | g10914 | [
0.0341656357049942,
0.0343007929623127,
0.019672060385346413,
0.001585523597896099,
0.0528210774064064,
0.0673496350646019,
-0.012530718930065632,
0.02496270462870598,
0.008230444975197315,
-0.027208048850297928,
0.0038961467798799276,
0.013048993423581123,
-0.01896713487803936,
0.01320854... |
<p>Generically an Abelian Fractional Quantum Hall Systems is described by chiral scalar fields $\hat{\Phi}^{\ }_{i}(t,x)$ with $i=1,\ldots,N$ and a Hamiltonian of the form
$
\hat{H}^{\ }_{0}:=
\int\limits_{0}^{L}
\mathrm{d}x\,
\frac{1}{4\pi}
\partial^{\ }_{x}\hat{\Phi}^{\mathsf{T}}
\;V\;
\partial^{\ }_{x}\hat{\Phi}
\;
$
with with $V$ a $N\times N$ symmetric and positive definite matrix.
The chiral scalar fields obey (up to Klein Factors) the following anticommutation
relations: $
\left[
\hat{\Phi}^{\ }_{i}(t,x ),
\hat{\Phi}^{\ }_{j}(t,x')
\right]=
-
\mathrm{i}\pi
K^{-1}_{ij}
\;\mathrm{sgn}(x-x')
$.
Here $K$ is a $N\times N$ symmetric and invertible matrix with
integer-valued matrix elements. We define electron operators of our theory by:
$\widehat{\Psi}^{\dagger}_{T}(t,x)
:=\
:e^{-\mathrm{i}\,T_{i}\,K^{\ }_{ij}\,\hat{\Phi}^{\ }_{j}(t,x)}:
\;,$
where the integer-valued $N$-dimensional vector $T$ determines
the charge (and statistics) of the operator. </p>
<p>Now we can introduce backscattering terms of the form:
$\hat{H}^{\ }_{\mathrm{int}}:=
-
\int\limits_{0}^{L}
\mathrm{d}x
\sum_{T\in\mathbb{L}}
h^{\ }_{T}(x)
:
\cos
\Big(
T^{\mathsf{T}} K\,\hat \Phi(x)
+
\alpha^{\ }_{T}(x)
\Big)
:.$
The real functions $h^{\ }_{T}(x)\geq0$ and
$0\leq\alpha^{\ }_{T}(x)\leq2\pi$ encode information about
the disorder along the edge when position dependent. </p>
<p><em>This text was partially taken from <a href="http://arxiv.org/abs/1106.3989" rel="nofollow">http://arxiv.org/abs/1106.3989</a>.</em></p>
<p><strong>Questions:</strong> </p>
<p>Does this type of backscattering also include inelastic backscattering?
Given a particular backscattering term how do I see whether the backscattering is inelastic or elastic? If inelastic backscattering is also allowed where does the excess momentum go? Are these terms "physical"? </p>
<p>I am looking forward to your responses. </p> | g10915 | [
0.017357366159558296,
0.014823614619672298,
-0.028067104518413544,
0.007367107085883617,
0.05704408511519432,
-0.04307110235095024,
0.06254642456769943,
-0.03203408792614937,
0.03453239053487778,
-0.00934770330786705,
-0.0462888702750206,
0.08814749866724014,
-0.05158700793981552,
-0.00630... |
<p>A <a href="http://en.wikipedia.org/wiki/Zener_diode" rel="nofollow">Zener diode</a> is used as a voltage stabilizer. The graph of current vs voltage of Zener diode clearly shows that there is a constant voltage across Zener after the breakdown voltage as the current increases rapidly.</p>
<p>But, I would like to know what happens at the microscopic level inside the Zener which creates a constant voltage across it.</p> | g10916 | [
0.015904998406767845,
0.04579286277294159,
-0.03225512430071831,
0.02628166414797306,
0.0020043211989104748,
-0.05302610248327255,
0.04995522275567055,
0.005398681852966547,
-0.04348983243107796,
-0.005401540547609329,
-0.03182143345475197,
0.012814320623874664,
-0.00909354817122221,
0.006... |
<p>I have two questions on mirrors.</p>
<ol>
<li>I’ve read that in the past quality mirrors were coated with silver but that today vacuum evaporated coatings of aluminum are the accepted standard. When I look at the <a href="http://en.wikipedia.org/wiki/Reflectivity">reflectance vs. wavelength plot</a>,</li>
</ol>
<p><img src="http://i.stack.imgur.com/pu0m0.png" alt="enter image description here"></p>
<p>I see that silver has a higher reflectance than aluminum. So why use aluminum instead of silver? If one wants the highest quality mirrors I assume that cost is secondary, so what is the physics that I am missing here?</p>
<ol>
<li>Why are the more “technical mirrors” (I am not sure what this means but I assume more precise?) front-surfaced instead of back-surfaced?</li>
</ol> | g10917 | [
0.07525885105133057,
-0.0025946549139916897,
0.0030464620795100927,
0.02307170070707798,
0.07416392117738724,
0.012559296563267708,
0.022280210629105568,
0.035491373389959335,
-0.0035355843137949705,
0.034181974828243256,
0.024082042276859283,
0.054937101900577545,
0.046322792768478394,
0.... |
<p>Can someone explain with a concrete example of how can I can check whether a quantum mechanical operator is bounded or <a href="http://en.wikipedia.org/wiki/Unbounded_operator" rel="nofollow">unbounded</a>?</p>
<p>EDIT: For example., I would like to check whether $\hat p=-i\hbar\frac{\partial}{\partial x}$ is bounded or not. </p> | g10918 | [
0.01364570390433073,
0.01397079136222601,
0.012139477767050266,
-0.024065382778644562,
0.02522655762732029,
-0.07389195263385773,
0.027862586081027985,
0.009974963963031769,
-0.0054514650255441666,
-0.029406074434518814,
0.0032861577346920967,
0.026350080966949463,
0.035563502460718155,
0.... |
<p>Is it possible to find out the generic eigenvalue spectrum of the non-Hermitian operator $L_x+iL_y$, without using any representation?</p> | g10919 | [
-0.03925574570894241,
0.000004939384325552965,
-0.010943405330181122,
-0.06152259558439255,
0.02546151913702488,
-0.03241753950715065,
-0.009548225440084934,
0.0009444031747989357,
-0.012933661229908466,
0.03719816729426384,
-0.015022246167063713,
0.0553203783929348,
0.019862603396177292,
... |
<p>I am introducing myself to fluids through some physics textbooks. Most of them start with <a href="http://en.wikipedia.org/wiki/Pascal%27s_law" rel="nofollow">Pascal's principle</a>.:</p>
<blockquote>
<p>If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount</p>
</blockquote>
<p>Then there's the example of the hydraulic lift that makes use of Pascal's principle.</p>
<p>What I don't get is why the fluid used in the example is always a liquid (and not a gas for example)? I know that liquids are incompressible but what happens if you use a gas to lift the car? Does part of the input force (or pressure) get wasted in increasing the temperature of the gas and therefore making it less effective? How much would the car get lifted if at all?</p> | g10920 | [
0.01603424921631813,
0.011854983866214752,
0.017513902857899666,
-0.03906235471367836,
0.027425456792116165,
-0.006480164360255003,
-0.03650326654314995,
0.04389789327979088,
-0.08541779965162277,
-0.0028549411799758673,
0.012106334790587425,
-0.032761164009571075,
-0.031025303527712822,
-... |
<p>Given, $M_4 = \Sigma \times C$,
How do you get an effective theory by studying maps
$\Sigma \rightarrow M_4$ .
Technically, the physics in one manifold is supposed to gossip about the overall affairs of people in the entire manifold at least up to a certain limit.
To be direct I am trying to know how the algebra bundles come about as a twist in $\sigma$ models. I am sure my question is not well posed, as usual feel free to edit it.</p> | g10921 | [
-0.010738666169345379,
-0.0001901841751532629,
-0.006601485889405012,
-0.010571072809398174,
0.019949691370129585,
0.0037843147292733192,
0.04798423498868942,
0.03152184560894966,
-0.051292210817337036,
0.038563620299100876,
-0.026334576308727264,
0.0016095144674181938,
-0.007593441288918257... |
<p>From Michael on <a href="http://skeptics.stackexchange.com/questions/1664/it-is-not-the-voltage-that-kills-you-it-is-the-current/1677#1677">Skeptics Stackexchange</a>:</p>
<blockquote>
<p>How about a wire that's grounded?
Safe to touch, right? WRONG.</p>
<blockquote>
<pre><code> ________________ 30 amps -> ________________
| |
+ |
220V Load
- |
|______(YOU ARE HERE)______<- 30 amps________|
|
Ground
</code></pre>
</blockquote>
<p>The wire you touched was not only at 0
volts, but also grounded, and yet, you
are feeling pretty shitty in this
diagram. You have ceased to be as a
human, and you are now a part of a
circuit, functioning as part of a
return leg (pictured above) or as a
"parallel path to ground" (not
pictured above.)</p>
</blockquote>
<p>I don't get how this can work. If the wire is at 0 volts and you are at 0 volts, then there is no potential difference and hence I'd expect no current. Is this physics correct?</p> | g10922 | [
0.0833834856748581,
-0.009212740696966648,
-0.0027185429353266954,
-0.001962798647582531,
0.08910326659679413,
0.044862326234579086,
0.042961400002241135,
0.0517231822013855,
-0.006068244110792875,
-0.01580287329852581,
0.011859734542667866,
0.04027431830763817,
-0.03825491666793823,
-0.01... |
<p>Searching for this on google proved to be quite tedious, but I reckon that someone working with crystals a lot might know this off the top of his head:</p>
<p>Is there a good source that lists the Madelung constants for a variety of geometries? I'd be particularly interested in that for a NaCl-type 110 surface.</p> | g10923 | [
0.0054552797228097916,
0.049583639949560165,
-0.0050258818082511425,
-0.07577940821647644,
-0.017095480114221573,
-0.007269120309501886,
0.053510986268520355,
0.019997920840978622,
-0.03128259629011154,
0.04045792669057846,
-0.02890394814312458,
0.015306966379284859,
0.0487968884408474,
-0... |
<p>During solar system formation, many bodies achieved hydrostatic equilibrium, a spherical shape where their self gravitational force was balanced by internal pressure. Many also achieved differentiation, where a body is seperated into layers of different density (core, mantle and crust). Differentiation was mainly the result of the melting heat from isotopes like Al 26 and Fe 60, with half lives of ~10^6 years, coupled with the body's gravity. My question is, can differentiation occur without the presence of radioactive isotopes? Can a body's mass alone create sufficient gravitational force to induce melting, say by the attraction of meteoroids, and develop differentiation?
Of course stars are differentiated, but I'm interested in their orbiting bodies.</p> | g10924 | [
-0.013264372944831848,
0.021145030856132507,
-0.002697056159377098,
-0.033460456877946854,
0.0310579314827919,
0.028398683294653893,
-0.01584143191576004,
-0.023812243714928627,
0.010041054338216782,
-0.04147420451045036,
-0.04228448495268822,
0.012500915676355362,
0.07470410317182541,
-0.... |
<p>Can ${g}^{\mu\sigma}{\Gamma}^{\rho}_{\sigma\nu}$ be written as ${\Gamma}^{\mu\rho}_{\nu}$? If so how come this symbol never appears in any GR book?</p> | g10925 | [
-0.02144860476255417,
-0.025278661400079727,
0.018255967646837234,
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0.04874487966299057,
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0.004949497990310192,
0.019058415666222572,
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0.037928029894828796,
0.033371780067682266,
0.03848433867096901,
-0.00043310452019795775,
-0.... |
<p>Shlomo Sternberg (math professor at Harvard) wrote a book called "Group theory and physics". On p156 (<a href="http://books.google.com/books?id=k2Fp3JA93oYC&pg=PA156" rel="nofollow">link</a>) there's a strange offhand comment:</p>
<p>"Experiments done in 1964 by Fitch and Cronin seem to indicate that CP is not conserved. I do not fully understand the issues involved in the correct interpretation of this experiment, which clearly shows that CP and CPT are not <em>both</em> conserved. It follows from the locality axioms, that quantum field theory implies that CPT is a symmetry of nature, and hence that CP is violated in the Fitch-Cronin experiment. But other, group theoretical, hypotheses might favor CP. I have my own views on the subject, which I will not expand on in this book."</p>
<p>I cannot find any other publication where Prof. Sternberg elaborated his views on this. But he obviously suspects that CP is an exact symmetry while CPT is not.</p>
<p>So the question is: (1) Does the Fitch-Cronin experiment (or any other) disprove CP? Or does it only "show that CP and CPT are not <em>both</em> conserved"? (2) If the latter, is it remotely plausible that CPT is false? (How universal is the CPT theorem? Does it hold even in string theory?)</p> | g10926 | [
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<p>In <a href="http://en.wikipedia.org/wiki/Gravity" rel="nofollow">Wikipedia</a> it's stated that <em>"[..] gravity, is the natural phenomenon by which physical bodies appear to attract each other with a force proportional to their masses"</em>.</p>
<p>Then I found many examples regarding free fall and gravity, such as "<em>a hammer vs feather</em>", "<em>an elephant vs a mouse</em>", etc.
They all say that in the absence of another external force (like air-friction/resistance), the hammer, the elephant, the mouse and even the tiny feather, will all fall freely with the same speed and at a constant acceleration (like $1g$).</p>
<p>Ok, I think I can understand that, it makes sense.</p>
<p>Now my question regarding these "experiments":</p>
<ul>
<li>let's suppose that we have an object $E=$ Earth (with its mass ~$6 \times 10^{24}$ $kg$)</li>
<li>let's suppose that I have other 3 distinct objects: $O_1=$ a planet with a mass $5.99 \times 10^{24}$kg, $O_2=$ an elephant and $O_3=$ feather.</li>
</ul>
<p>If we imagine that the all 3 objects are situated above the Earth at a distance $d= 250$ miles, and if we suppose that there is no air-resistance or any other external force to stop their free-fall, then what will happen with ours objects? Which will fall first, second, third?</p>
<p>My understanding is that the $O_1$'s inner force will fight to attract the Earth toward it. The Earth will do the same thing but, having a mass just "a bit" larger than $O_1$, the Earth will eventually win and, will eventually attract the $O_1$ toward it ("down" to Earth).
The same thing will happen with the elephant and with the mouse/feather, except that the elephant having a larger mass than the mouse (or feather), will fight just a bit more than the others and will decelerate its fall with $0.00000...1\%$ (let's say) comparing with the mouse/feather (which seems almost the same thing if we ignore few hundred decimals).</p>
<p>Am I completely wrong about this story ?</p> | g10927 | [
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<p>This is a follow-up question to "<a href="http://physics.stackexchange.com/questions/7922/in-quantum-mechanics-why-do-the-probabilities-of-the-possible-outcomes-of-a-meas">In QM, why do the probabilities ... add up to 1?</a>".</p>
<p>No actual measurement is perfect. While theorists may ignore this, experimenters know well enough that in many runs of a given experiment no outcome is obtained. (The efficiency of many real-world detectors is rather low.) This means that in order to make the probabilities of the possible outcomes of a measurement add up to 1, one discards (does not consider) all those experiments in which no outcome is obtained.</p>
<p>Quantum mechanics thus allows us to predict the probabilities of measurement outcomes <em>on condition that</em> there is an outcome. But is there anything in quantum mechanics — after all, the theoretical framework of contemporary physics — that allows us to predict that a measurement, which is about to be made, will have an outcome? Does quantum mechanics allow us to formulate causally sufficient conditions for the occurrence of an outcome (no matter which) or the success of a measurement?</p> | g10928 | [
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