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<p>I fear my textbook is teaching an incorrect concept of <a href="http://en.wikipedia.org/wiki/Work_%28physics%29" rel="nofollow">Work</a>. I am very frustrated right now since I was struggling to understand the concept in the way that was explained in the textbook and my instincts tell me it is wrong. Also note that this is a online highschool course, so the textbooks are not published per se and it is common for these courses to be incorrect.</p>
<p><strong>background information:</strong></p>
<p>Here is the definition of work given in the textbook:</p>
<p><code>We say that work is done on an object when a force that is exerted on it causes a displacement.</code></p>
<hr>
<p>I'll give two examples here as I consider both to be incorrect.</p>
<p><strong>Textbook Example 1:</strong></p>
<p>If a 10 N force causes an object to be displaced 5.0 m, what is the work done on the object?</p>
<p><strong>Textbook Answer:</strong> </p>
<p>W = Fd</p>
<p>W = 10 N x 5.0 m</p>
<p>W = 50 Nm</p>
<p>W = 50 J</p>
<hr>
<p><strong>Textbook Example 2:</strong></p>
<p>A 0.0782 kg hockey puck was shot with an acceleration of 2.8 m/s2 over extremely fast ice (we will assume that it is frictionless). How much work was done when the puck passed the 12.0 m mark and the 112.0 m mark? </p>
<p><strong>Textbook Answer:</strong></p>
<p>In order to determine the work done, we need to know the distance traveled (which we do) and the force (which we do not). However, because we know the mass of the puck and its acceleration, then we can determine the force and then, use this force to determine the work.</p>
<p>Fnet = ma</p>
<p>Fapplied = 0.0782 kg x 2.8 m/s2</p>
<p>F = 0.218 96 kg/m/s2</p>
<p>F = 0.218 96 N</p>
<p>F = 0.22 N</p>
<p>Now we know the force exerted on the puck, we can now determine the work done on the puck as it passes the 12.0 m mark.</p>
<p>W = Fd</p>
<p>W = 0.22 N x 12.0 m</p>
<p>W = 2.64 Nm</p>
<p>W = 2.6 J</p>
<p>We can also determine the work done on the puck as it passes the 112.0 m mark.</p>
<p>W = Fd</p>
<p>W = 0.22 N x 112.0 m</p>
<p>W = 24.64 Nm</p>
<p>W = 25 J</p>
<p>Notice that the distance the puck travels affects the amount of work done! </p>
<hr>
<p><strong>Why I think these are incorrect:</strong></p>
<p>I believe these answers are incorrect because of what the unit Joules means to me. Newton meters would appear to be a meaningless unit of the distance in the calculation was the distance traveled by the object since friction is not considered and any distance might be possible that being the case. As well, it means that, as shown in the hockey puck problem, the resulting joules may be found to be thousands of Joules because the object is sliding on a frictionless surface, but there is no more a transfer of energy--and work IS defined as a transfer of energy. In the puck problem it doesn't make sense for the resulting Joules to be more after a longer distance since there is no more energy transferred. Once the puck is accelerated, it will remain at a constant velocity and its kinetic energy will be constant (since it is a frictionless surface). Therefore, the Joules of energy transfer should remain the same after the acceleration.</p>
<p>I think that Work should really be defined as: <strong>the distance the force was exerted over</strong> as opposed to <strong>the distance the object traveled</strong>. Please correct me if I'm wrong. If I'm right, I will send the link of this question to the head of the science department or school as it is a huge oversight to teach a completely wrong concept to students (including myself).</p> | 3,180 | [
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<p>I recently remembered that someone worked out <a href="http://faculty.washington.edu/jcramer/BBSound_2013.html" rel="nofollow">what the big bang sounded like</a> and that got me thinking...</p>
<p>About 377,000 years after the Big Bang, electrons became bound to nuclei to form neutral atoms. Because of (?) this, the mean free path of photons became effectively infinite, i.e. the universe became transparent to radiation.</p>
<p><strong>What would this have <em>looked</em> like?</strong></p>
<p>More precisely I could ask: did the sky suddenly become dark, or was the amount of radiation basically same after as before?</p>
<p>What would the distribution and timescale have been like? In a perfectly homogeneous universe it would happen at the same rate everywhere. Did the universe have any structure at this point? Would you have been able to see blotches of lighter and darker patches of the sky (assuming you were within one of the more transparent patches), these being proto-shapes of galactic filaments perhaps, or the noisy grit of the CMB? Or would it just be a vague cloud at any scale, and in any part of the spectrum?</p> | 3,181 | [
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<p>How did Nima Arkani-Hamed come up with the following nonzero space-like commutation relation in string theory?</p>
<p>$$\left\langle \left| \left[ \hat{\mathcal{O}}(\vec{x}), \hat{\mathcal{O}}(\vec{y}) \right] \right|^2 \right\rangle \sim \exp \left( -\frac{k\left| \vec{x} - \vec{y} \right|^{D-2}}{8\pi G} \right)$$</p>
<p>His explanation isn't too clear.</p> | 3,182 | [
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<p>In free space, the linear momentum density of an EM wave is given by the Poynting vector $\vec S$ over the speed of light squared, $\vec g=\frac{\vec S}{c^2}$.</p>
<p>In a medium, $S$ is generally not directed along the wave vector $\vec k$. If $v_g$ is the group velocity and $W$ is the density of EM wave energy, then $S = v_g W$. On the other hand, it is often assumed that the momentum of a photon is still $\hbar\vec k$. What is the macroscopic momentum density of EM wave in an anisotropic medium such as magnetized plasma ?</p> | 3,183 | [
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<h3>Cost per Power Capacity</h3>
<p>For a number of energy technologies I'd like to know what the minimum costs are to install a given power capacity. Are there any such comparisons available?</p>
<h3>Research Spendings</h3>
<p>Technologies often own much to publicly funded research efforts so I'm also interested in estimations regarding development cost of the technologies themselves. Of course foundations to a given technology may have been readily available, so let's limit this to spendings on larger research projects <em>dedicated</em> to developing a technology and motivated by the need of energy. I'm aware it's unlikely any exhaustive comparison in full detail exists but maybe something related?</p>
<p>(Maybe someone should tag this question 'energy-economics' -- I'm lacking the reputation)</p> | 3,184 | [
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<p>My 8-year-old daughter's school report says that she's good at understanding the basic science she's doing, but she's having trouble seeing how experimental results lead to conclusions. Specifically, it says she struggles to appreciate how changing parameters in an experiment can be used to prove or rule out a hypothesis.</p>
<p>I'm a biologist/chemist by degree, and I found it hard to think of any parameter-based experiments in my field that we could do at home that wouldn't bore a child. They like creepy-crawlies well enough, but biology experiments tend to need time and repeat observations to get a result. Chemistry tends to need more specialist equipment and reagents.</p>
<p>I've got a couple of books of home science, and I scoured the net as well, but most of the living-room science experiments I found that were suitable for an 8-year-old were more demonstrations than actual experiments. There were few parameters: nothing you could really vary.</p>
<p>So can any of your learned people suggest some simple, parameter-based physics experiments we could do at home or in the garden, without specialist equipment, which involve science at a level an eight-year-old could engage with?</p> | 3,185 | [
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<p>Imagine a variation on the double slit experiment. I'll describe it in 2D using the $x-y$ plane. The $x$-axis is impenetrable other than the two slits, which are positioned at $(-1,0)$ and $(+1,0)$. Detectors will be positioned in the upper half plane, say along $y=10$ or something. Here's the twist: the negative $y$-axis is impenetrable as well. Two sources will be needed, say one at $(+1,-\infty)$ and the other at $(-1,\infty)$. These sources create plane waves impinging on the two slits.</p>
<p>The basic question: will one observe simple interference at the detectors?</p>
<p>For photons, I think the answer is obviously yes. (Maxwell's equations in free space are linear and there's no problem "superposing" or simply summing the contributions of each of the slits.)</p>
<p>But what if one uses electrons as the source? The wall separating the two sources clearly seems to preclude a single electron from taking both paths to the destination, as would be the case in the usual setup. That suggests you'll have to have more than one electron at a time involved to achieve interference. But electrons interact electromagnetically, and that interaction may wreck any interference pattern. In contrast, photons don't interact, so simple interference occurs there.</p>
<p>Alternatively, one might think that interference will still occur with single electrons, since there is no way to determine which source the electron came from. Intuitively, I lean toward this answer being correct, but if it is, how would you understand the underlying Hilbert space and Hamiltonian of such a system? I don't think it can possibly be understood as a single electron system with a single wave function throughout space.</p>
<p>Or can it?</p> | 3,186 | [
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<p>In the linear sigma model, the Lagrangian is given by </p>
<p>$$ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) +\frac{1}{2}\mu^2\sum_{i=1}^{N}\left(\phi^i\right)^2-\frac{\lambda}{4}\left(\sum_{i=1}^{N}\left(\phi^i\right)^2\right)^2 \tag{11.65} $$
(for example see Peskin & Schroeder page 349).</p>
<p>When perturbatively computing the effective action for this Lagrangian the derivative $ \frac{\delta^2\mathcal{L}}{\delta\phi^k(x)\delta\phi^l(x)} $ needs to be computed. (for instance, Eq. (11.67) in P&S): </p>
<p>$$
\frac{\delta^2\mathcal{L}}{\delta\phi^k(x)\delta\phi^l(x)} ~=~ -\partial^2\delta^{kl} +\mu^2\delta^{kl}-\lambda\left[\phi^i\phi^i\delta^{kl}+2\phi^k\phi^l\right].\tag{11.67}$$</p>
<p>My question is, how is one supposed to handle the derivative term?</p>
<p>This seems to be completely implicit in the presentation of P&S, but from what I could gather it should go like so:</p>
<p>1) Because we are computing the effective action, $\mathcal{L}$ is actually under an integral and we can replace $\left(\partial_\mu\phi^i\right)\left(\partial_\mu\phi^i\right)$ with $-\left(\partial^\mu\partial_\mu\phi^i\right)\phi^i=-\left(\partial^2\phi^i\right)\phi^i$ using Stokes' theorem.</p>
<p>2) Then when performing the first derivative I get $\frac{\delta}{\delta\phi^l}\left[-\left(\partial^2\phi^i\right)\phi^i\right]=-\partial^2\phi^l$.</p>
<p>3) It is the second derivative I get stuck at, for as far as I can see, $\frac{\delta}{\delta\phi^k}\left[-\partial^2\phi^l\right]=0$, for there is only dependence on the 2nd derivative of $\phi^l$ and not $\phi^l$ itself. If, as is usual in field theory, the field and its derivatives are treated as independent dynamical variables, then the second derivative should also be an independent dynamical variable. How is it explained then, that the result of this computation should be $\frac{\delta}{\delta\phi^k}\left[-\partial^2\phi^l\right]=-\delta^{kl}\partial^2$?</p> | 3,187 | [
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<p>I know that for parallel RLC circuits, the $Q$ factor is given by: </p>
<p>$$ Q = R \sqrt {\frac{L}{C}} $$</p>
<p>But now suppose it is connected in series to a resistor $R_2$ and capacitor $C_2$. Would the $Q$ factor be changed?</p>
<p><img src="http://i.stack.imgur.com/dpLzs.png" alt="enter image description here"></p> | 3,188 | [
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<p>The mass of air bubble in any medium is considered as negative. Is the air bubble is massless. I m in confusion. can we not neglect the mass of air bubble in material medium. But i have found in many books the mass of air bubble in material medium as negative. please discuss.....</p> | 3,189 | [
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<p>I put a nucleus in a magnetic field. It spins. Does the electric charge distribution remain homogeneous or does the charge redistribute? Can this be measured? Will accept reference as answer.</p> | 3,190 | [
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<p>There are a lot of different models of inflation in cosmology, with plenty of different features and issues.</p>
<p>I read that the problem with some models is that they introduce a never-ending inflation, but I can't find an example of it.</p>
<p>Anyway, I thought that indefinite inflation could only happen in a true de-Sitter space, while if we have a quasi-de-Sitter space, it would always be an end at some point. Eventually after a very long stage of inflation, but still - there is an end.</p>
<p>So, am I correct? Is the problem of never ending inflation a tough one? Is it a problem that often arise?</p> | 3,191 | [
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<p>Newton's theory of gravity supports "gravity waves" in that moving objects cause changing gravitational fields. For example, two bodies rotating around their center of mass will have a stronger gravitational field when they are longitudinally oriented than when they are transverse oriented. Given two masses of mass $M$ orbiting on a circle of radius $r$, at a distance $d$ from an observer, the strength of the attraction is:
$$\begin{matrix}
F_{min} &=& \frac{2GM}{d^2},\\
F_{max} &=& \frac{2GM}{d^2}\frac{1+r^2/d^2}{(1-r^2/d^2)^2}
\end{matrix}$$</p>
<p><img src="http://i.stack.imgur.com/dIjHy.jpg" alt="enter image description here"></p>
<p>This should be detectable at long distance.</p>
<p>My question is this: With that sort of gravity, would the laser interferometer based gravity wave detectors be able to detect the gravity wave? An example is the <a href="http://en.wikipedia.org/wiki/LIGO" rel="nofollow">LIGO, Laster Interferometer Gravitational-wave Observatory</a>.</p> | 3,192 | [
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<p>I have trouble imagining how two point-particles can have different properties.</p>
<p>And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume?</p>
<p>Not only that, but it can also detect all fields without having any structure.
Maybe it can check curvature of spacetime to account for gravity, but how can a point contain the information of what the other fields-vectors are? This seems to mean that also the information/volume in space is infinite.</p>
<p>Mathematically, a point cant have any intrinsic structure, so how does physics which is a mathematical theory explain this?</p> | 3,193 | [
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<p>I just learnt what absorption and emission spectrum are. And our teacher introduced us to what <a href="https://en.wikipedia.org/wiki/Quantum_dot" rel="nofollow">Quantum Dots</a> are. And showed us their absorption and emission spectra.
And they look something like this
<img src="http://i.stack.imgur.com/CuGuK.gif" alt="enter image description here"></p>
<p>I understand that the peaks of the spectra is supposed to match the transition energy in the QD. But why does the absorbance increase as the wavelength decreases? I am not aware of any QD transition energy at that wavelength. And this seems to be true for all the QD spectra I googled. Is it dependent on other substances in the solution or the film?</p>
<p>Please correct if I am mistaken.</p> | 3,194 | [
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<p>Some of the major challenges that heralded the need for quantum mechanics we're explaining the photo-electric effect, the double-slit experiment, and electrons behavior in semi conductors.</p>
<ol>
<li><p>What are some of the predictions we can expect to see from a theory of quantum gravity?</p></li>
<li><p>What types of experiments have shown the necessity for a quantum gravity theory?</p></li>
</ol> | 3,195 | [
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<p>I am not professional physicist; but I am curious about Stephen Hawking's "imaginary time". It would be better to elaborate exactly what it is. I am not confused because of the word "imaginary" but I find it confusing to imagine a two dimensional "plane time". If we express time in a plane instead of a one dimensional axis, then what does the movement of an observer along the imaginary axis signify physically?
</p> | 77 | [
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<p>If a plane is flying at a constant speed at some altitude like 5-6 km and it releases a bomb:</p>
<ul>
<li>does the bomb move forward at the same horizontal speed as the airplane?</li>
<li>or does its horizontal speed decrease due to drag and no thrust to push it forward?</li>
<li>or does it outrun the airplane because it's more aerodynamic than the airplane and the gravity is adding to its horizontal speed?</li>
<li>or does it depend on the bomb? If so, what would be the typical scenario?</li>
</ul>
<p>I looked at a demonstration on Wolfram Alpha (<a href="http://demonstrations.wolfram.com/TrajectoryOfABomb/" rel="nofollow">http://demonstrations.wolfram.com/TrajectoryOfABomb/</a> - requires a plugin) and it looks like the bomb moves forward at the exact same speed as the plane, and I don't know if that's accurate.</p> | 3,196 | [
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<p>In my laboratory, I have already obtained a polarization-entangled photon source. It was created via the Spontaneous Parametric Downconversion process of 2 BBO crystals. The next question is that would I be able to use this source to create qubits? If so, how?</p>
<p>Thank you in advance.</p> | 3,197 | [
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<p>I'm having some trouble of getting a grasp of this term, so I hoped someone could enlighten me.</p>
<p>As far as I understand, the idea behind it is, that cells have different cycles in their "life", and in each cycle there is different phases where the radiosensitivity is different compared to the others. Then, by using radiation you can sync the cycles of tumour cells for example, so they start in the same phase, then you wait until you reach a good radiosensitive phase, and then irradiate again.
But unfortunately because cells are so different, the speed of which they go through different phases are not the same, so therefore it can be difficult to sync the phases as wished.</p>
<p>That is how I currently understand it. Is that totally wrong, or...?</p> | 3,198 | [
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<p>I have to give a 10 minute physics talk that have to involve a fair bit of mathematics -- i.e. not just qualitative/handwaving material to some undergrads. I have wasted the last 3 hours looking for appropriate topics and have pretty much lost the will to live. Any suggestions would be greatly appreciated.</p> | 3,199 | [
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<p>I understand an electric quadrupole moment is forbidden in the standard electron theory. In <a href="http://arxiv.org/abs/gr-qc/0412064v3">this paper</a> considering general relativistic corrections (Kerr-Newman metric around the electron), however, there is a claim that it could be on the order of $Q=-124 \, \mathrm{eb}$. That seems crazy large to me, but I can't find any published upper limits to refute it. Surely someone has tested this? Maybe it's hidden in some dipole moment data? If not, is anyone planning to measure it soon?</p> | 3,200 | [
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<p>The question inspired by an upcoming <a href="http://physics.berkeley.edu/index.php?option=com_dept_management&act=events&Itemid=444&task=view&id=1235">colloquim at UCB</a>.</p>
<p>A naive interpretation of quark asymptotic freedom seems to imply that at high enough energies they should be weakly interacting. On the other hand, the quark-gluon plasma accessible experimentally via heavy ion collisions is claimed to be close to an ideal liquid. There is also a lot of excitement about this because of AdS/CFT correspondence (see <a href="http://physics.stackexchange.com/q/2026/1739">this</a> and related questions at Physics.SE). </p>
<p>Shouldn't the quark-gluon matter phase diagram be close to an ideal gas at least in some
low-density & high-temperature region?
In other words, is quasi-independent quark gas picture of QCD ever valid? And is there a simple reason why?</p> | 3,201 | [
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<p>What does it mean to say that mass "approaches infinity"?</p>
<p>I have read that mass of a body increases with the speed and when the body reaches the speed of light, the mass becomes infinity. </p>
<p>What exactly does it mean to say that the mass "approaches infinity" or "becomes infinity"? I am not able to get a picture of "infinite mass" in my mind.</p> | 3,202 | [
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<p>In the double slit interference pattern for the wave of an electron, what will happen if I make the slits to be smaller than the size of an electron ? Will I still observe an interference pattern on the opposite side of the screen or no electron will be able to cross the slit? </p>
<p>If no, then how can a quantum prisoner escape ?</p> | 3,203 | [
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<p>Or more specifically, does the CMB radiation have an observer effect on us in our local system due to gravitational lensing? Acknowledging this effect, if any would be nearly negligible I have an also: would this same effect be "felt" in the void spaces between galaxies?</p> | 3,204 | [
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<p>This paper describes black holes as space flowing inward (the rotating hole also twists in a weird way):</p>
<p><a href="http://arxiv.org/abs/gr-qc/0411060" rel="nofollow">http://arxiv.org/abs/gr-qc/0411060</a></p>
<p>The proper time given by the objects is the same as special relativity except for that fact that they are on a treadmill:</p>
<p>$$ds^2 = dt^2-|v - w|^2$$</p>
<p>where t is "time" in the sense that the (static) geometry is translationally invariant along t, v is the velocity of you, w is the velocity of space. w is a function of the coordinates. The rotating black hole adds a "twist" term that I don't understand well; space itself doesn't rotate!</p>
<p>Is it possible to describe arbitrary spacetimes (i.e. two unequal mass, unequal spin neutron stars colliding to become a black hole) in this fashion? Of course, there may be more than one right answer as you no longer have time translational invariance. What kind of coordinates singularities/caustics would such a description entail? </p> | 3,205 | [
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<p>How does the uncertainty principle limit the accuracy of atomic clocks. I know line width and measurement time are important but not exactly why? </p> | 3,206 | [
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<p>Layman here,</p>
<p>Stumbling through some physics stack posts and started reading the Wikipedia for the <a href="http://en.wikipedia.org/wiki/Timeline_of_the_Big_Bang">chronology of the big bang</a>. In it, it states</p>
<blockquote>
<p>The very earliest universe was so hot, or energetic, that initially no
matter particles existed or could exist except perhaps fleetingly, and
the forces we see around us today were believed to be merged into one
unified force. Space-time itself expanded during an inflationary epoch
due to the immensity of the energies involved. Gradually the immense
energies cooled – still to a temperature inconceivably hot compared to
any we see around us now, but sufficiently to allow forces to
gradually undergo symmetry breaking, a kind of repeated condensation
from one status quo to another, leading finally to the separation of
the strong force from the electroweak force and the first particles.</p>
</blockquote>
<p>Where is the "immense energy" going to when it is "cooled"? Is there now no "immense energy"?</p> | 801 | [
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<p>During its formation, the Sun contracted under the force of its own gravity, until hydrostatic equilibrium was established. In this contraction, its temperature increased from $3\times10^4\text{ K}$ to $6\times10^6\text{ K}$. How to find the energy radiated during this contraction?</p> | 3,207 | [
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<p>Is gauge pressure <a href="http://en.wikipedia.org/wiki/Pressure_measurement#Absolute.2C_gauge_and_differential_pressures_-_zero_reference" rel="nofollow">always zero-referenced against ambient air pressure?</a> Or is it referenced against the sum of all pressures acting on a fluid, which usually just happens to be ambient air pressure?</p>
<p>To clarify, here is a potential question: Imagine a container with two immiscible fluids of different densities. 1atm of atmospheric pressure is acting at the surface of the first fluid. An object at the deepest point of the first fluid has a gauge pressure of 3atm, so the absolute pressure at this depth is 4atm. If the absolute pressure of the deepest point of the second fluid is 8atm, what would the gauge pressure of the object be if it was submerged to the deepest point of the second fluid?</p>
<p>If gauge pressure is always zero-referenced to air pressure, the gauge pressure would be 8atm - 1atm = 7atm. However, if it accounts for the sum of the pressure from all the fluids above it, then the gauge pressure would be 8atm - 1atm - 3atm = 4atm. Which is the correct gauge pressure?</p> | 3,208 | [
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<p>When a short circuit occurs it's obvious that there is fire. How come electric energy turns out to be heat energy? What causes the conductors to get hot when short circuit is present.</p> | 3,209 | [
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<p>I'm currently learning the mathematical framework for General Relativity, and I'm trying to prove that the Lie derivative of the Riemann curvature tensor is zero along a killing vector.</p>
<p>With the following notation for covariant differentiation, $A_{a||b} $ (instead of $\nabla_b A_a$ ), I have the following:</p>
<p>$\it\unicode{xA3}_\xi R_{amsq} = R_{amsq||x} \xi ^x + R_{xmsq} \xi^x{}_{||a} + R_{axsq} \xi^x{}_{||m} + R_{amxq} \xi^x{}_{||s} + R_{amsx} \xi^x{}_{||q}$.</p>
<p>I suspect that I need to invoke the second Bianchi identity. However, before I can do this, I need to somehow get this into a different form. There has to be some property of either killing vectors or maybe covariant derivatives that I'm forgetting/failed to learn. Any help would be appreciated.</p> | 3,210 | [
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<p>Yes, this is a homework question, but I've already failed to solve it enough times that the online system hosting it isn't going to give me any marks, so I figure it's a good time to stop hitting the wall and actually ask. The question is as below:</p>
<blockquote>
<p>A spectroscopist uses a spectrometer that has a grating with 600 grooves/mm. This grating can illuminate a CCD with a range of 1228 nm, in the 5th order (m = 5). One day, she buys a larger format CCD -- one larger than her old one by a factor of 3. What spectral range of wavelengths can be measured when she now obscures the 4th order lines (m = 4) with her larger CCD?</p>
</blockquote>
<p>Came out with answers like $1.228 \times 10^{-6}$ and $4.17 \times 10^{-7}$ so far because I have no idea what the process is. Hints would suffice.</p>
<p>Basically, everything I've tried so far revolves around a formula: $m\lambda=d\sin\theta$.</p>
<p>What did I do so far:</p>
<ul>
<li>$5(1.228\times10^{-9})=\frac{0.001}{600}\times\sin\theta$ to get $\sin\theta$ and then put it back in to replace $\sin\theta$ as $\frac{4}{5}\sin\theta$. Then I went and searched up a diffraction grating to find out that the orders are not equidistant.</li>
<li>$5(1.228\times10^{-9})=\frac{0.001}{600}\times\sin 90^\circ$ where I assumed the angle is perpendicular.</li>
<li>Other stuff I can't remember</li>
</ul>
<p>I feel like I understood the wavelength portion of this incorrectly, but I have no idea why.</p> | 3,211 | [
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] |
<p>In this paper about <a href="http://dx.doi.org/10.1109/IROS.2006.282433" rel="nofollow">Backstepping controll of a quadrotor helicopter</a> an algorithm for control is described, but I have hit a dead end.</p>
<p>In equation 15 it is described the part of state space for the angular and translation motion of a rigid body.</p>
<p><img src="http://i.stack.imgur.com/MSmYq.png" alt="Equation 15"></p>
<p>The author states $R_r$ is the rotation velocity matrix between Earth fixed reference frame and Body fixed reference frame. I assume the "between" means a rotation from body coordinates to earth coordinates.</p>
<p>$R_t$ is the translation velocity matrix between Earth fixed reference frame and Body fixed reference frame. I also assume the "between" means a rotation from body coordinates to earth coordinates.</p>
<p>One question is how do I calculate the value of the angular acceleration described by the partial derivative of $\dot{\phi}$ and $\dot{\theta}$. The author does not state and I would like to know if it is numerically or if it is analytically possible.</p>
<p>The other question is in which referential is $\dot{\zeta}$ and why did the author make a rotation and "derotation" on $Kt$? From the paper $G$ is a vector with the $z$ element set to $g=9.81$.</p>
<p>Last question, more like a curiosity, would anybody give me a pointer to state-space formulation? I do not follow how the author composed the state space system.</p> | 3,212 | [
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<p>I was watching a movie. A spaceship was forced into "warp speed". The co-ordinates could not be set. The spaceships trajectory was that of a nearby sun. Forcing the spaceship to power down was the solution. Now out of "warp speed" and with no computer aid (steering etc) the spaceship was seen to be spinning toward the sun trapped in its gravitation field.
My question is, would the spaceship (typically aerodynamically shaped) spin toward the surface? My opinion is no. The spaceship would just fall flat due to the surface area provided at the bottom of the fuselage </p> | 3,213 | [
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<p>I'm very new the topic of SPPs and have been trying to understand this particular method of exciting surface plasmons using a 1D periodic grating of grooves, with distance $a$ between each groove. If the light incident on the grating is at an angle $\theta$ from the normal and has wavevector ${\bf k}$, then apparently if this condition is met:</p>
<p>$\beta = k \sin\theta \pm\nu g$</p>
<p>where $\beta$ is corresponding SPP wavevector, $g$ is the lattice constant $2 \pi/a$ and $\nu={1,2,3,...}$ then SPP excitation is possible.</p>
<p>I haven't ever really had a formal course in optics, so my question is where this condition comes from. It seems like Fraunhofer diffraction, but only for the light being diffracted at a $90^\circ$ angle to the normal. Most books don't state how they get this result, they just say it's because of the grating "roughness" which really confuses me.</p>
<p>Any help would be greatly appreciated.</p> | 3,214 | [
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<p><strong>Intro:</strong> Few hours ago, there was a storm. We heard some constant banging which couldn't be explained by thundering. Then we found out, it was a sewer lid jumping. Maybe it's normal in other parts of the world, but for me it was like the first time in my life.</p>
<p>I've captured <a href="http://youtu.be/v88wnXY0UjM" rel="nofollow">the video</a>.</p>
<p><strong>The question</strong> is what was causing this to happen. It's kind of clear that it was air pressure so strong that it was capable of lifting this metal lid. But where did this air pressure appear? </p>
<ul>
<li>Wind blowing into the sewer? If so, would it be so strong to lift the lid? And shouldn't be the sewers protected against wind somehow?</li>
<li>Water filling the sewer so quickly, that it made the air pressure this strong? Would this really be the easiest way for the air to leave the sewer?</li>
<li>Something else (such as bored sewer worked :) )?</li>
</ul> | 3,215 | [
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<p>What determines what pitch an object such as a bell or tuning fork produces when struck? I have heard that the box in the "king's chamber" of the great pyramid at Giza is tuned to 438 Hz. I know that in hand-bell choirs, the bigger the bell, the lower the tone, but I have noticed that size does not seem to be the determining factor in a bell's tone.</p> | 3,216 | [
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<p>From what I understand, an object entering the atmosphere will start to burn up from the tremendous resistance of the atmosphere. Presumably, for asteroids under a certain size, they will burn up completely and never impact the surface of the earth. </p>
<p>Do we have a way of determining the minimum size needed for actual impact? </p>
<p>If so, roughly what is the size and how does it compare to the average size of asteroids that pass by us regularly?</p> | 3,217 | [
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<p>My Question is how to explicitly move into the "Coulomb gauge" in Yang-Mills theory.</p>
<p>Using the answer provided by QMechanic, one can move into the "temporal gauge" for Yang-Mills fields: <a href="http://physics.stackexchange.com/questions/33133/gauge-fixing-choice-for-the-gauge-field-a-0">Gauge fixing choice for the gauge field $A_0$</a> .</p>
<p>Once in this gauge, to move into the "Coulomb gauge," does one take an explicitly <em>time independent</em> $g$, and demand that it satisfy:
$$
i\partial^\mu (g^{-1}(\partial_\mu - i A_{\mu})g) = 0
$$ </p>
<p>Thus giving $\partial ^{\mu} A'_{\mu} = 0$? Are we guaranteed that such a $g$ exists? Thanks.</p> | 3,218 | [
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<p>I have been studying decoherence in quantum mechanics (not in qft, and don't know how it is described there) and renormalization in QFT and statistical field theory, I found at first a similarity between the two procedures: on one side decoherence tells us to trace over the degrees of freedom we don't monitor, in some way intrinsically unknown to recover a classical picture picture from quantum mechanics, on the other side by renormalizing we also integrate over "our ignorance" but this time, the U.V. physics or high energy modes to get the infra-red physics we observe. Beyond the technical similarity (taking a trace, for discrete Kadanoff-Wilson transformations) it feels that in both cases we are forced to do these procedures because we start from a wrong picture where we separate the free object (purely quantum in the first case, with bare parameters in the second one) and then calculate the effects of the interactions, that are responsible from what we, observers, see, classical and infra-red physics. This where it comes to me that some interested links between the two concepts may exist or be pointed out (and also wonder what decoherence becomes in QFT).</p>
<p>I still see one huge asymmetry between the two, decoherence is dynamical, it has a typical time of decay, where renormalization is static.</p>
<p>I hope I could explain my interrogation clearly, and some interesting comments will come.</p> | 3,219 | [
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<p>I don't understand why when first launched Space X's Dragon capsule had to orbit the Earth many times in order to match up with the <a href="http://en.wikipedia.org/wiki/International_Space_Station">ISS</a>? Was this purely to match it's speed, or to get closer (as in altitude) to the ISS?</p>
<p>In the stages when it gets to about 200m, it seemed like it was able to go directly up to the ISS, how come it couldn't do that the entire way.</p>
<p>(Additionally, in Scifi movies you see smaller shuttles able to go directly to space stations in orbit, is that type if travel not possible?)</p> | 3,220 | [
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<p>I've seen questions on how small you can make a tokamak. But I haven't yet seen any "physical" upper limit on the tokamak design.</p>
<p>If you take a wind turbine for example, doubling the linear dimensions will increase the sweep area by a factor 4 but the structural mass with a factor 8, which clearly explains why you don't want to make (convensional) wind turbines above certain dimensions.</p>
<p>With a tokamak, I imagine that if you double the linear dimensions, the plasma volume (and hence the power production) will increase eightfold, whereas area that you have to protect against fast neutrons will only quadruple. So once you master the tokamak technology, you would only need to scale it up appropriately to bring down capital costs.</p>
<p>What do I miss? What cannot be scaled up easily in a tokamak?</p> | 3,221 | [
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<p>Can someone write down the <a href="http://en.wikipedia.org/wiki/Boltzmann_equation" rel="nofollow">Boltzmann equation</a>, not neglecting any of the variables of the involved functions and integrals? Specifically, how to concisely capture the "primed" variables in a sensible manner?</p> | 3,222 | [
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<p>The Pressure of a static spherical object (say star), which has the Schwarzchild metric outside it, satisfies the following differential equation called the <a href="http://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" rel="nofollow">TOV equation</a>.</p>
<p>$$\frac{\mbox{d}P}{\mbox{d}r}=-\left(P+\rho\right)\frac{m(r)+4\pi r^3P }{r\left(r-2m(r)\right)}$$ </p>
<p>Here $\rho$ the mass density is a function of $r$, and $m(r)=4 \pi\int_0^r\rho(r^{\prime}){r^{\prime}}^2 d{r^{\prime}} $. </p>
<p>How do I solve this equation? Wald states the solution as follows; but Please could someone suggest <strong>how I solve this analytically?</strong></p>
<p>($\rho_0$ is the density at the radius of the star $R$, and $M$ is the total mass $M=4 \pi\int_0^R\rho(r^{\prime}){r^{\prime}}^2 d{r^{\prime}} $)</p>
<p>$$P(r)=\rho_0\left(\frac{\left(1-\frac {2M}{R}\right)^\frac12 - \left(1-\frac {2Mr^2 }{R^3 }\right)^\frac12 }{\left(1-\frac {2Mr^2 }{R^3 }\right)^\frac12 -3\left(1-\frac{2M}{R}\right)^\frac12 }\right)$$ </p> | 3,223 | [
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<p>I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I don't understand is: Why are the $\mu B$ parts not diagonal? If the Hamiltonian is $\vec{\mu} \cdot \vec{B}$, why aren't all the components just diagonal? How is this matrix built systematically? Can someone please explain?</p>
<blockquote>
<p>We now consider the effect of a uniform magnetic field $\mathbf{B} = B\hat{z}$ on the hyperfine levels of the ${}^2 S_{1/2}$ ground state of hydrogen. Initially, we will neglect the effect of the nuclear (proton) magnetic moment. The energy eigenstates for the Hamiltonian describing the hyperfine interaction are also eigenstates of the operators $\{F^2, F_z, I^2, S^2\}$. Therefor if we write out a matrix for the hyperfine Hamiltonian $H_\text{hf}$ in the coupled basis $\lvert Fm_F\rangle$, it is diagonal. However, the Hamiltonian $H_B$ for the interaction of the magnetic moment of the electron with the external magnetic field,</p>
<p>$$H_B = -\mathbf{\mu}_e\cdot\mathbf{B} = 2\mu_B B S_z/\hbar,\tag{4.20}$$</p>
<p>is diagonal in the uncoupled basis $\lvert(SI)m_S, m_I\rangle$, made up of eigenstates of the operators $\{I^2, I_z, S^2, S_z\}$. We can write the matrix elements of the Hamiltonian in the coupled basis by relating the uncoupled to the coupled basis. (We could also carry out the analysis in the uncoupled basis, if we so chose.)</p>
<p>The relationship between the coupled $\lvert Fm_F\rangle$ and uncoupled $\lvert(SI)m_Sm_I\rangle$ bases (see the discussion of the Clebsch-Gordan expansions in Chapter 3) is</p>
<p>$$\begin{align}
\lvert 1,1\rangle &= \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2}\tfrac{1}{2}\rangle,\tag{4.21a} \\
\lvert 1,0\rangle &= \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle + \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21b} \\
\lvert 1,-1\rangle &= \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2},-\tfrac{1}{2}\rangle,\tag{4.21c} \\
\lvert 0,0\rangle &= \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle - \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21d}
\end{align}$$</p>
<p>Employing the hyperfine energy shift formula (2.28) and Eq. (4.20), one finds for the matrix of the overall Hamiltonian $H_\text{hf} + H_B$ in the coupled basis</p>
<p>$$H = \begin{pmatrix}
\frac{A}{4} + \mu_B B & 0 & 0 & 0 \\
0 & \frac{A}{4} - \mu_B B & 0 & 0 \\
0 & 0 & \frac{A}{4} & \mu_B B \\
0 & 0 & \mu_B B & -\frac{3A}{4}
\end{pmatrix},\tag{4.22}$$</p>
<p>where we order the states $(\lvert 1,1\rangle, \lvert 1,-1\rangle, \lvert 1,0\rangle, \lvert 0,0\rangle)$.</p>
</blockquote>
<p>And for Eq. (2.28) the other part is</p>
<blockquote>
<p>$$\Delta E_F = \frac{1}{2}AK + B\frac{\frac{3}{2}K(K + 1) - 2I(I + 1)J(J + 1)}{2I(2I - 1)2J(2J - 1)},\tag{2.28}$$</p>
<p>where $K = F(F + 1) - I(I + 1) - J(J + 1)$. Here the constants $A$ and $B$ characterize the strengths of the magnetic-dipole and the electric-quadrupole interaction, respectively. $B$ is zero unless $I$ and $J$ are both greater than $1/2$.</p>
</blockquote> | 3,224 | [
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<p>I know that photons and electrons and such are said to have a wave particle duality, but what does that mean for a photon? When light strikes an object, are many photons emitted, enough to draw infinitely many rays, is only one emitted, or something in between?</p>
<p>In particular, I'm having trouble with thin film interference:
<img src="http://i.stack.imgur.com/E53nS.gif" alt="enter image description here"></p>
<p>The two resulting rays are said to constructively interfere, which is confusing to me. The two rays are clearly parallel, but not coinciding, so how do the two interfere at all? I think my problem is that I imagine light to be a single ray, with a linear oscillating magnetic field- what is the proper way to address these rays?
Are they photons? Or are they small instances of a wave front? I've heard Huygens' Principle, but in this case we present single rays at the end, so I'm led to believe they really ARE rays, in which case they would be photons, and the interference problem would be a result of the wave/particle duality.
The only other thought I've had with regards to the interference is that, as opposed to looking at the rays as one dimensional rays, they could be some kind of representation of a wave 'centered' around that vector, but that doesn't make sense either.
I know it's a heavy question, but it's really confusing me.</p> | 3,225 | [
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<p>The question is not so much about the theorem, but more about what it means in this context: <a href="http://en.wikipedia.org/wiki/Berry_connection_and_curvature#Applications_in_crystals" rel="nofollow">see this link</a>.</p>
<p>So yes, because of Bloch's theorem the Hamiltonian eigenstates in a crystalline system can be written as
\begin{align}
\psi_{n,\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}u_{n,\vec{k}}(\vec{r}),
\end{align}
and so the Berry connection can be defined:
\begin{align}
A_{n}(\vec{k})=i\langle n(\vec{k})|\nabla_{\vec{k}}|n(\vec{k})\rangle,
\end{align}
but what in the world is $|n(\vec{k})\rangle$? </p>
<p>I've read a few articles on topological insulators and they always seem to start off with the Bloch wavefunction $e^{i\vec{k}\cdot\vec{r}} u_k(\vec{r})$, and then somehow they magically get the ket $|u(\vec{k})\rangle$ from which the Berry connection is defined... is $|u(\vec{k})\rangle$ the column vector comprised of the Fourier coefficients of $u_\vec{k}(\vec{r})$ w.r.t. $e^{i\vec{G}\cdot\vec{r}}$ or what?</p> | 3,226 | [
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<p>Why does it get hotter (feel hotter) in a sauna when one pours water over the hot stones?</p>
<p><a href="http://en.wikipedia.org/wiki/Sauna">Wikipedia says</a> that the water condenses onto the skin, but the actual air humidity is so low that I doubt anything is condensing there. The water (sweat) evaporates from skin instead.</p> | 3,227 | [
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<p>If I had a flexible tube sealed at both ends and I submerged it in water (held vertical) Would the bottom half of the tube compress and the top half expand? What would the pressure in the tube be? Say its a 2" length of tube with the bottom being in 12" of water</p> | 3,228 | [
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<p>In the two Phys.SE questions <a href="http://physics.stackexchange.com/q/466">What is the mechanism behind the slowdown of light/photons in a transparent medium?</a> and <a href="http://physics.stackexchange.com/q/7437">Why glass is transparent?</a> transparent media were discussed. But I'd like to clarify one detail: is a photon absorbed (and delayed) by the medium and then reemitted, or scattered instantly? </p>
<p>Is e.g. a laser beam still coherent after passing glass? As medium molecules are disordered, this should distort the phase of photons taking different paths.</p> | 3,229 | [
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<p>Can someone explain in both layman's terms and also technically why when I pulled my glass filled with liquid soda from the freezer, the liquid soda quickly froze?</p>
<p>Doesn't this violate the 2nd law of thermodynamics since heat moved away from the glass with soda and to the ambient?</p>
<p>Thanks</p> | 3,230 | [
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<p>In CE configuration of NPN transistor, collector emitter junction is reverse biased.
But how is a N-N junction reverse biased?</p> | 3,231 | [
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] |
<blockquote>
<p>The fact that photons emitted from an electric-dipole active atom
cannot be spatially localized better than to the near-field zone of
the atom is seen as the origin of genuine superluminality.</p>
</blockquote>
<p><a href="http://www.ncbi.nlm.nih.gov/pubmed/11309086" rel="nofollow">http://www.ncbi.nlm.nih.gov/pubmed/11309086</a></p>
<p>Question: What are superluminal interactions? No detailed information available on the web. I know that electromagnetic waves cannot be spatially localized better that to the near-field zone of the location where they are created. </p> | 3,232 | [
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<p>I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as:
\begin{equation}
\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}
\end{equation}
where the $\mathbf{1}$ represents the trace. However, I do not understand why we cannot further decompose the traceless part into a symmetric and an antisymmetric part.</p>
<p>In order to understand my logic: A general tensor $\varphi^i$ transforms as:
\begin{equation}
\varphi^i \rightarrow U^i{}_j \varphi^j
\end{equation}
whereas $\varphi_i$ transforms as:
\begin{equation}
\varphi_i \rightarrow (U^*)_i{}^j \varphi_j
\end{equation}
where $U \in \mathrm{SU(3)}$ is a $3 \times 3$ matrix. Now, I will let $S^i{}_j$ denote the traceless part of $T^i{}_j$ (i.e. $S^i{}_j$ has dimensions $\mathbf{8}$) and we can decompose this in the "symmetric" and "antisymmetric" part as usual:
\begin{equation}
S^i{}_j = \frac{1}{2}(S^i{}_j + S_j{}^i) + \frac{1}{2}(S^i{}_j - S_j{}^i)
\end{equation}
Then under an $\mathrm{SU(3)}$ transformation:
\begin{equation}
S^i{}_j + S_j{}^i \rightarrow U^i{}_k (U^*)_j{}^l S^k{}_l + U^i{}_k (U^*)_j{}^l S^k{}_l = U^i{}_k (U^*)_j{}^l (S^i{}_j + S_j{}^i)
\end{equation}
and:
\begin{equation}
S^i{}_j - S_j{}^i \rightarrow U^i{}_k (U^*)_j{}^l S^k{}_l - U^i{}_k (U^*)_j{}^l S^k{}_l = U^i{}_k (U^*)_j{}^l (S^i{}_j - S_j{}^i)
\end{equation}
Therefore, the symmetric part keeps its symmetry and the antisymmetric part keeps its antisymmetry. Thus two invariant subspaces are created and the representation is reducible? To sum up, I would think we decompose $T^i{}_j$ as:
\begin{equation}
\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{3} \oplus \mathbf{5} \oplus \mathbf{1}
\end{equation}
where $\mathbf{3}$ denotes the dimensions of the antisymmetric part and $\mathbf{5}$ denotes the dimensions of the symmetric part. Where am I going wrong?</p>
<p>Edit: I got my convention from "Invariances in Physics and Group Theory" by Jean-Bernard Zuber:</p>
<p><img src="http://i.stack.imgur.com/JduAD.jpg" alt="enter image description here"></p> | 3,233 | [
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<p>In a <a href="http://en.wikipedia.org/wiki/Cathode_ray_tube" rel="nofollow">CRT</a>, where do the ejected electrons go after they cause fluorescence on the screen, have they lost most of their energy, or do they actually go through the glass?</p> | 3,234 | [
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<p>The metric for the BTZ black hole is</p>
<p>$ds^2=-N^2dt^2+N^{-2}dr^2+r^2(N^\phi dt +d\phi)^2$</p>
<p>where $N^2=-M+\frac{r^2}{l^2}+\frac{J^2}{4r^2}$ and $N^\phi=-\frac{J}{2r^2}$.</p>
<p>It is often said that BTZ black hole is asymptotically AdS$_3$, but if I take $r\rightarrow \infty$ limit, then the BTZ metric, to the leading order of each component, becomes</p>
<p>$ds^2\rightarrow-\frac{r^2}{l^2}dt^2+\frac{l^2}{r^2}dr^2+r^2d\phi^2-Jdtd\phi,$</p>
<p>while the AdS$_3$ metric reads, asymptotically,</p>
<p>$ds^2_{AdS_3}\rightarrow-\frac{r^2}{l^2}dt^2+\frac{l^2}{r^2}dr^2+r^2d\phi^2.$</p>
<p>My question is that since the off-diagonal term $-Jdtd\phi$ doesn't approach to zero at infinity, how can one claim that BTZ is asymptotically AdS$_3$? </p> | 3,235 | [
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<p>I'm trying to simulate the degenerative Anderson model. So depending on an energy difference first orbital and afterwards spin magnetism occurs. First i try to solve an easier ansatz with a limitation of only two orbitals to $l_z = \pm 1$.
In this case i end up with a set of 4 equations:</p>
<p>$\Delta \tan^{-1} (1/\pi * n_{m \sigma}) = E_0-E_f+(U-J)n_{\overline{m}\sigma}+U(n_{1\overline{\sigma}}+n_{2\overline{\sigma}})$ </p>
<p>Where here $\sigma$ is the direction of the spin of certain electron in the orbital $m$, here only $1,2$ and the overline always is the opposite. U is a Hubbard potential and $J$ an intraatomic exchange.
So we gathered a self consistent set of non-linear equation and i want to find solution in terms of $n_{m \sigma}$ satisfying all 4 equations. I already did some simulation and were able to plot it on dependancy of the energy difference with respect to the fermi energy, which is my parameter. So i change it in a certain range and one finds magnetic solution for $E_0 < E_f$.</p>
<p>Is there a good method to solve such a problem with not too much calculation power? And in the end i want to play around with both potentials and see how the number of electron is changing for each situation. </p>
<p>Thanks for any advice,</p>
<p>/no</p>
<p>PS: i already asked a question concerning to my code i wrote in SO, but i was rethinking the method of mine for solving this problem. I did an iterative attempt and reformulated the equation for a root searching algorithm and varied the energy difference $E_0 - E_f$. The programming stuff is on <a href="http://stackoverflow.com/questions/6269212/ipython-solve-set-of-self-consistent-equation">ipython</a> plus a result of one simualtion run.</p> | 3,236 | [
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<p>I have a hemispherical bowl in which I roll a small particle around the edge, starting from the top at point <em>A</em> with a velocity $v_o$. It travels halfway around the sphere and reaches point <em>B</em>, which is a vertical distance <em>h</em> below <em>A</em>, with a velocity $v_f$. Point <em>A</em> is a radial distance of $r_o$ from the vertical centerline and point <em>B</em> is a radial distance of $r$ from the vertical centerline. There is no friction. The goal is to solve for the angle, $\theta$, between the horizontal and the velocity $v_f$.</p>
<p>Here is a diagram of the problem scenario:</p>
<p><img src="http://i.imgur.com/57qgEHI.png" alt="diagram of sphere"></p>
<p>My solution relies on the assumption that angular momentum only relies on the velocities in the plane perpendicular to the vertical centerline. Is that a safe assumption? Also, when dealing with energies, is rotational KE and linear KE the same? Should I be taking RKE into account?</p>
<hr>
<p>$$
L_o=L_f
$$
$$
mr_ov_o=mrv_f\cos \theta
$$
$$
\theta = \arccos(\dfrac {mr_ov_o}{mrv_f}) = \arccos(\dfrac {r_ov_o}{rv_f})
$$</p>
<hr>
<p>$$
KE_o + PE_o = KE_f
$$
$$
\frac 12 mv_o^2 + mgh = \frac 12 mv_f^2
$$
$$
v_o^2 + 2gh = v_f^2
$$
$$
\sqrt {v_o^2 + 2gh} = v_f
$$</p>
<hr>
<p>$$
\theta = \arccos(\dfrac {r_ov_o}{rv_f}) = \arccos(\dfrac {r_ov_o}{r\sqrt {v_o^2 + 2gh}})
$$</p> | 3,237 | [
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<p>Why can’t information travel faster than the speed of light, if the two endpoints to and from which the information is being sent are moving relatively to each other, as long as the information travels slower than the inverse of the velocity which the planets are moving apart at.</p>
<p>According to the velocity addition formula: w=(u−v)/(1−uv/c2), as long as u < 1/v, then w is positive and the laws of causality are not violated.</p>
<p>Is there a reason then that information cannot travel faster than the speed of light between two points moving relatively to each other. Also, couldn’t information travel an infinite number times the speed of light, if two points were not moving relatively to each other.</p> | 3,238 | [
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<p>I have read explanations of this but haven't really understood. Given a spacetime $(M,g)$ I have read that if I represent the metric in some coordinates $(x,y,z,t)$ as $g(x,y,z,t)$ and then in another coordinate system as $g'(x',y',z',t'),$ that $g'(x,y,z,t)$ (now using the old coordinates) will also solve the Einstein equations. Now $g$ and $g'$ are two different metrics on the manifold and so <em>should</em> predict different physics, but somehow they are the same? If I measure the distance between two points on the manifold I should get different answers if the metrics are different, shouldn't I? This is the part I don't really understand. Is it something like $(M,g')$ isn't a solution, but only $(M',g')$ where $M'$ is diffeomorphic to $M$?</p>
<p>To give an explicit example, let's look at the Schwarzschild metric: \begin{equation}
ds^2=-\left(1-\frac{2GM}{c^2 r}\right)c^2dt^2+\left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2+r^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \;.
\end{equation} Apparently (according to <a href="http://faculty.luther.edu/~macdonal/HoleArgument.pdf" rel="nofollow">http://faculty.luther.edu/~macdonal/HoleArgument.pdf</a>) \begin{equation}
ds^2=-\left(1-\frac{2GM}{c^2 f(r)}\right)c^2dt^2+\left(1-\frac{2GM}{c^2 f(r)}\right)^{-1}f'(r)^2dr^2+f(r)^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \;
\end{equation}
is also a solution for any diffeomorphism $f$ though distances aren't the same. Of course when deriving the Schwarzschild metric it seems (according to Carroll's book) $(r,\theta,\phi,t)$ are just symbols that you only interpret after you find the metric, which is confusing. Note I didn't do a change of coordinates here, I changed the actual metric. Are they really both solutions on the same manifold?</p> | 3,239 | [
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<p>While reading Hobsen et al.'s "General Relativity: An Introduction for Physicists", I came across a bit confusing derivation. Multiplying the 4-force and 4-velocity, the following derivation can be made</p>
<p>$
\boldsymbol{u} \cdot \boldsymbol{f} = \boldsymbol{u} \cdot {d\boldsymbol{p} \over d\tau}
= \boldsymbol{u} \cdot ({dm_0 \over d\tau}\boldsymbol{u} + m_0{d\boldsymbol{u} \over d\tau}) = c^2 {dm_0 \over d\tau} + m_0 \boldsymbol{u} \cdot {d\boldsymbol{u} \over d\tau} = c^2 {dm_0 \over d\tau}
$</p>
<p>After this derivation, the authors make the following conclusion:</p>
<blockquote>
<p>where we have (twice) used the fact that $\boldsymbol{u} \cdot \boldsymbol{u} = c^2$. Thus, we see that in special relativity the action of a force can alter the rest mass fo a particle! A force that preserves the rest mass is called a pure force and must satisfy $\boldsymbol{u} \cdot \boldsymbol{f} = 0$</p>
</blockquote>
<p>But I have the following objections and questions about this derivation:</p>
<ol>
<li><p>The rest mass is by definition a constant, so it should have been considered a constant while differentiating.</p></li>
<li><p>If we go back to Newton's second law, which is still valid under the special theory of relativity (though with some correction), the mass is the resistance of a body to changes in velocity, i.e. the larger the mass is, the stronger the force we need to change its velocity. But a non-free force seems to contradict this basic concept when $dm_0 \over d\tau$ is negative, because this means that the force is reducing the resistance of the body towards the force. As a funny comparison, imagine that the harder you push a heavy box, the lighter it becomes (which is obviously not the case even in Newtonian mechanics, not to mention that special relativity predicts the opposite, i.e. the faster the body is, the harder it becomes to increase its velocity)!! </p></li>
<li><p>Unless the mass is being converted to energy or transferred somewhere else (which is not inferred from the derivation, as the derivation comes straightforward from the force equation without depending on any other equation), where is the mass going?! Isn't this contradictory to the conservation of mass an energy law?</p></li>
<li><p>If we assumed in this derivation that the rest mass is variable, why didn't we do so in many other derivations in the special theory of relativity?</p></li>
<li><p>Do we have examples of such forces anyway? :-)</p></li>
</ol> | 3,240 | [
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<p>For example I have a dielectric solid with a small charged ball in it. And I have external electric field $E$. So what force is acting on this ball? </p>
<p>The field in dielectric is $\frac{E}{\epsilon}$, so the force should be $\frac{Eq}{\epsilon}$.</p>
<p>On the other hand. If I remove a small piece of dielectric then the field in the hole will be $E$. Now I put charge in this hole and the force is $Eq$. The hole is just for charged ball and there is no free space.</p>
<p>So what is the force? </p> | 3,241 | [
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<p><em>How does water carve rock?</em></p>
<p><em>And more generally, how does a soft material carve a hard material?</em></p>
<p>Obviously it happens, but is it a continual process (every drop of soft water carries away a minute amount of hard material) or is it a stochastic thing (every once in awhile, the force of the water just happens to exceed what's necessary to snap off a grain of sand)? Or is it something else (the erosion is actually caused by hard pieces of grit carried in the water)? </p>
<p>UPDATE: <em>Grit certainly plays a major role in actual weathering, but is it <strong>necessary</strong>? Would pure water ever erode rock?</em></p> | 3,242 | [
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<p>The Morning Glory roll cloud (pictured below):</p>
<p><img src="http://i.stack.imgur.com/I5c0J.jpg" alt="enter image description here"></p>
<p>from <a href="http://en.wikipedia.org/wiki/Morning_Glory_cloud">this Wikipedia page</a>, which briefly describes the landform and meteorological causes, but states</p>
<blockquote>
<p>The Morning Glory cloud is not clearly understood because their rarity means they have little significance in terms of rainfall or climate</p>
</blockquote>
<p>is there any definitive physical theory as to how these meteorological phenomena form?</p> | 3,243 | [
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<p>I'm not sure whether this question is well defined, but I am interested in the volume of the (obviously not linear) subspace of qubits
$$
\left|\Psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta\, e^{i\varphi} \left|\downarrow\right\rangle$$
where $\varphi \in [0, 2\pi)$ and $\alpha$ and $\beta$ are fixed. Looking at the <a href="http://en.wikipedia.org/wiki/Bloch_sphere" rel="nofollow">Bloch sphere</a> representation
$$
\left|\Psi\right\rangle = \cos\left(\tfrac{\theta}{2}\right) \left|\uparrow\right\rangle + \sin\left(\tfrac{\theta}{2}\right) e^{i\varphi} \left|\downarrow\right\rangle$$</p>
<p>shows that this is a circle with $\theta=\mathrm{const}$ and radius $\sin\theta$, so the size of the space is just the circumference of this circle, $2\pi \sin\theta$.</p>
<p>On the other hand, the state only differs by an overall phase from, and is therefore equivalent to
$$
\left|\Psi\right\rangle = \beta\, \left|\downarrow\right\rangle + e^{-\varphi}\alpha \left|\uparrow\right\rangle$$
so this leads to another Bloch sphere representation</p>
<p>$$
\left|\Psi\right\rangle = \cos\left(\tfrac{\theta'}{2}\right) \left|\downarrow\right\rangle + \sin\left(\tfrac{\theta'}{2}\right) e^{-i\varphi} \left|\uparrow\right\rangle$$
where $\cos\left(\tfrac{\theta'}{2}\right) = \sin\left(\tfrac{\theta}{2}\right) $ which now leads me to think that this family of states form a circle of circumference $2\pi\cos\theta$. Is this whole line of reasoning flawed from the beginning?</p> | 3,244 | [
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<p>Consider the path $x^\mu(u)$ in Minkowski space; such that:</p>
<p>$$t = \frac{a}{c} \sinh(u) , \quad x = a \cosh(u) ,\quad y = 0 ,\quad z = 0 $$</p>
<p>where $a$ is a positive constant and $u$ is a parameter</p>
<p>Use equation:
$$ c \nabla \tau = ds = \sqrt{\eta_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} du $$
to find the proper time elapsed along the path starting from $u = 0$, as
a function of $u$.</p>
<p>I'm having quite a bit of trouble with using the notation if someone could help me please?</p>
<p>So far I have established:
$$ d_u x^{\mu} = \left(\frac {a}{c}\cosh(u),\, a\sinh(u),\,0,\,0\right)) $$</p>
<p>Then:</p>
<p>$$\eta_{\mu\nu} d_u x^{\mu} d_u x^{\nu} = \sum_{\mu =0}^{3} \sum_{\nu =0}^3 \eta_{\mu\nu} d_u x^{\mu} d_u x^{\nu} $$</p>
<p>Then I get a bit lost...</p> | 3,245 | [
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<p>If the lattice types are categorized according to the point group symmetries, then what is the difference, for example, between sc and bcc structures?</p> | 3,246 | [
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<p>A few days ago I went for a walk in the evening. We're having winter with a little snow and freezing temperatures. We're in a quiet, shallow valley with a train station about 1km from us. I heard a train coming so I wanted to wait for it to watch it arriving to the station. To my surprise, although I was hearing the sound coming from behind a hill, the train wasn't appearing. After several minutes, I gave up, and went back, and finally I saw the train arriving after another a few minutes. The train must have been several km away when I first heard it.</p>
<p>I watched this phenomenon later and I realized that also I could hear and understand people talking on much larger distances than usual.</p>
<p>This has not happened before, and my only idea is that it's because of the cold weather. I have two ideas how to explain it:</p>
<ol>
<li>Cold air propagates sound better for some reason.</li>
<li>We have a few cm of snow covered by ice crust, as we had freezing rain a few days ago. I guess this can mean sounds aren't absorbed by earth and are reflected instead, which makes them propagate further. (I'd say this is more probable than 1.)</li>
</ol>
<p>Is any of this reasonable, or is there another explanation?</p>
<p><sup>(I'm not a native speaker so please feel free to correct any language errors.)</sup></p> | 3,247 | [
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<p>What does the term <a href="http://en.wikipedia.org/wiki/Equations_of_motion" rel="nofollow">equation of motion</a> refer to? If I am asked a question of the form 'What is the equation of motion of this object?', what should I write?</p> | 3,248 | [
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<p>How does e.g. sodium chloride (aq) conduct electricity? By accepting electrons (unlikely since they already have a full outer shell)? But they can't be hopping around themselves, can they? I mean, if I have two poles made of metals inserted into a beaker with this solution, and I try to let a current go through from the pole through the solution to the other pole, the ion itself moving can't be conducting electricity, right? The metal pole won't accept such an ion instead of an electron...</p> | 3,249 | [
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<p>If in a uniform magnetic field a conducting rod moves on a set of joined friction less rails which are perpendicular to the direction of magnetic field, a potential is developed and current flown this current generates heat, the mechanical energy required to move the rod is converted to heat energy in wires.</p>
<p>Now if a battery of EMF equivalent to the induced emf is joined with opposite polarity, no current will flow and hence no heat will be dissipated. How in this case the mechanical energy required to move this rod is conserved ?</p> | 3,250 | [
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<p>In the following circuit (that has come up during a BJT DC analysis for what is worth), I'd like to calculate the potential at point B and point E.</p>
<p>My notes say that $V_B = -10 + 0.7 = -9.3V$. Trivial as it may seem, I can't understand it. The only way that I can think of is that the negative pole of the source is at $-10V$ and since there is a potential difference of $0.7V$ between the poles of the source, then the positive one must be at $-10 + 0.7 = -9.3V$.</p>
<p>I think that the presence of $-10V$ and GND is that confused me. What would be the potential of GND here ? Zero or something else ? If my understanding is sound, then the GND must have the same potential as $V_B$, which is $-9.3V$.</p>
<p>Any insights ?</p>
<p><img src="http://i.stack.imgur.com/pnnD8.png" alt="enter image description here"></p> | 3,251 | [
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<p>What is the maximum ratio in the rate of change in time in reference to object $A$ which is standing still and object $B$ which is moving at the speed of light?</p> | 3,252 | [
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] |
<p>Did big bang create gravity?
What role gravity is assumed to have played in the formation (starting from the big bang) of large structures of our universe and what other important physical mechanisms and processes probably led to the structure we observe today?</p> | 3,253 | [
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<p>I'm reading a book (Sun of Suns by Karl Schroeder) that the main location is a planet called Virga, which contains air, water, and floating chunks of rock, and has no or a very small amount of gravity. There is a main 'sun' at the center of the planet, which provides the heat for weather.</p>
<p>Could a 'planet' of this type exist?</p> | 3,254 | [
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<p>I have difficulty understanding exercise 24 in <a href="http://www.physics.leidenuniv.nl/dbpages/em1_test/em1_test.asp" rel="nofollow">this document</a>:</p>
<blockquote>
<p>Two parallel wires I and II that are near each other carry currents i and 3i both in the same direction. Compare the forces that the two wires exert on each other.</p>
<p>(a) Wire I exerts a stronger force on wire II than II exerts on I.</p>
<p>(b) Wire II exerts a stronger force on wire I than I exerts on II.</p>
<p>(c) The wires exert equal magnitude attractive forces on each other.</p>
<p>(d) The wires exert equal magnitude repulsive forces on each other.</p>
<p>(e) The wires exert no forces on each other. </p>
</blockquote>
<p>I think - if you use $F_m=IlB\sin \alpha$, which is Force on electric wire in uniform magnetic field - that $F_{II}=IlB\sin \alpha = 3IlB\sin \alpha > IlB\sin \alpha =F_{I}$
So answer would be b), but how is it possible because you have Newton's third law( the forces should be equal, but does it apply here) and there is not any magnetic field here.
So do I use this Lorentz's law or which law do I use?</p> | 3,255 | [
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<p>Why is the sign convention used in the derivation of the lens formula and yet used <em>again</em> when it is applied in numerical problems? Won't the whole idea of sign convention be eliminated if it is used twice?</p> | 3,256 | [
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<p>I am trying of resolve this exercise: Show that if $|\psi \rangle$ is an entangled state of two Qbits, then the application of a unitary operator of the form $U_1 \otimes U_2$ necessarily generates an entangled state.</p>
<p>Suppose that $(U_1 \otimes U_2)|\psi\rangle$ is not entangled then it must have the form </p>
<p>$$(U_1 \otimes U_2)|\psi\rangle = (a|0\rangle+b|1\rangle)\otimes (c|0\rangle+d|1\rangle),$$
but the unique way this will be true is when $|\psi\rangle$ is not entangled due to definition of linear operator.</p>
<p>My proof is bad? Help me please</p> | 3,257 | [
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<p>Would it violate any known laws of physics to construct a universe containing no mass, only energy?</p> | 3,258 | [
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<p>We've been discussing radioactive decay at school, and I grasped everything except for $\beta +$ decay. When I googled radioactive decay, I immediately found out they dumbed down radioactive decay for us, which is probably why they didn't care to explain what they did, they just showed some calculations. We have never discussed neutrino' s and antineutrino's, they leave that out of the equation, which is no problem since they have negligible mass and no charge. </p>
<p>So we've been taught that a proton and electron form a neutron, which I have also discovered is not true (I'm discovering a lot of new things :P). I learned that this is caused by the spontaneous (?) change from up to down quarks and vice versa. </p>
<p>However, at school I must keep to the 'rules' and by those rules I don't really understand $\beta +$ decay. I see $\beta -$ decay as follows: </p>
<p>An electron flees a neutron and leaves a proton. That's why you get an atom with a higher atomic number.</p>
<p>However, how would this work with beta + decay? Can it even be dumbed down to this kind of high school thinking?</p> | 3,259 | [
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<p>How quarks electric charge directly have been measured when quarks never directly observed in isolation? (Due to a phenomenon known as <a href="http://en.wikipedia.com/wiki/color_confinement" rel="nofollow">color confinement</a>.)</p> | 3,260 | [
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<p>I'm studying an interacting bose-einstein condensate using the energy functional proposed in this paper</p>
<p>K. Huang, C.N. Yang, Phys. Rev. <strong>105</strong></p>
<p>$$
E\left[\phi\right] = \int d^3\vec{r} \left\{-\frac{\hbar^2}{2m}|\nabla\phi|^2 + \phi^*\left[U(x,y,z)+\frac{g}{4}\left| \phi \right|^2 + \frac{g_2}{4}\nabla^2\left(\left|\phi\right|^2\right)\right]\phi \right\}
$$
where
$$
g = \frac{4\pi \hbar^2 a}{m} \qquad g_2 = \frac{4\pi\hbar^2 a^2(\frac{1}{3}a-\frac{1}{2}r_e)}{m}
$$
$a$ is the scattering lenght, $r_e$ the effective range of the potential.</p>
<p>1) why in the kinetic term is there the square of gradient of $\phi$ and not $\phi^*\nabla^2\phi$? </p>
<p>2) How can i get the relation between $a$ and $r_e$ in this special case? Authors use directly the relation for hard-sphere potential $r_e = 2/3a$ but i don't understand why. In the functional there isn't a hard-sphere potential...</p>
<p>3) I've calculated the dispersion relation for waves inside a condensate trapped in one dimension(i've written $\phi(x,y,z) = \frac{1}{\sqrt{\pi}\sigma}\varphi(z)e^{-(x^2+y^2)/(2\sigma^2)}$ and i've integrated the functional over $x,y$ )
$$
\omega = k\sqrt{\frac{1}{m}\left( n_0\tilde{\gamma} - \left( \tilde{\gamma_2}n_0-\frac{\hbar^2}{4m}\right) k^2\right)}
$$
where $\tilde{\gamma}$ and $\tilde{\gamma}_2$ depend on the confinement strength,
I'd like to check if there is consistence with experimental data. do you suggest me anything?</p> | 3,261 | [
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<p>We can get monopole $1/r$, dipole $1/r^2$, quadrupole $1/r^3$ and octupole $1/r^4$ potential falloff by placing opposite point charges at the corners of a point, line, square and cube, respectively. My book cryptically says "and so on", how does one get a $1/r^5$ and higher potential falloff with a finite number of point-charges in 3 dimensions?</p> | 3,262 | [
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<p><img src="http://i.imgur.com/VG7izgO.jpg" alt=""></p>
<p>The problem is to find the current on the capacitor. $I''$ should be correct, but I don't know how to construct the formula for $I'$.
I managed to get the value for $I_c$ using Thevenin and Norton equivalents, and they're the same, so that should be correct.</p> | 3,263 | [
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] |
<p><img src="http://i.stack.imgur.com/Oo2mp.png" alt="enter image description here"></p>
<p>Can anyone point me to a derivation of this expression? $n_s$ is the number of bosons in a state.</p> | 3,264 | [
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<p>Normally we write a Bloch Hamiltonian $H(\mathbf{k})$ for the bulk and determine the spectrum which gives us various bands i.e we basically obtain $E=E(\mathbf{k})$ for the bulk only.</p>
<p>Also in the real space if we solve a tight-binding model, we get the energy eigenvalues which have both edge modes and the bulk spectrum within it, but this does not give us k-dependence. </p>
<p>However in many papers they plot the bulk spectra along with the edge modes with k-dependence. For example see Fig 1 in</p>
<blockquote>
<p>C. L. Kane, and E. J. Mele. “<a href="http://dx.doi.org/10.1103/PhysRevLett.95.226801" rel="nofollow">Quantum spin Hall effect in graphene</a>.” <em>Physical Review Letters</em> <strong>95</strong>, no. 22 (2005): 226801. (<a href="http://arxiv.org/abs/cond-mat/0411737" rel="nofollow">arXiv</a>)</p>
</blockquote>
<p>My question is that how are they numerically determining the k-dependence of edge states?</p> | 3,265 | [
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<p>I wish to pursue a career that somehow involves programming, electronics, and physics. What are such careers? </p>
<p>Also, I have heard of some 1 year post-graduate diplomas/courses for specialization in Physics. Which universities/ colleges offer them?</p> | 3,266 | [
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<p>A Lagrangian is given by,
$$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$
$$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$
where V (A) now includes nonlinear terms and E is the energy which is found by taking the appropriate Legendre transform of the Lagrangian and evaluating it at the upper turning point of an oscillation, $A_{max}$.
Now using the potential $V= \phi^2-\phi^3+\frac{\phi^4}{4}$, and $\phi=A(t)e^\frac{-r^2}{R^2}$we can write,
$$V(A)= (1+\frac{d}{2R^2})A^2-\left(\frac{2}{3}\right)^\frac{d}{2} A^3+ \frac{A^4}{2^\frac{d+4}{2}}$$
$$V''(A)= (2+\frac{d}{R^2})-6\left(\frac{2}{3}\right)^\frac{d}{2} A+ 3\frac{A^2}{2^\frac{d}{2}}$$ </p>
<blockquote>
<p>For $d=2$, they got $E_{\infty}=4.44$ and $d=3$ they found the value $E_{\infty}=39.69$, but how? <strong>Why do we write here $E_{\infty}$?</strong> For more information <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">please check equations 13 and 14 in the link</a> </p>
</blockquote> | 3,267 | [
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<p>How can one show that the action of dilaton in the String Background Fields must be of the form: $ S_\Phi = \frac1{4\pi} \int d^2 \sigma \sqrt{h} R(h) \Phi(X) $?</p>
<p>Thank you.</p> | 3,268 | [
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<p>I'm a physics student and I'm attending an introductory course of particle physics. My professor stated that, in center of mass frame, the $\nu_\mu e^- \to \nu_\mu e^-$ elastic scattering has an isotropic angular distribution, while the $\bar{\nu}_\mu e^- \to \bar{\nu}_\mu e^-$ scattering has not.</p>
<p>I can't figure why this should be true. Any help would be appreciated.</p> | 3,269 | [
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<p>I have a Hamiltonian:</p>
<p>$$H=\dot qp - L = \frac 1 2 m\dot q^2+kq^2\frac 1 2 - aq$$</p>
<p>In a system with one coordinate $q$ (where $L$ is the Lagrangian). One of the Hamilton equations is:</p>
<p>$$\dot q =-\frac {\partial H} {\partial p}$$</p>
<p>But when I try to derive $H$ with respect to $p$, I get very confused. What is the derivative of $q$ with respect to $p=m\dot q$, for instance? When I boil it right down, my confusion stems from the fact that I realize I don't know what that partial derivative means. A partial derivative of a multi-variable function should be taken with respect to an index (you just specify which variable, thought of as a "slot" in the function, you're deriving with respect to). I suppose I'm not clear on what multi-variable function $H$ represents (I mean, $q$ and $\dot q$ are functions of $t$, so you could say it's a one variable function...), or how I should interpret $p$ as a variable.</p>
<p>I have similar difficulties with the equation $\dot p=\frac {\partial H} {\partial q}$, although I think I can understand $\frac {\partial H} {\partial t} = \frac {dL} {dt}$. The left hand side should give $m\dot q\ddot q + kq\dot q - a\dot q$, right?</p> | 3,270 | [
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<p>I have a doubt over the Kelvin and Planck's statement of thermodynamics' second law, in particular applied to a cycle. Let's take a Carnot cycle as an example, and let's call the first two transformations (the isotherm and the adiabatic) done. Now, isn't it obvious that the machine has to give up heat to go back to the initial state? Isn't it something that follows from the fact that the cycle has to be continuous?</p> | 3,271 | [
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<p>This question is actually related to my earlier question ("<a href="http://astronomy.stackexchange.com/questions/948/what-exactly-is-the-definition-of-motion-and-its-relation-to-machs-conjecture">what is motion</a>").
The fact that objects move a lot in the universe and that the universe is expanding, can imply that gravity is a repulsive force that increases with distance.. so the farthest objects repel us more.</p>
<p>This can still explain several existing observations, e.g., why does the apple fall? </p>
<p>Motion is the result of such repulsion. Two objects unlucky enough not to be moving relative to each other get squished due to the repulsion of the rest of the universe around them. The earth repels the apple less than the stars so it is pushed towards the earth. </p>
<p>Furthermore, it can explain the expanding universe without the need for dark energy.</p>
<p>This could be demonstrated in a thought experiment. If we take a lot of same-charge particles (with small mass) such as electrons and lock them in a large box at a low enough temperature. The mutual repulsion of the particles may cause similar motion as if due to gravitational attraction. </p>
<p>Another experiment would be to measure the slight changes in our weight during day and night when the sun and earth align (if their masses are large enough to detect the feeble change in repulsion). </p>
<p>[EDIT: the question in the original form may not have been clear. It is "can we model".. with a yes/no answer and why (not). If downvoting, please justify. </p> | 3,272 | [
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<p>If time stops at the event horizon, can we ever detect two black holes merging? In other words, if you are a short distance away, would you encounter a <em>spherically symmetric</em> gravitational field, or a <em>dipole</em> field? </p> | 765 | [
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<p>I'm studding special relativity and there is a famous example where there is a moving train witch has a light source on its floor and a mirror attached to its roof.</p>
<p>an observer in the train sees a pulse of light leaves the source vertically and goes to the mirror and comes back to the source. </p>
<p>the example says that an observer on the earth should see the path of light as a triangle like figure "b" in this image:
<img src="http://i.stack.imgur.com/UdADX.png" alt="enter image description here"></p>
<p>but I don't understand why?! If someone shoot a ball like this, it's logic because the ball will have an extra velocity in the direction of x-axis but here we know that the pulse is going upward with the velocity c, so if the observer on the earth see the light is going right with the train , it should have an extra velocity Vx in the direction of x, and so the total velocity will be equal to (c^2 + Vx ^2)^(1/2) witch is larger than c.</p>
<p>it will be solved with saying that the light will go on the L direction (shown in the figure) with the total c velocity, but how we know that the observer on the earth should see this? why not this : " the light goes upward with the velocity c and so wont come back to the light source in the train. "</p> | 3,273 | [
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<blockquote>
<p>A car is driving down a mountain ($v=90 km/h=25 m/s$, when the driver realizes that brakes aren't working. He try to lose velocity going up an inclined ($20°$) plane, with a friction coefficient of $k=0.60$. How many meters will it take to halt?</p>
</blockquote>
<p>I've tried as following ($s$ is the request):
$$K=\frac{mv^2}{2}$$</p>
<p>At the end, the potential energy gained is:
$$U=mgh=mg\cdot s\cdot sin \alpha$$</p>
<p>In the mainwhile the energy lost due to the friction is:
$$L_f=F \cdot s=mg \cdot cos(\alpha) \cdot s$$</p>
<p>But the work done by non conservative forces (friction) is also:
$$L_f=U-K$$</p>
<p>And I have:
$$mg \cdot cos(\alpha) \cdot s=mg\cdot s\cdot sin \alpha-\frac{mv^2}{2}$$
$$g \cdot cos(\alpha) \cdot s=g\cdot s\cdot sin \alpha-\frac{v^2}{2}$$
$$9.22s=3.35s-312.5$$
But I get a negative time.
What's wrong? I'm sure that there is a stupid error, but I can't find it.</p>
<p>The correct result (reported on the textbook) is 120 m. </p> | 3,274 | [
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<p>I'm trying to understand why when you have a muon decay event, the energy of the electron peaks near the maximum kinematically allowed value. Is there an intuitive explanation for why this is the case or is it one of those things where the explanation is that it's just how the math works out?</p> | 3,275 | [
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<p>This has been a really great confusion for me now ....</p>
<p>Many places i have read in books that when a potential difference is applied across the ends of a wire a constant electric field is generated inside it which drives the current through it...</p>
<p>My question is how is this electric field generated ?? Why is it constant in magnitude throughout the wire ?? What is the mechanism of flow of current inside the wire ??</p>
<p>And</p>
<p>What all thing happen inside the conductor just after closing the switch and before the low of constant current through it ?? </p>
<p>give detailed answers ??</p> | 78 | [
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