title: Positive Mass Theorem and Geodesic Deviation

text: This is a thought I had a while ago, and I was wondering if it was satisfactory as a physicist's proof of the positive mass theorem. The positive mass theorem was proven by Schoen and Yau using complicated methods that don't work in 8 dimensions or more, and by Witten using other complicated methods that don't work for non-spin manifolds. Recently Choquet- Bruhat published a proof for all dimensions, which I did not read in detail. To see that you can't get zero mass or negative mass, view the space-time in the ADM rest frame, and consider viewing the spacetime from a slowly accelerated frame going to the right. This introduces a Rindler horizon somewhere far to the left. As you continue accelerating, the whole thing falls into your horizon. If you like, you can imagine that the horizon is an enormous black hole far, far away from everything else. The horizon starts out flat and far away before the thing falls in, and ends up flat and far away after. If the total mass is negative, it is easy to see that the total geodesic flow on the outer boundary brings area in, meaning that the horizon scrunched up a little bit. This is even easier to see if you have a black hole far away, it just gets smaller because it absorbed the negative mass. But this contradicts the area theorem. There is an argument for the positive mass theorem in a recent paper by Penrose which is similar. Questions:   1. Does this argument prove positive mass?   2. Does this mean that the positive mass theorem holds assuming only the weak energy condition?


title: Quantum Number of a Tennis Ball

text: A tennis player has a tennis ball container with a single ball in it (it normally holds three). He shakes the tennis ball horizontally back and forth, so that the ball bounces between the two ends. We model the tennis ball as a quantum particle in a box. The questions: what is the quantum number n for this ball? If the ball were to absorb a photon and jump to the next energy level, what should the energy (in eV) of that photon be? For both of these questions, I am confused about how to apply quantum mechanical principles to the tennis ball. For the former, I suppose that a quantum number of n would make sense...if we model it as a particle in the box, the probability distribution across the container would match that of a particle at the n=2 energy level (remember--we are moving back and forth, and therefore the particle would be most likely to be at one of the ends and not in the middle). Would this be the correct reasoning? Is there something else I'm missing? For the latter question, I would use $$E_n=\frac{n^2\hbar^2\pi^2}{2ma^2}$$ where m is the mass of the tennis ball and a is the length of our container. Say that we are now in the $n=2$ energy level. To go to $n=3$, we have to apply an energy of $\Delta E=E_3-E_2$ to make that jump. Is this the correct procedure for both of these questions? I'm just having a hard time applying quantum mechanical thinking to these macroscopic objects. Thank you in advance.


title: Normalizing the free particle wave function

text: One way to normalize the free particle wave function > "is to replace the the boundary condition $\psi(\pm{\frac{a}{2}}) = 0$ [for > the infinite well] by periodic boundary conditions expressed in the form > $\psi(x)=\psi(x+a)$" \-- _Quantum Physics_ , S. Gasiorowicz How does this work? What does this mean physically? Or more precisely, why does this approximation suffice? I understand that this makes the wavefunction square-integrable (when integrated from $x=0$ to $x=a$) hence normalizable. Thanks.


title: How do you obtain the commutation relations at non-equal times (for the edge of a fractional quantum Hall state)?

text: The edge of a fractional quantum Hall state is an example of a chiral Luttinger liquid. Take, for the sake of simplicity, the edge of the Laughlin state. The Hamiltonian is: $$H = \frac{2\pi}{\nu}\frac{v_c}{2} \int_{\textrm{edge}} dx \rho(x)^2 $$ Here $\nu$ is the filling fraction, which is constant, $v_c$ is the velocity of the edge mode and $\rho$ is the charge density operator. You can think of this Hamiltonian as a delta-function interaction $V(x,x') = \delta(x-x')$. Together with this Hamiltonian there is also the commutation relation of the field $\rho$: $$ [\rho(x),\rho(x')] = i\frac{\nu}{2\pi} \partial_x \delta(x-x')$$ I haven't gone through the exercise myself, but I presume this is derived by going to momentum space, obtaining the canonical momenta through Hamilton's equation of motion and performing a canonical quantization. These equal-time commutation relations together with the Heisenberg equations of motion lead to: $$\partial_t \rho(x,t) = i[H,\rho(x,t)] = v_c \partial_x\rho(x,t) $$ This demonstrates that the edge is chiral, since $(\partial_t - v_c\partial_x)\rho(x,t) = 0$ and therefore the correlator involving $\rho(x,t)$ (or any other correlator) is a function of $t+x/v_c$ alone (hence the name "chiral" and "left-moving"). There are also particle excitations (e.g. the electron) which are generated through vertex operator $\Psi(x,t)$ (these can be motivated through bosonization and/or conformal field theory, but I won't go into that). In any case these field operators have the following equal-time commutation relations with the current operator: $$[\rho(x),\Psi(x')] = Q \Psi(x')\delta(x-x')$$ Here $Q$ is the charge of the operator $\Psi$ with respect to the charge density operator $\rho$. This charge is then of course the electric charge. My question is now: how do you generalize the commutation relations to non- equal time? What is: $$[\rho(x,t),\rho(x',t')] = \ldots $$ $$[\rho(x,t),\Psi(x',t')] = \ldots $$ ?


title: Hydrostatics: Log floating in water near a dam

text: So I am studying for a final and can't seem to solve this. There is a log floating in water and I need to find its weight. The question I have is what parts of the volume of the log count when summing the forces in the Y axis. ![enter image description here](http://i.stack.imgur.com/108N2.png) What i have now for AREA alone is $r^2 - .25\pi r^2$ pushes down and $.5\pi r^2 + r^2$ pushes up. however this does not take into account the air above the log in the upper right quadrant. is the force pushing up taking into account the half circle below the water and also the $2r^2$ above the horizontal of the log? The actual question for this image says: A log is stuck against a dam as shown in the diagram. given the radius of the log of 1.4 m and the length of the log into the page, 10 m find the weight of the log in kN.


title: Numerical algorithms to generate a random wavefunction from a thermal ensemble

text: I am seeking an algorithm to generate a random wavefunction = $\sum {c_i |\varphi _i\rangle }$ from a thermal ensemble, whose density matrix $\rho \sim e^{-\beta H}$, without the need to diagonalize the Hamiltonian. One possible method is to generate a random number for each basis and compare it with $e^{-\beta \langle\varphi _i|H|\varphi _i \rangle }$ and if it is lower we assign the corresponding $c_i$ to complex number with norm = 1 and totally random phase and otherwise make it just zero. Another possible algorithm to assign all $c_i$ to random complex numbers of norm one and then evaluate $e^{- \frac{\beta}{2} H}|\psi \rangle$. Are there other algorithms? Edit: I now tried the second method with evaluating $e^{- \frac{\beta}{2} H}|\psi \rangle$ by propagating the wavefunction in imaginary time and it works fine! So I recommend this one.


title: Zeta-function regularization in QFT for heat kernels

text: When one is doing zeta-function regularization of the heat-kernel for QFT then one is doing these following steps,   * the integral over the imaginary time   * taking the trace of the heat-kernel or the short-distance limit    * the space-time volume integral I would like to know as to what justifies the sequence in which these steps need to be done. I have generally seen it being done in the order as listed above but its more or less clear from working with examples that the answer clearly depends on which order on does it. Or is the art of regularization about being able to choose the "right" sequence depending on the situation? But is there an argument as to why there should be only one particular sequence which will generate a finite answer? Also in some examples it seems that the order needs to be changed whether one is regularizing the zero-modes of the theory or not. But that doesn't seem consistent! That would effectively mean doing a different regularization for different terms of the same expression!


title: Studying from "Berkeley Physics Course Volume I - Mechanics"

text: I'm a first year physics student and the main source for our Physics I course is "Berkeley Physics Course - Volume I." I'm having a hard time understanding this book because it assumes a pretty high level of previous knowledge, and the mathematical level is also advanced. Is anyone familiar with a site that assists with this book in particular, or some kind of accompanying source?


title: What is the medium that allows magnetic fields *or any forcefield* to exist?

text: Magnetic fields are obvious distortions.. of.. something, but what exactly are they distortions of? Massive objects produce curvatures/gradients in space- time resulting in what we observe as gravity.. what is the equivalent explanation for magnetic/electric fields?


title: What's the core difference between the electric and magnetic forces?

text: I require only a simple answer. One sentence is enough... (It's for high school physics)


title: If transported back to the 18th century could you solve the Longitude Problem without an accurate clock?

text: Seeing an interesting BBC article today at http://www.bbc.co.uk/news/science- environment-23514521 about the Longitude Problem, I wondered if it could have been solved, in a way practical at the time (the 18th century), by any means other than the solution eventually found. So if fanciful, but possibly entertaining and instructive, questions are allowed here, suppose you were transported back to the early 18th century and forbidden from disclosing any fundamental physics unknown at the time, explicitly or otherwise. Would you, knowing what we do today about classical physics and astronomy etc, have any ideas for bagging that prize money, aside from building a sufficiently accurate clock?


title: How to solve Poisson equation in electrodynamics?

text: I learned electrodynamics. According to the vector potential determination, $$ \mathbf B = [\nabla \times \mathbf A ], $$ Coulomb gauge, $$ \nabla \mathbf A = 0, $$ and one of Maxwell's equations, $$ [\nabla \times \mathbf B ] = \frac{1}{c}4\pi \mathbf j, $$ I can assume, that $$ [\nabla \times \mathbf B ] = \nabla (\nabla \mathbf A) - \Delta \mathbf A = -\Delta \mathbf A = \frac{1}{c}4 \pi \mathbf j. $$ How to prove that the one of the solutions of this equation is solution like newtonian potential, $$ \mathbf A = \frac{1}{c}\int \limits_{V} \frac{\mathbf j (r) d^{3}\mathbf r}{|\mathbf r - \mathbf r_{0}|}? $$


title: shifting from mathematics to physics

text: I am a postgraduate in mathematics. I studied physics during my B.Sc.studies.I want to go for further studies in physics particularly in theoretical physics. I am in a job and cant afford regular classroom teaching. Could anyone tell me something about some distance education programs? or are there programs for mathematics graduates to work in theoretical physics areas like string theory?


title: Does thin film interference (anti-reflective coating) let more light through?

text: The theory of an anti-reflective coating is that the reflected light off the coating and the reflected light off the substrate is 180 degrees out of phase, causing destructive interference and subsequently no light is 'reflected'. But how does this process allow more light to pass through the substrate. Because although the reflected light is causing destructive interference, before the light leaves the coating, it is still reflected, meaning that light has still been lost.


title: Sliding force less or equal?

text: Why is the force required to slide a magnet off a steel plate A LOT less than the force required to directly pull it off? The force required to pull the magnet can be: 20lb While the force required to slide the magnet can be: 1lb more/less. Why is that?


title: Poynting theorem and entering power

text: I refer to the time-domain version of the Poyinting theorem in electro- magnetism: $- \displaystyle \oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{S} - \int_V \mathbf{E} \cdot \mathbf{J}_i \ dV = \int_V \frac{\partial}{\partial t} \frac{\mu |\mathbf{H}|^2}{2} \ dV + \int_V \frac{\partial}{\partial t} \frac{\epsilon |\mathbf{E}|^2}{2} \ dV + \int_V \sigma |\mathbf{E}|^2 dV$ where $S$ is the boundary surface of the integration volume $V$. With $\mathbf{J}_i$ I mean impressed currents (imposed by some field source, like antennas) and with $\mathbf{J}$ I mean induced currents (induced on some conductor, if present, into $V$). The product $\mathbf{E} \cdot \mathbf{J}_i$ must be negative if $\mathbf{J}_i$ is the current that generates $\mathbf{E}$. But if some current $\mathbf{J}_i$ generates power into $V$, some power should enter $V$ to feed $\mathbf{J}_i$. If there is a Poynting vector entering $V$ and providing power to $\mathbf{J}_i$, can we consider both (the Poynting vector and the current $\mathbf{J}_i$) as increasing the power into $V$, and so both negative? Should not the Poynting vector entering $V$ be considered the only source of power in $V$? As an example we can consider a volume $V$ enclosing an antenna. There will be a term $\mathbf{E} \cdot \mathbf{J}_i$ due to the radiation; there will be a Poynting vector _exiting_ from the volume (radiated power); but what about the Poyting vector related to the feed line of the antenna? It is _entering_ the volume and it has a different form than the previous one, because it is the Poyting vector of a transmission line and not of a radiated field. How is it represented in the above equation? Thank you anyway!


title: How to verify gauge invariance of an amplitude

text: I have calculated a tree level amplitude for Compton scattering (${e\left(p\right)+\gamma\left(k\right)\to e\left(p\prime\right)+\gamma\left(k\prime\right)}$): $${ i\mathcal{M}=M_{\mu\nu}\epsilon^{*\mu}\left(k\prime\right)\epsilon^{\nu}\left(k\right)\textrm{.} }$$ How should I go about trying to verify that it is gauge invariant?


title: Does the termination point(level) influence water flow from a pipe

text: I get water to my home from a nearby Tank A at a certain height above ground level. I have a 1" pipe through which I get this water to my home.. I leave this water into my well by connecting a 1" tube to this pipe. Reason for question: I have seen water pressure vary(lower- more flow) depending upon the height I hold my pipe(on the outlet side) when I empty my fish tank, keeping how deep the pipe is immersed on the other side a constant. Now, if I elongate and leave the pipe from Tank A at deeper level below ground level should the flow rate increase?(will i get more water in the same time) as apposed to a shorter pipe which still goes into my well. or simply: Will I empty my overhead tank on 2nd floor quicker if I use the tap on the ground floor instead of the tap on the second floor. The taps being same size.


title: What highest amount of energy can we get from a laser?

text: Can we get high energy from laser like fusion and fission reactions ? what is highest energy we can get from laser ?


title: Do quarks really exist?

text: Title says it all. Have they been seen in experiment or are they just theoretical things? Do quarks really exist?


title: Reflection, transmission, absorption...how to calculate them?

text: I was wondering whether there is an equation that enables me to calculate the reflection, transmission, absorption and polarization, when the electric field everywhere is given? Consider this: You have solved the full Mie scattering process, so incident field, the field in the sphere and the scattered field are known. How can one calculate those quantities then?


title: Why does the car clock dim down so much when the headlight is turned on?

text: Even on a current new 2012 car, when the green LED clock inside the car is quite bright, but when the car's headlight is turned on, the clock dims down to only about 1/4 of its brightness, which makes the time hard to see. I thought the clock requires merely a watt or even less for its brightness, and the car stereo which output at least 30 watt of music (if comparing a 30 watt speaker for the PC to the car stereo), won't be less loud when the headlight is turned on. Why does the clock dim down so much? Is it explained by Ohm's law? Also, can't it be "parallel" instead of "in series" with the headlight, so that the brightness is not affected?


title: Inclined conveyor belt mystery

text: **The situation** An inclined conveyor belt with topmost point $h$ height above the ground receives sand at a constant rate $X$ from a container at a negligible height above the lowermost point of the conveyor belt. There is sufficient friction on the conveyor belt so that the sand stops almost immediately after coming in contact with the belt. **The question** On trying to find the minimum force required to maintain this situation, miraculously (after making some appropriate assumptions) the force needed at the bottom most point for the sand to start moving and the force required to maintain the motion of the sand on the conveyor belt above that point have to be same for the total force exerted by the conveyor belt to be minimum. Mathematically it is easy to reach to this conclusion but I can't understand why this happens physically.


title: Formulation of general relativity

text: EDIT: I think I can pinpoint my confusion a bit better. Here comes my updated question (I'm not sure what the standard way of doing things is - please let me know if I should delete the old version). The major change is that I removed focus from the third question which probably is a purely mathematical question (in the notation below, it asks what properties of (M,T) together with (M,T) being consistent, forces (M,T) to be unique.). Say that a pair (M, T) is _consistent_ if it satisfies the Einstein field equations. Here M is a manifold with a metric g (from which one can define its Ricci tensor among other things) and T is a map from M to tensors living in the same space of tensors as the Ricci tensor (I ignore units for now). I put absolutely no other restrictions on (M, T). Now relativity makes perfect sense to me as a mathematical statement: it distinguishes some pairs (M, T) as 'consistent'. Therefore, if we had a way to map "our world" to a pair (M,T) at least we could theoretically check whether (M,T) is consistent or not. My problem is that I do not at all understand how to do this. To begin with, which set do I choose for M? I think I can answer this question myself. I take this set to be the set of intuitive descriptions(that I can make intuitive sense of) of events in the world. For example, E := "(a particular point in) Stockholm on 08:00, Jan 24, 2013". This description I could understand intuitively, and at least theoretically (if time permits) I could go there to check the theory if it makes a statement about E. Another kind of description, given already another description F, could be G:="the event I get to by using rocket R, travelling for time T according to this watch I bring with me, from G", where R and T are intuitive descriptions. Please let me know if this choice of set is inappropriate. In this case I have no problem of turning the set M into a manifold, not yet with a metric. Finally (and here is my confusion): I am at a point p (constructed as above). What (intuitively described) experiments do I perform to find the metric tensor at p, respectively the stress-energy tensor at p? I cannot come up with two different answers for these two tensors - and in this case the theory is not a very interesting one, since then it just predicts that two identical (in the intuitive sense) experiments are the same. If I try to get an answer to this from e.g. Wikipedia I get lost in a deep tree of coordinate-dependent definitions which in some places appear to assume I already have intuitive sense for both mass and metric, and that I have intuitive sense for these being the related as relativity predicts they should be. I'm hoping there are two distinct intuitively described experiments I could perform, which relativity predicts should have the same result. FINAL EDIT: I have received many useful comments, and the answer by Ron Maimon answers my initial question, which was "what is a suitable set to choose for M when trying to map 'our world' to a pair (M,T)?". It seems a definition such as I suggest above "should work", as should that described by Ron in his answer. Furthermore, Ron points out(I think) that it is an assumption of the theory that any such labeling should give the same results. Since my initial question is answered I accept Ron's answer and will possibly come back with my further question "how to intuitively understand the metric and stress energy tensors in terms of experiments any person with sufficient degree of common sense and superhuman abilities (by which I just mean, can reach high accelerations, is not so heavy as to affect the stress-energy tensor in significant ways et.c. Equivalently, superhuman abilities would not be needed in case the speed of light was something like 10 meters per second) could perform?" if I am able to formulate it in a precise way. OLD VERSION (not needed for the question): As far as I understand, general relativity states that _the world is a manifold M, and M is completely described by the Einstein field equations._ This already appears as an incomplete statement, and I'll explain why I think so. Before that:   * what is a complete statement of general relativity, possibly including undefined terms (so in my attempt above, the "is" in "the world is a manifold" and "Einstein field equations" are undefined terms)? Now to why this does not make sense to me. The Einstein field equations state that two tensors (it is not necessary to define tensor for my confusion to arise) agree _at every point_. This seems to presuppose that the set of points making up the manifold is already given. Thus:   * what is a good description of the set of points of M? For clarity, my definition of manifold M says for one thing that M is a _set_. For the first question, it appears to me that there need to be some extra assumptions, since one could conceivably think of a 'world' without matter which should be completely 'flat', and also of our world which is not. These cases should clearly be different.   * Exactly what data determines a 'theory'? (meaning that M is completely determined from this data - again I would like to have complete description but am happy with undefined terms as long as it is clear that they are such) Ideally, the second and third questions should be answered by any answer to the first question, but I added the latter questions to indicate what confuses me particularly.


title: Will a gas keep forever in a "perfect" flask?

text: I've been wondering about the porosity of materials, I know that, for example the air comes out of tires/balloons because (besides having huge gaps on the rim contact area/knot) they are made of a porous material, but it has nothing to do with molecules, because the porosity appears in a much larger scale. Now, let's say we have a flask made of a solid material, and let's say it's "perfect", I mean, the molecules are perfectly aligned, and it has a normal thickness like 1mm or so, if we fill this flask with a gas and put it in a vacuum chamber*, will the gas come out over time? I don't know if all elements have "alignable" molecules, but you have the idea. *-let's say it's an inert chamber, no gravity, radiation and stuff. I have a few possibilities I could think of:   * It depends on the size of the gas molecule and the gaps between the molecules on the solid material. If it fits in, then yes.   * It isn't possible because the forces keeping the molecules on the solid material together would also keep the gas molecules from transpassing it.   * If the above is correct, after some time the flask itself would disintegrate (?) depending on how strong is the force keeping the molecules together. I know that in normal conditions, objects are always "losing" molecules, but will it happen in this hypothetical chamber also?


title: Analyzing the motion of a ball rolling without slipping inside a hemispherical bowl

text: Consider a solid ball of radius $r$ and mass $m$ rolling without slipping in a hemispherical bowl of radius $R$ (simple back and forth motion). Now, I assume the oscillations are small and so the small angle approximation holds. I wish to find the period of oscillation and I analyze the motion in two ways, first using conservation of energy and secondly using dynamics. However, I receive two inconsistent answers. One or both of the solutions must be wrong, but I cannot figure out which one and more importantly, I cannot figure out why. **Method 1** : We write the energy conservation equation for the ball $mgh + \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = Constant$ from the center of mass, we take the height as $h = R-(R-r)cos\theta$ where $\theta$ is the angle from the vertical. Applying the no slip condition $v = r\omega$ and taking the moment of inertia for a solid sphere $I = \frac{2}{5}mr^2$ we can write the energy equation as $mg(R-(R-r)cos\theta) + \frac{7}{10}mr^2\omega^2 = Constant$ Differentiating with respect to time: $mg(R-r)sin\theta\cdot\omega + \frac{7}{5}mr^2\omega\cdot\alpha = 0$ taking the small angle approximation $sin\theta = \theta$ we get $g(R-r)\theta + \frac{7}{5}r^2\alpha=0$ $-\frac{5g(R-r)}{7r^2}\theta = \alpha$ from which we can get $T = 2\pi\sqrt{\frac{7r^2}{5g(R-r)}}$ **Method 2** : The only torque acting on the ball at any point in its motion is the friction force $f$. So we can write $\tau = I\alpha = fr$ again using the rolling condition $a = r\alpha$ and the moment of inertia for a solid sphere, $\frac{2}{5}ma = f$ The net force acting on the system is the tangential component of gravity and the force of friction, so $F = ma = mgsin\theta - f$ $\frac{7}{5}a = gsin\theta$ taking the small angle approximation and converting $a$ to $\alpha$ we get $\alpha = \frac{5g}{7r}\theta$ and a corresponding period of $T = 2\pi\sqrt{\frac{7r}{5g}}$ Now the solutions are very different and I would appreciate it if someone would point out where I went wrong.


title: What is the relation between General Relativity and Newtonian Mechanics?

text: What is the relationship of General Relativity and Newtonian Mechanics? Namely, which laws does GR replace of Newtonian Mechanics, and which laws of Newtonian Mechanics are incorporated into it. Or is GR a complete replacement and overhaul?


title: significance of maxima and minima of time varying kinetic energy of a system

text: Consider a system of particles where the kinetic energy of the system is varying with time. I'd like to know the significance (or meaning) of the time derivative of the kinetic energy being zero at a point. What is the significance of time instances where the kinetic energy has maxima and minima ?


title: Limitations on how far one can travel in the universe

text: Someone once incorrectly told me that, given the speed of light is the speed limit of the universe, aliens would have to live for hundreds of years if they are to travel distances of hundreds of light years to reach Earth. In a "special relativistic" and non-expanding universe however, this is not the case. As velocity approaches the speed of light, say $v = 0.999c$, then we have $\gamma = \frac{1}{\sqrt{1-\frac{(0.999c)^2}{c^2}}} = \frac{1}{\sqrt{1-\frac{0.998001c^2}{c^2}}} = 22.37$ Let us assume that an alien wishes to travel 100 light years from his planet to Earth. If the alien is travelling at $v = 0.999c$, he will observe the distance between his planet and the Earth to contract, and will measure the contracted distance to be: $Distance = \frac{100ly}{\gamma} = \frac{100ly}{22.37} = 4.47$ Light years. The Alien will be able to travel this distance in a time of : $Time = distance/speed = 4.47/0.999 = 4.47 years$ It is easy to show that as the alien's speed increases, the time taken to travel the 100 light year distance approaches 0. It can thus be shown that thanks to length contraction and time dilation of special relativity, all parts of a special relativistic universe are accessible to an observer with a finite life time. We however don't live in a purely special relativistic universe. We live in an expanding universe. Given the universe is expanding, are some parts of the universe no longer theoretically accessible to observers with finite life times?


title: Step in a proof that $\textrm{div} \ \mathbf{B} = 0$ from Biot-Savart's law

text: Notation: The magnetic field $\mathbf{B}$ generated by a point charge $e$ moving with velocity $\mathbf{v}$ is given by Biot-Savart's law $$\mathbf{B} = \frac{\mu_0 e\ \mathbf{v} \wedge \mathbf{r}}{4\pi r^3}$$ where $\mathbf{r}$ is the vector from the charge to the point at which the field is measured, $r = \left| \mathbf{r} \right|$, and $\wedge$ denotes vector product. Question: According to my book: Since $\mathbf{r}\, / \, r^3 = - \,\textrm{grad} \left(1\, /\, r \right)$, we have $$\textrm{div} \left( \mathbf{v} \wedge \frac{\mathbf{r}}{r}\right) = \mathbf{v} \wedge \textrm{curl} \left( \textrm{grad} \frac{1}{r}\right) = 0.$$ What I have is $$\textrm{div} \left( \mathbf{v} \wedge \frac{\mathbf{r}}{r}\right) = \mathbf{v} \, . \, \textrm{curl} \left( \textrm{grad} \frac{1}{r}\right) - \left( \textrm{grad} \frac{1}{r} \right) \, . \, \textrm{curl} \ \mathbf{v},$$ so I think there is a typo in the book ($ \, \wedge$ should be $\, . \,)$. However, I still don't know how to go from my equation to the correct one. Is it because $\textrm{curl} \ \mathbf{v} = 0$? If so, why? I understand that the curl of a gradient vanishes identically.


title: If a planet were cut in half but stayed hemispherical, how hot would the ocean on the flat side be?

text: Some time ago I asked a question about gravity on a hemispherical planet. What would gravity be like on a hemispherical planet? Would the water all boil away at first, quickly cooling the core of the planet? Would the ocean boil for centuries?


title: model for flexible stick

text: I'm trying to model a flexible stick with a partial differential equation. I want one of the ends to be fixed and the other end to swing. Do you guys know of any good models I can use? Any references would be appreciated.


title: Is it humanly possible to change Earth's axis?

text: Just what the title states with the qualification that the change must be affected without using other celestial bodies as mentioned in the Clarke/Baxter SF 'Sunstorm'. Obviously given the momentum of Earth it would require a major something to break the inertia. But something like the way an ice-skater whilst spinning on their own axis can apply a touch of heel/toe at the correct point to change the way they spin. For instance, change the rate at which our oceans move with the spin of Earth ... This is a 'curiosity' question rather than a real-life problem so please feel free to vote to close.


title: Electric Field: distributed uniformly in one infinity tape of length

text: > _One charge density surface is distributed uniformly in one infinity tape of > length with $2a$ width from distance $d$. Determine the Electric Field in > the point perpendicular from the distance $d$ of the centre of tape._ Answer: > $\frac{\rho_s}{\pi\epsilon_0}\tan^{-1}(a/d)$ PS.: I tried every ways (Coumlomb Law and Gauss Theorem) Gauss Theorem (I guess closer) Consider cylindrical surface: $Q = \int_v \rho_v dv = \rho_s.2a.L$ (Call L as infinity length) $Q = \oint_s \vec{D}.d\vec{S} = D_r . 2\pi.d.L$ $\Rightarrow E = \large\frac{\rho_s}{\pi\epsilon_0}\frac{a}{d}$ Coulomb http://i.stack.imgur.com/Ks4gC.jpg The image woth more than thousand words. Please I'd like some advise to solve this problem. http://i.stack.imgur.com/Ks4gC.jpg


title: Is there really no meaning in potential energy and potential?

text: I have been told all my physics life that potential energy between two mass/charge has no meaning and only their difference has meaning. The same goes for electric potential, only the difference matter. Perhaps I am not understanding it correctly, but before I talk about masses, let's talk about potential energy/potential associated with two charges. I am not sure where had I seen it, but a long time ago I was presented with a problem like this. Let's say I have a +Q and a -Q. What is the potential energy between them? The change in potential energy is the negative work done by the conservative force namely (vector sign and dot product got rid of, since the cosine 1) $\Delta U = -\int_{a}^{b} k(Q)(-Q) \frac{1}{r^2} = -k(Q)(-Q) \left. \frac{-1}{r} \right |_{a}^{b} = -k(Q)(-Q) \left (\frac{1}{b} - \frac{1}{a} \right) $ Now I assume that I brought it from infinitely far, so that 1/a = 0 In that case I am left with $\delta U = \frac{kQQ}{b}$ Here are my questions 1) A long time ago, I saw a formula looking EXACTLY like what I just did there, but it doesn't concern with the _change_ , it's just gives me the potential energy. Now the formula I remember was $U = -\frac{kQq}{r}$ where there is a negative. I thought this formula already takes care of the signs? Or am I wrong? 2) Kinda the same concept. If I tell you some charge (not telling you the sign) has a electric field and you have another charge (which is the test particle, not telling you the sign again even though it is conventional to use + charge ) somewhere in that field. I tell you that the electric potential at that point (not the difference) is K (where K is positive number). What can you conclude, if anything? What if it were negative? Suppose I tell you suddenly that the charges are the same signs, and I give you a location in which the electric potential is positive (I think it has to be). What does it mean? EDIT: Let me also just clarify a bit that I was taught that electric potential (not difference, I stress again) is the work that someone does to bring a charge from infinity to some point whereever. In my book however, it's defined as $V = \int_{R}^{\infty} \vec{E} \cdot \vec{ds}$


title: Relative change in drift velocity

text: Consider an ensemble of electrons which all experienced a collision at time $t=0$. Let $n(t)$ denote the number of electrons in this ensemble. Assume that the number of electrons $\mathrm{d}n$ from this ensemble experiencing collisions in a time $\mathrm{d}t$ is proportional to $n$, i.e. $$\mathrm d n = -an(t) \mathrm d t,$$ for some consant of proportionality $a$. **Edit.** Also assume that at $t=0$ an electric field is switched off. Why is the relative change in drift velocity $\frac{\mathrm d |\langle \mathbf v \rangle|}{|\langle \mathbf v \rangle|}$ equal to the relative change in the number of electrons which have not yet experienced a collision$-\frac{\mathrm d n}{n}$? **Edit.** Noting that the thermal drift velocity is always zero, the contribution of an electron, which experience a collision after $t=0$, to the average velocity will vanish as soon as it experiences a collision. Because then it loses the drift velocity that was induced by the electric field.


title: What is the difference between S/cm and Scm-1?

text: Conductivity is noted in S.cm-1 in this http://pubs.acs.org/doi/abs/10.1021/ja502765n i wanted to compare it to the conductivity values listed on wikipedia for common materials. i could not find information on what the difference was through google's broken search engine.


title: Net work output of an engine performing Carnot cycle?

text: My problem gives me a Carnot cycle heat engine with water as its working fluid, with $T_H$, $T_L$, and the fact that it starts from saturated liquid to saturated vapor in the heating process. I need to find the net work output of this engine. So my solution goes like this: $$\eta_{HE} = 1 - \frac {T_L}{T_H}$$ So I get $\eta_{HE}$. And then I also know that $$\eta_{HE} = \frac {W_{HE}}{Q_H}$$ Since my goal is $W_{HE}$, I know I'll need to solve for $Q_H$ first. Since I have the state and temperatures of water at both states, I can get $h_H$ and $h_L$. I know that $h_L - h_H$ is some quantity of energy, and it must be either $Q_H$ or $Q_L$. The problem is, I don't know which it is. And I assume it is $h_L - h_H$ because it is heated first and then cooled. I hope this is correct. I know that the change in enthalpy is the energy either released or absorbed by the system. In this case, since the temperature moves from higher to lower, it seems to me that the system is absorbing energy and hence this must be $Q_H$. But it also makes sense to me to say that $h_L - h_H$ is the energy released by the system (as in an exothermic reaction), so it can be $Q_L$ too, because that is the heat rejected by the system. So my question is, for the general case, which is it? Or is it situational? Does its being $Q_H$ or $Q_L$ depend on the resulting sign when I do my calculations? For this problem, the sign of $h_L - h_H$ that I computed is positive. Then this looks like heat entering the system and so it is $Q_H$. Am I correct? Or, am I completely on the wrong track and should be chasing down another solution?


title: Can the motion of a cracking whip be described as the interference of two waves?

text: I was watching a whip crack in slow motion and I noticed that the motion of the whip could be described using two different circular descriptions. 1) the user circles the whip around over his head, creating a rotation with a period. 2) next, he jerks the whip, causing a different and faster movement of the whip. the new movement appears to travel down the whip (which is already moving) until it reaches the tip. At this point, the tip creates a loud crack, which is associated with a supersonic movement. In my mind, the two different motions cause an visual effect similar to a (constructive?) interference pattern. Can the cracking of a whip be described as the interference of two waves? If so, then what is going on?


title: In a universe that is expanding at a constant rate, do objects that are attracted to each other feel a force opposite to their attraction?

text: In this article, the authors make the claim (pg **44** ) that "Expansion by itself—that is, a coasting expansion neither accelerating nor decelerating—produces no force." I'm having a hard time convincing myself of this. I think about two bonded atoms and the space between them expanding. While I don't doubt that the structure remains intact, it seems to me that the attracting force will always have to play "catch-up" against the expansion which is pushing the particles apart. Where am I going wrong in thinking of expansion as a force which acts counter to the attraction? Does anyone have a metaphor handy? Another way of stating my question: with large enough (but not accelerating) rate of expansion, would we get to a point where molecules broke apart? EDIT: In this related question, the agreed-upon answer seems to be that the expansion of space manifests itself as a force.


title: What is the intuition behind the direction of torque?

text: I've seen the other posts on this question, but unfortunately I'm still having difficulty understanding the meaning of the direction of the torque vector. It makes intuitive sense that the magnitude of the torque exerted on an object is dependent on the length of the lever arm and the angle between the lever arm and the applied force. What I don't understand is what the direction of the torque vector is describing. How can I relate the direction of the torque vector to a real world example (that is not an example involving a wrench + screw and the screw "unscrewing" in the direction of the torque...which doesn't give me any more of an intuition on what torque means). Thanks!


title: Møller scattering: twisted?

text: I am studying the Møller scattering, but I don't know how to get the twisted diagram from the S-matrix. Has anybody a good explanation?


title: If one were to move withing a tesseract, would the direction of gravity be different within each cell

text: If you unfold a tesseract into 3D space you get a cross shape (basically). Animated, it looks like the bottom most cube becomes inverted from it's 4D orientation. Would that mean that the pull of gravity within that cell would be opposite of the other cells when in 4D?


title: Where is the mistake in the following DC circuit analysis?

text: In the following circuit, I'd like to calculate the potential difference $V_{BA}$. One way is to solve the circuit, by calculating all the currents $I_1, I_2, I_3$. The system of equations is: $\\{I_1 + I_2+I_3 = 0, 6 - 2 I_2 +2 I_1=0, 12 - 2I_3 + 2I_2 = 0\\}$. The solution is $(I_1, I_2, I_3) = (-4, -1, 5) mA$. Then $V_{BA} = 2 I_3 = 2 \times 5 = 10V.$ I'd like to solve the same problem via a different route. I put the GND at the negative pole of $-6V$ source. Then for point A, I write: $\frac{0-V_A}{2} + \frac{V_A - 6}{2} + \frac{18-V_A}{2} = 0$, giving me $V_A = 12V$. Then $V_{BA} = V_B - V_A = 18 - 12 = 6V$, that is clearly wrong. Where is my mistake ? ![enter image description here](http://i.stack.imgur.com/Y5oPS.png)


title: Can a photon be made to orbit a known (or undiscovered theoretical) body?

text: Can a photon through some process be made to orbit a celestial or any other object? Two follow-up questions.   1. Can this orbit be described as the photon crossing its own path.   2. Will this wave-function be effected by positive interference. To the effect of increasing frequency?


title: how far away can something be from the earth and still be in orbit?

text: Just as the title asks, How far away can, say, a satellite be and still be in "orbit" ? How about for a given velocity? _Fun Facts_ 200 miles (320 km) up is about the minimum to avoid atmospheric interference. The Hubble space telescope orbits at an altitude of 380 miles (600 km) or so. _potentially helpful numbers_ mass of Earth = 5.97219 × 1024 kilograms mass of the Moon = 7.34767309 × 1022 kilograms distance (earth, moon) = 238,900 miles (384,400 km)


title: What force particle mediates electric fields and magnetic fields?

text: The force carrier for magnetic fields and electric fields are supposedly photons. I don't get it: 1) Wouldn't that mean that a charged particle (e.g. an electron or even a polarized H2O molecule) would constantly be losing endergy from sending out photons? 2) Wouldn't that mean that an electric field is inseparable from a magnetic field, as photons have both - and that one can't have one without the other? 3) Would it be possible, then, to determine the wavelength of magnetic-field- mediating photons? If so, what is the wavelength - is it random or constant? 4) How can a photon (which has momentum) from one electrically charged particle to an oppositely charged particle cause these particles to _be pulled toward each other_ \- or how can a magnetic field cause an electrically charged moving particle to experience a force _perpendicular to_ the source of the magnetic field if a particle with a non-zero mass moving between the two is the mediator of that force? If "virtual photons" are involved, please explain why they work differently from regular photons.


title: Projectile motion question: vertical and horizontal graph

text: I've have got some vertical and horizontal distances for a projectile-like motion. In order to work out the trajectory, why is it better to plot on the x-axis, "horizontal distance^2", and on the y axis, "vertical distance"?


title: Physics related Podcasts

text: > **Possible Duplicate:**   >  Are there any good audio recordings of educational physics material? In the same way that was already asked about good books of Physics in this StackExchange, I would like to know good physics podcasts! What are the most informative and enjoyable to hear?


title: How does charge flowing between emf terminals reduce voltage difference?

text: I'm currently learning what electromotive force is and while reading my book's description of an ideal source of emf, I had difficulty understanding what these sentences mean: > The nonelectrostatic force maintains the potential difference between the > terminals. If it were not present, charge would flow between the terminals > until the potential difference was zero. I don't quite get what it means by this. How does the nonelectrostatic force "maintain" the potential difference between the terminals? If it wasn't present and only the electric force remained, why would the potential difference decrease to zero? The only thing I could think of was that the electric force from the E field inside the source of emf would still be exerting forces on charges from high potential to low potential, turning its electric potential energy into kinetic energy by doing work on the charges. I still don't know how that would reduce the voltage difference across both of its terminals to zero though. Regarding the ideal source of emf, the book assumes there is a positive and a negative terminal with the positive terminal at a higher potential and the nonelectrostatic force going from low to high potential while the electric force goes from high to low, just in case that helps clear any ambiguity.


title: A quantum finite square well

text: I have a problem with the argument of a finite square well. ![enter image description here](http://i.stack.imgur.com/HVL3H.png) ![enter image description here](http://i.stack.imgur.com/O1rSN.png) The stuff I read has mentioned that the Curvature " second derivative " is opposite sign of the wave function only when the E larger than V where E = energy of the particle V = potential.This is how we obtained oscillatory solution (when E is larger than V) ( I understood this part ) And For E smaller than V ,it is true that we obtain exponential solutions, but can we use the same argument I used above? Is it true that as u = ( exp(-ikx ) decays the curvature has the same sign, so the negative gradient starts getting to be more positive but less and less positive until it tails down to infinity? since now second derivative u depends on the value of u and the difference between E and V. Is it a correct way of interpreting this phenomena? Appreciate any idea or help from every one.


title: Calculating Electric Field from a Potential Difference w/o Charge

text: So here is my homework question: > _Two long cylindrical shells of metal (radii $r_1$ and $r_2$, $r_2 > r_1$) > are arranged coaxially. The plates are maintained at the potential > difference $\Delta\phi$. The region between the shells is filled with a > medium of conductivity $g$. Use Ohm's law, $J = gE$, to calculate the > electric current between unit lengths of the shells._ So I understand how to go from the current density to the actual current, but I need to find the electric field $E$ in order to find the current density $J$. I am not given the charges on each cylinder. How do I calculate the $E$ field when I am only given a potential difference? I know $\Delta\phi = \int E\,\mathrm{d}\ell$ but I feel like that doesn't help me much.


title: Transparence of an infinite square well?

text: What does it mean by an infinite square well being **transparent**? I have been doing the calculation of the infinite square well and I came up with an answer $T = 1$ where $T$ for Transmission coefficient. But I can't really tell what it actually means in terms of physics. I would imagine a particle to be trapped in a infinite square well to be the inner electron very close to a Big nucleus, So what does it mean by transparent in this context? Does it mean No other wave function can interact with this electron? If there is an infinite potential, why would wavefunction still be able to pass through it but not getting bounce off the edges? of the well? ![enter image description here](http://i.stack.imgur.com/DgX2U.png) ![enter image description here](http://i.stack.imgur.com/3Ha97.png)


title: Can you lift yourself standing on a wooden block using pulley mounted on block?

text: I am new to this site. Please help me figure out this problem. Is it possible to lift oneself like shown in schematic? Waiting for good suggestions. Chirag![Reference Schematic](http://i.stack.imgur.com/W0jHx.jpg)


title: Did relativity make Newtonian mechanics obsolete?

text: Did Einstein completely prove Newton wrong? If so, why we apply Newtonian mechanics even today? Because Newton said that time is absolute and Einstein suggested it relative? So, if fundamentals are conflicting, how can both of them be true at a time?


title: How to describe hadron-hadron strong interactions from QCD?

text: I'm trying to set up the problem of deriving the thermodynamics of hadronic matter. I know how to proceed in the case of an effective description such as mean field (Walecka/linear sigma model) but I'm trying to start from QCD. I do not wish to actually obtain the partition function, but I want to at least set up the complete problem. In the mean field approximation, we usually write baryon and meson fields and baryon/meson couplings as the interactions, meaning that we also bypass the problem of describing the hadrons as QCD bound states. On the extreme oposite, I know how to proceed in the case of pure quark matter (it's just the "QCD Lagrangian"). How do I write the full Lagrangian? How do I represent the hadrons and the hadron-hadron strong (well, and EM) interaction without shifting to an effective approximation? Can it be done with quarks and gluons as the basic degrees of freedom? I don't have any compromise with the "practical usefulness" of the result and I'm not particularly interested on the peculiarities of the dynamics of the bound states.


title: Relative velocities of boats and a stone thrown by the boat

text: > **Question:** A police boat is chasing a boat with criminals along a > straight river by moving against the stream. The speed of the river stream > is 3 miles per hour, the speed of the boat with criminals relative to the > river is 30 miles per hour, and the police boat is 4 miles per hour faster > than the boat with criminals. > > Currently the criminals are ahead of the police, and horizontally throw a > stone at the police boat at a speed 16 miles per hour relative to their boat > (i.e. relative to the boat of criminals). > > What is the horizontal velocity of the stone relative to the police boat and > to the river bank? You need to state what the origin and the positive > direction of motion are. Choose the direction of the river stream to be the positive direction and choose the origin to be in front of criminals' boat. Denote the velocity of the river stream by $\dot{x}_R$ Denote the velocity of the police boat by $\dot{x}_P$ Denote the velocity of the criminal boat by $\dot{x}_C$. Denote the velocity of the stone thrown by $\dot{x}_S$. From the question we have that $\dot{x}_R = 3$, and we also have that the velocity of the criminal boat relative to the river stream is \begin{align*} \dot{x}_C - \dot{x}_R &= -30 \\\ \implies \dot{x}_C - 3 &= -30 \\\ \implies \dot{x}_C &= -27 \end{align*} Note that the minus 30 is because from the point of view of the river, the criminal boat is travelling in the negative direction. Now from $\dot{x}_C$ I can calculate $\dot{x}_P$, since \begin{align*} \dot{x}_P &= \dot{x}_C + (-4) \\\ &= -27 - 4 \\\ &= -31 \end{align*} Note that the minus 4 is because the police boat is 4 mph faster, but in the negative direction. Also from $\dot{x}_C$ I can calculate $\dot{x}_S$. From the question we have that \begin{align*} \dot{x}_S - \dot{x}_C &= 16 \\\ \implies \dot{x}_S - (-27) &= 16 \\\ \implies \dot{x}_S &= -11 \end{align*} I was wondering if my solution was right, even though $\dot{x}_S = -11$ is negative, even though it moves in the positive direction.


title: fraction of magnetic energy stored outside a solenoid

text: If I have a long solenoid, e.g. length $l$ and radius $r$ with $l = kr$, where k >> 1, with a nonpermeable (e.g. air) core, how much of the magnetic energy is stored outside as compared to inside? If I go by the Wikipedia article on solenoids, $B = \frac{\mu Ni}{l} \to H = \frac{Ni}{l}$ inside the coil, so the energy density should be $\frac{1}{2}\mu H^2$, and therefore the total energy inside the coil is $\frac{\mu N^2i^2}{2l^2}l\pi r^2 = \frac{\mu N^2 i^2}{2l}\pi r^2$. The inductance is supposed to be $L = \frac{\mu N^2 A}{l} = \frac{\mu N^2 \pi r^2}{l}$, and if you compute energy $\frac{1}{2} Li^2$ you get $\frac{\mu N^2 i^2 \pi r^2}{2l}$ which is the same answer as the energy inside the coil calculated above, which relies on approximations, so we can't subtract these and figure out the energy stored outside. Are there practical rules of thumb?


title: Birds inside of a closed box

text: Suppose there's a closed box with birds inside, on top of a scale. If the birds started flying inside, would the box get lighter?


title: Weighing a flying bird

text: Let thing of a bird standing still in a box on top of a weighing machine that shows a mass $m_0$. Now, imagine that the bird is flying, still in the same box and the same weighing machine shows a mass $m_1$. As the bird when flying, is applying a force towards the weighing machine, could we deduce that $m_0 = m_1$? I'm asking this question because saying that these masses are equal makes me as uncomfortable as saying that they are not equal and can't figure the right answer. So does $m_0 = m_1$ and why?


title: Bird flying in a cage

text: Assume that you are holding a cage containing a bird. Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage?   Does it make a difference whether the cage is completely closed or it has rods to let air pass?


title: Why is a nucleus isotropic?

text: I believe in Neutron Scattering the neutrons after hitting a nucleus can bounce in any of 360*3 dimensions -> 1080 degrees? Why is this so? Shouldn't it only bounce "off" the neutron in approximately the same "direction" that it came in such as when a particle bounces off a mirror -> because of the cross-section ...


title: Proof that QCD is the theory describing strong interactions?

text: I would like to ask what are the experimental evidences that led to the conclusion that QCD is the right theory to describe strong interactions. I know that some of the key point are the decay of $\pi_{0}$ and the measurement of Jets but I'd love to see a full answer to this question. Is there a still a chance to Regge theory nowadays?


title: How useful is programming in physics?

text: I have been wondering recently how useful programming is to a physicist. It seems fairly useful (simulations are a lot cheaper than the actual thing in many cases) in some areas (say space programs), but in other areas (for example at CERN) it seems pointless. So is it useful and perhaps how do you use it?


title: Why do we need a metric to define gradient?

text: For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} \right)$. In which way do we need a metric? But some people did tell me only on a Riemann manifold (with a metric) can we define the gradient.


title: Newtonian Mechanics problem?

text: > The ball of mass $m$ is given a speed of $v_a = \sqrt{3gr}$ at position $A$. > When it reaches $B$, the cord hits the small peg $P$ after which the ball > describes a smaller circular path. Determine the position $x$ of $P$ so that > the ball will be able to reach point $C$. ![enter image description here](http://i.stack.imgur.com/58MPI.png) **My Work** Let $T$ represent kinetic energy and $V$ potential energy. Let $O$ be the datum for gravitational potential energy. $T_1 + V_1 = T_2 + V_2$ $1/2m(3gr) - mgr = mg(r-x)$ $3/2r - r = r-x$ $x = r-3/2r + r$ $x = 1/2r$ However the correct answer is $x = 2/3r$. What's my mistake? Thanks a lot.


title: Why do electricity generators have to work harder for higher loads?

text: Maybe this is a silly question, but why does, say, a gasoline-powered AC generator have to use more gas depending on the load? Let's say I have a 120VAC generator and either a 1A or a 10A load, and assume it can handle 1200W without issue. For 10A, the generator uses more gas and works harder compared to 1A. Is this because the 10A is passively causing a magnetic field that makes the generator shaft physically more difficult to spin? If so, why does the magnetic field oppose the generators movement if the current is flowing in the direction the generator is trying to make it go to begin with? (Sorry, I know that is probably silly but I have a very limited understanding of electricity and magnetism.) Or is it because the circuitry in the generator is actively sensing voltage drop due to higher load, and increasing the throttle to maintain a constant output voltage? Sort of like I'd imagine a water pump would have to work harder to maintain a constant pressure in a system with a leak in it or with flow. (Related: If so, is this type of sensing _necessary_ for a generator to limit the voltage to low loads, or is it just a bonus feature to increase efficiency by not running at full throttle all the time?) Or is it some combination of both? Or are both of those somehow the same thing? (It seems like it must be a little bit of the former; because the generator stalls under too high loads, which I guess means something is resisting the spin - unless the generator just actively shuts itself down?) What is the process from "higher load" => "higher fuel usage"?


title: Law of equipartition

text: Law of equipartition predicts the heat capacity of gases correctly. It assumes that inter-molecular attraction in gases is negligible (which is true). But for solids, inter-molecular attraction is not negligible, the, how come it still predicts the correct value for molar heat capacity? How can we ignore the potential energy due to inter-molecular attraction in solids? Each oscillator has two degree of freedom (kinetic and potential). Solid has oscillators in all the 3 dimension. So, total degree for freedom will be $6$. Using equipartition, the internal energy of one mole a solid will be $ U = 6 \times \frac{1}{2}RT = 3RT$. Heat capacity, $C = \dfrac{dQ}{dT} = \dfrac{dU + PdV}{dT} = \dfrac{dU}{dT}$ $(\because V \text{ is constant})$ $\implies C = \dfrac{d}{dT}[3RT] = 3R \approx 25 J mol^{-1} K^{-1}$ Which is a very close approximation of the actual values. My question is why is this prediction in agreement with experimentally determined values, when we have ignored the potential energy due to inter- molecular attraction. Which is not negligible for solids?


title: 5D Ricci Curvature

text: As part of a hw problem for a class, we're supposed to be deriving the equivalence given in equation 2.3 of this paper http://arxiv.org/abs/1107.5563. I was wondering if there is some special relation involving the Ricci Curvature in 5d's relationship to one in 4d. Since with a general metric like the one given in 2.1, calculating the Christoffel symbols would seem to be an enormous and not particularly smart idea.


title: Asymptotic Freedom - Qualitative Explanation

text: I am doing a (mostly qualitative) course on Particle Physics, and am confused about the concept of asymptotic freedom. The lecture notes basically say that a quark may experience no force/be "unbound" temporarily as a result of a collision. (due to properties of the strong force) Is there all there is to it? Later, it mentioned that asymptotic freedom is important in electron-positron annihilation into hadrons. Were there no asymptotic freedom, the cross section of the process would be different. I can't see how this follows on from what was said above. So I am seeking an qualitative explanation of this concept, and perhaps something about its consequences as well.


title: Electric Potential of a Cube Made of Point Charges

text: I'm trying to find the potential energy of multiple geometric shapes made entirely out of point charges. This particular shape is a cube made out of two different point charges, A and B, each separated by a constant distance. I'm not entirely sure how to go about this problem but I'm pretty sure that the equations V=Ed and E=kQ/r^2. I'm pretty sure you just add up the charges but I have no idea how to do this in a cube form..


title: Electromagnetic spectrum

text: I understand that the electromagnetic spectrum is made up of different frequencies of light waves, but is this true in all cases such as with longer wave frequencies? "such as with microwaves". sometimes I get the impression that with microwave ovens for example use waves of electrons, so my question is, is there some kind of threshold where somewhere between infrared to microwaves does the frequency makes the conversion from light to electrons? (and if so, how?) or is it always light and has some other correlation such as light influencing the electrons thru the air?


title: Why does lightning cause sound?

text: If I understand correctly, lightning is the discharge of electricity from the atmosphere into the planet. However, if I switch on a lamp, the wires are not causing thunder (or any audible sound). I've also heard that the thunder comes from lightning breaking the sound barrier. This sounds weird to me since I would assume that lightning would be traveling at, well, light speed, so I'm not sure how the threshold could be crossed. How does lightning cause thunder?


title: Total Resistance of Infinite Resistor Grid?

text: The problem of the infinite resistor grid is very common. The solution for the resistance between any 2 nodes in an infinite resistor lattice is all over the internet. My question is somewhat similar but more pragmatic. If we had a grid that was very large but yet finite... Then what would be the average voltage drop across a given grid for a given current density? For arguments sake, a grid in the region of say 4000 by 4000. Maybe it would be safe to assume an infinite grid(?) Very interesting Q. Can anyone shed any light?


title: Bungee jump physics

text: Question: A bungee jumper jumps from a bridge. The length of the loose rope is 30 m. When the jumper reach the lowest point possible, the rope stretches 10 m. What is the final stretch of the rope, when the oscillation of the rope stops? Mechanical energy loss is null. Hook law applies for the elastic. I think this problem is unsolvable, because we need mass or the coefficient of the rope?


title: Is there a measure of internal energy flow?

text: A system might have internal energy and/or kinetic energy. Kinetic energy in classical mechanics is a form of energy the object has, only because of its relative movement to other objects. If you have a system like an oscillator, then you have a system where the amount of internal energy (energy of the spring, say) and the kinetic energy oscillate periodically. The amount of energy that is flowing from "internal energy" to "kinetic energy" and back therefore also meassures how this system is different from a system with might have the same amount of energy in total, but all its energy only stored in form of internal energy. Is there a reasonable meassure of this difference, that is the ''energy-form flow''? Maybe such a meassure might also be related to the entropy, because if there is kinetic energy around, I expect the entropy to be higher than otherwise. Then I was thinking mabye that something like $\frac{dE_{kin}}{dt}$ might therefore be relevant. But the only quantity which directly puts energy and time together is the action. However, I'm not sure if the action has an illustative meaning, which goes beyond its definition, anyway.


title: boundary conditions for liquid with surface tension

text: so one uses equations of motion to describe liquids (e.g. Navier–Stokes equations). These are equations for $\vec{v}(\vec{r},t)$ with boundary conditions on the surface $S$ of the liquid (e.g. $\vec{v}(\vec{r}\in S,t) = \vec{0}$). How should one incorporate surface tension $\sigma$ in these equations/boundary conditions? It seems, only boundary conditions must change, and $\Delta p = \sigma (1/R_1 + 1/R_2)$ is the first thing that comes to mind, but how to get $1/R$ from $\vec{v}(\vec{r},t)$?


title: The gauge-invariance of the probability current

text: It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\\ \phi\to\phi-\frac{\partial \chi}{\partial t}\\\ \psi\to \psi \exp\left(\frac{iq\chi}{\hbar}\right)\end{cases}$$ The Schrodinger equation $$\left[-\frac{\hbar^2}{2m}\left(\nabla-\frac{iq\vec A}{\hbar}\right)^2+q\phi \right]\psi=i\hbar\frac{\partial}{\partial t}\psi$$ gives back the same equation. How does it follow that the probability current is gauge invariant?


title: Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?

text: Quantum fields are presented as operator-valued _distributions_ , so that the _operators_ in the theory are linear functionals of some test function space. This works well for free fields, giving us a particular form for VEVs, in which the 2-point connected correlation function determines the whole theory, and a trivial S-matrix. For interacting fields, however, the VEVs and the statistics of S-matrix observables that are predicted by the theory depend not only on the test functions (generally given for the S-matrix in terms of pure, improper wave- number modes of the free field at $t=\pm\infty$) but also on the "energy scale of measurements". One obtains a different functional form at different energy scales. The energy scales at which one is measuring are already encoded in the wave- numbers associated with the test functions, so it seems that the test functions play extra duty, to determine the functional form of the VEVs (at least in their S-matrix form) both by linearly smearing the quantum field and, more indirectly, by determining the renormalization scale. As a consequence, the VEVs, the Wightman functions, seem clearly to be nonlinear functionals of the test functions for interacting QFTs. I've worried at this for a while. The argument as put brings into question the Wightman axioms' insistence that a quantum field must be a linear map from the test function space to the space of operators, although I suppose the effect would be only logarithmic in some measure of the frequencies of the test functions. The Haag-Kastler additivity axiom also seems problematic. I'm not aware of weakening this particular aspect of the Wightman or Haag-Kastler axioms so as to be able to construct models of the axioms that parallel empirically successful QFT having been discussed in the literature, but have they? [Note that dropping the linearity of the map from test functions to operators in the algebra of observables does not affect the linearity of the algebra. There's still a perfectly good probability interpretation of a state over a *-algebra and a perfectly good Hilbert space, but the relationship of measurements to space-time is somewhat modified.] EDIT: I'll _attempt_ to make the Question slightly more accessible. It turns out that what follows makes it significantly longer. The Wightman functions (effectively a different name for Vacuum Expectation Values (VEVs)) allow the reconstruction of a Wightman (quantum) field. That is, given the vacuum state over the algebra one can construct the Wightman functions, $$W(x_1,x_2,...x_n)=\left<0\right|\hat\phi(x_1)\hat\phi(x_2)...\hat\phi(x_n)\left|0\right>,$$ but also, given the Wightman functions one can construct the algebra. Wightman functions satisfy somewhat arcane relationships, so that just any function $VV(x_1,x_2,...,x_n)$ is almost certainly not a Wightman function and cannot be used to reconstruct a Wightman field. The object $\hat\phi(x)$ is not an operator in the algebra of observables, it is an _operator-valued distribution_ , so that if we have a well-behaved "test function" $f(x)$, where "well-behaved" most often means that it is smooth (infinitely differentiable) and has a smooth fourier transform, then $\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x$ _is_ an operator. This is often called "smearing" with the test function $f(x)$. The test function $f(x)$ effectively tells us what relative amplification we apply to the field at each point in space-time for a given measurement, and its fourier transform $\tilde f(k)$ tells us what relative amplification we apply to the field at each point in momentum space. In signal analysis the same object is called a "window function", which can be looked up on Wikipedia. One can talk about the bandwidth of a measurement, or say that a signal transformation is high-pass or low-pass. If it were not for non-commutativity of measurement operators at time-like separation, quantum field theory would be _exactly_ what one would want for a mathematical formalism for modeling stochastic signal processing. If one only talks about measurements that are space-like separated --admittedly a tight constraint--, QFT _is_ a stochastic signal processing formalism. In contrast to $\hat\phi(x)$, $\hat\phi_f$ is not a singular mathematical object; the difference is comparable to the difference between a smooth function and the Dirac delta function. In terms of these operators, the Wightman functions become $$W(f_1,f_2,...,f_n)=\left<0\right|\hat\phi_{f_1}\hat\phi_{f_2}...\hat\phi_{f_n}\left|0\right>,$$ which is linear in all the test functions. When we move to the practical formalisms of interacting QFT, however, the VEVs predicted by the bare theory are infinite. Renormalization fixes that in a Lorentz invariant way, but it leaves the VEVs a function of the energy scale $\mu$ of the measurements that are involved in an experiment. Thus, one finds that the Wightman functions are now a function of $\mu$ as well as of the points $x_i$, $W_\mu(x_1,x_2,...x_n)$, or, in terms of operators, $W_\mu(f_1,f_2,...f_n)$. [It's perhaps worth noting that we don't know whether these $W_\mu(...)$ satisfy the relationships required for them to be Wightman functions, so that we could reconstruct a Wightman field, but I've never seen a discussion of perturbative QFT conducted in such terms.] The Wightman functions in terms of positions do not encode energy scales, but they are _not_ observables, they are only a template for constructing observables. The fourier transforms of the test functions, $\tilde f_i(k)$, however, already determine various energy scales at which the measurements made in an experiment operate, potentially in _much_ more detail than a single number $\mu$. We could just say that the energy scale of measurements $\mu$ is separate from the energy scales determined by the test functions $f_i$, but it seems more natural _to me_ to say that $\mu$ is a functional of the test functions $f_i$, in which case we can write $W'(f_1,f_2,...f_n)=W_\mu(f_1,f_2,...f_n)$, where $W'$ is now a non- linear functional of the $f_i$. I suppose we expect $W'$ to be a symmetric function of each of the $f_i$. For a more-or-less trivial example, suppose we have two ordinary quantized Klein-Gordon fields $\hat\phi_f$ and $\hat\xi_f$, not necessarily having the same mass spectrum, then we could construct a quantum field $\hat\Phi_f=\hat\phi_f+\hat\xi_{f+\epsilon f^2}$ (where the square $f^2$ is $[f^2](x)=[f(x)]^2$, so that $\hat\Phi_f$ satisfies microcausality), so that $\hat\Phi:\mathcal{S}\rightarrow\mathcal{A};\hat\Phi:f\mapsto\hat\Phi_f$ is a nonlinear map from the test function space into the algebra of observables. This is still a Gaussian field, so not interacting. Already there start to be significant complications. One of the most fundamental Wightman axioms is the restriction to positive spectrum, which in terms of test functions is a projection to forward light-cone components of the test functions. Only components of $\tilde f(k)$ for which $k_\alpha$ is in the forward light-cone have any effect on measurement results. In the above, however, forward light- cone components of $f^2(x)$ contribute, which in fourier space, where $f^2(x)$ is a convolution, means that negative frequency components of $f(x)$ will contribute. Although this means that this kind of nonlinearity apparently does not satisfy positivity of the energy, it's not clear that this means such a system is not **_stable_** , which is the (non-axiomatic) reason why we insist on positivity of the energy as an axiom. Which, whoops, brings into question positivity of the energy, one of the _big_ axioms, which I asked about here. Now, of course, I'm trying out doing research here as if it was a virtual blackboard. Many of the ideas above ought to be rubbed out, different ideas tried and equally or more vigorously discarded. I've got lots of pieces of paper that have discarded ideas on them that you don't want to see, because I hope you have your own discarded ideas, as anyone who doesn't get too fixed in their ways should, most of which I probably don't want to see (not that I'm not a little fixed in my ways, but I've discarded _some_ ideas). The trouble is that I can't see how we can go forward from me just typing away, I think the feedback loop may be too long for this to work. Still, the alternative is to write a paper and get feedback six months later. This introduces Questions that I may try to put on meta. By the way, making a Question Favorite counts for no Reps, but it's much appreciated.


title: How sound intensity (dB) and sound pressure level (dB) are related?

text: Source: http://www.engineeringtoolbox.com/sound-power-intensity-pressure- d_57.html Both sound intesity and pressure level are measured in dB. Given a specific sound, are these two dB values the same?


title: Duality and Fourier Transforms

text: I read that $(FF(f))(x)=2\pi f(-x)$, where $F$ is the Fourier transform _and_ $F(f(x-a))(k)=\exp(-ika) X(k)$ where $X(k)=F(f(x))$ _implies_ $F(\exp(iax)f(x))(k)=X(k-a)$. But I don't see how that is done... I am quite happy with getting $F^{-1}X(k-a)=\exp(iax)f(x)$ by brute force calculation. I would like to see how to use duality though.


title: What happens to a Luttinger liquid under time reversal?

text: Suppose you a have an ordinary Luttinger liquid with   $$ H = \int dx \sum _{\eta= \pm 1 , \sigma =\uparrow,\downarrow } \psi^\dagger_{\eta, \sigma} (x) (-i v \eta \partial _x) \psi _{\eta,\sigma} (x). $$ You then bosonize it using $$\psi_{\eta,\sigma}=\frac{1}{\sqrt{2 \pi \alpha}} F_{\eta ,\sigma} e^{-i \phi_{\eta, \sigma}}$$ where F is the Klein factor and $\phi$ a bosonic field. My question is, what happens to F and $\phi$ under time reversal? The problem here is that the usual result for fermions, $$T c_{i,\uparrow} T^{-1}= c_{i,\downarrow}$$ $$T c_{i,\downarrow} T^{-1}=- c_{i,\uparrow}$$ where T is the time reversal operator, is not so obvious when there is separation between the Klein factor and the phase. I mean, why would $\phi_\uparrow$ turn into $\phi_\downarrow$ ?


title: Dirac spinor and Weyl spinor

text: How can it be shown that the Dirac spinor is the direct sum of a right handed Weyl spinor and a left handed Weyl spinor? EDIT:- Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed Weyl spinors. Their transformation properties are known. When I put these two spinors in a column and construct a four-component column which is a direct sum of $\psi_L$ and $\psi_R$ i.e., $\psi_D=\psi_L\oplus\psi_R$. This I **define** to be the dirac spinor. Right? Since it is a direct sum under Lorentz transformation, the corresponding Lorentz transformation matrix is diagonal. Right? Then it is easy to show that, it satisfies the Dirac equation in Chiral basis. Right? This is possible because we started from the definition of left-handed and right handed weyl spinors and their transformation properties are known. Right? This is explicitly carried out in the book by Lewis Ryder. But suppose I start the other way around. I solve the dirac equation in chiral basis. Then no one tells me that the upper two components are really left-handed and lower two are really right-handed. Suppose I take this chiral basis solution of dirac equation and now take that to be my **definition** of dirac spinor. Then how can I show the opposite, that it is made up of two irreps of Lorentz group i.e., $\psi_D=\psi_L\oplus\psi_R$ ?


title: What's the definition of the time ordering operator for more than two particles?

text: For two particles, $\langle {\mathcal T} a(t_1) a^\dagger (t_2) \rangle = \langle a(t_1) a^\dagger (t_2)\rangle \theta (t_1-t_2) + \xi \langle a^\dagger (t_2)a(t_1) \rangle \theta (t_2-t_1)$ with $\xi$ is a plus sign for bosons and a minus sign for fermions. How would I write, for example, $\langle {\mathcal T} a(t_1) a^\dagger (t_2) a(t_3) a^\dagger (t_4) \rangle$ ?


title: Quantum entanglement vs classical analogy

text: Consider that we have two balls, one white and one black, and two distant observers A and B with closed eyes. We give the first ball to the observer A and the second ball to the observer B. The observers don't know the exact color (state) of their balls, they know only the probability of having one or another color, until they look at them (measure). If the observer A looks at his ball he will see its color, which is white, so he immediately knows the color of the second ball. Lets call this “classical entanglement”. My question is: What is the difference between this “classical entanglement” and the quantum entanglement, for example, of two entangled electrons with opposite spins states? Can this analogy be used to explain the quantum entanglement?


title: Time in special relativity and quantum mechanics

text: The time is treated differently in special relativity and quantum mechanics. What is the exact difference and why relativistic quantum mechanics (Dirac equation etc.) works?


title: I noticed today that it is easier to open a soda bottle(standard 20 oz) upside-down, why is this?

text: Am I correct in assuming it is the weight of all that liquid + gravity?


title: If you place a spring on a neodymium hard-drive magnet, it appears to vibrate in slow-motion. Why is that so?

text: By chance(playing around really) I saw that a spring(mainly from a pen) placed on a neodymium hard-disk magnet(and then flicked by your finger at the top) makes a nice-effect (see youtube video ). It appears to oscillate in slow- motion(looks like tornado). Of course, "slow-motion" is purposely simplistic and unscientific - I am very far from a physicist. Here's the video - http://www.youtube.com/watch?v=n0OJQ1iXZg0&feature=channel_video_title I was too impatient in the video though, I should have zoomed in on the spring and waited. Sorry about that.. Here's a page about the magnets used: http://www.reuk.co.uk/Hard-Disk-Drive- Magnets-For-Wind-Turbines.htm Here are the polarities, plus a horizontal profile below: ![enter image description here](http://i.stack.imgur.com/7krCW.jpg) ![enter image description here](http://i.stack.imgur.com/fPLEl.png) More details: You really want to use a retractable pen spring, the thin kind. And Hard-drive magnets are key - I think it doesn't work with others. I think it's partly because of the 4-poles of a neodymium magnet. i.e, it's actually two-magnets-in-one. Cigarette lighters also have a long delicate magnet, which is good but too tipsy.   LBNL, supposedly you can stack these magnets, but they seem impossible to separate from the backing-piece. I appreciate any tips or advice.


Will timbre/quality be different if two different people play the same guitar? Assume that frequency/pitch and amplitude are same.


title: Confused with stress, strain and linear thermal expansion

text: > Four rods A, B, C, D of same length and material but of different radii r, > 2r , 3r and 4r respectively are held between two rigid walls. The > temperature of all rods is increased by same amount. If the rods do not > bend, then which of these are correct: > >   1. The stress in the rods are in the ratio 1 : 2 : 3 : 4. >   2. The force on the rod exerted by the wall are in the ratio 1 : 2 : 3 : > 4. >   3. The energy stored in the rods due to elasticity are in the ratio 1 : 2 > : 3 : 4. >   4. The strains produced in the rods are in the ratio 1 : 2 : 3 : 4. > _Four rods A, B, C, D of same length and material_ => Same Youngs Modulus, Same coefficient of linear expansion, Same Length. Also, _The temperature of all rods is increased by same amount._ Before answering the above question I've few other questions:   1. Suppose rods were not held between two rigid walls. Then there would have been change in length. In that case, would there be stress? Intuitively it feels like stress would be zero, as there seems to be no restoring forces developed. But Stress = Youngs modulus * strain. Strain is definitely not zero. So, stress should not be zero. Confused!   2. No say, rods were held between two rigid walls. Assuming rods are not bending. There length will not change. So, strain would be zero. Stress = Youngs modulus * strain. So, Stress must be zero. But intuitively it seems there will be stress, because there will be restoring forces in the rod pushing walls away. Again confused! Now coming back to the original problem. The above two confusions are causing trouble. But just going by intuition. There will be stress developed even though there is no strain. But stress = Restoring Force/Area. Here areas for all rods are different pi*r^2, because r is different. But Restoring force is same. So, the ratio must be 1/1 : 1/4 : 1/9 : 1/16. Right? Surprisingly answers are 3,4. There is lot of confusion. Kindly clarify


title: An electron is an excitation of the electron field. So when we observe a higgs boson that means we've excited the higgs field?

text: See this related question: If particles are excitations what are their fields? I ask this question because, according to a lecture, the higgs boson was frozen into a "matrix" at some point before recombination (otherwise we would not have atoms). Is this considered a field in the same way an electron field is a field? It was described that higgs was everywhere ... Is that another way of saying there is Higgs field? This means the electron interacts with the higgs field and not with excitations of the higgs field? Or probably Higgs bosons are only excited momentarily. Was the electron field also created like the higgs field at some point? See here https://www.youtube.com/watch?v=hL2BLAOVbDA


title: producing distilled water using sun energy

text: My home town (Iran, Lamerd) has very hot summer (always > 45+ degree) with dry weather and sharp sun shine. I used some black tubes under glassy box to heat up water. It was a natural thermal siphon and did not need pimping. definite volume of salty water in the container was connected to the tube glassy box. the empty space of the container was filled with hot humid. I have just connected this humid to a condenser. finally I had the distilled water but production rate was slow. Is that economically feasible to scale it up and be used for drinking water?![My home made apparatus](http://i.stack.imgur.com/V0Qqz.jpg)


title: Reflectivity of a glowing-hot metal surface

text: When a polished piece of metal (or steel in particular) is heated to incandescence, how do its reflective properties change? Given a mirror-like surface, would the object temporarily cease to act like a mirror at a certain temperature? Would the specular reflection of light turn into diffuse reflection; in other words, would the surface temporarily become matte/dull? I am not asking about permanent changes to the object such as melting, but rather can there be a temporary change in the reflectivity only while the the object is hot, that disappears when the object is cooled down? Background of the question: I have a laser scanner that usually has trouble scanning shiny surfaces with high specular/ low diffuse reflectivity; and it seems to work a lot better for hot steel pieces than for cold ones. Glowing steel also _looks_ less shiny to me than cold steel, but of course that is hard to tell with all the glowing


title: Center of mass problem

text: According to the definition of potential energy, we use $U= mhg . $ In the figure below , **A thin uniform rod of mass m and length h is positioned vertically above an anchored frictionless pivot point.** Why does a author say that potential energy is $U =\dfrac{mgh}2$ why not $U =mgh$ ![enter image description here](http://i.stack.imgur.com/DfGcJ.png) Similar problem I faced on a problem, when I had to find out the potential energy for lifting water from a hole with height h. > So my problem is, when should I consider center of mass and when I should > not?


title: Flat space current conservation sign confusion

text: It is said that in Minkowski spacetime, the current conservation law for the number current $N^\mu$ where $N^0$ is the number density and $N^i, i=1,2,3$ is the particle flux in the $x^i $ direction, is given by $$\partial_\mu N^\mu=0 ....................(*)$$ What I don't understand is he following. I would have thought that $$\partial_t N^0=-\nabla\cdot \vec N$$ where $\vec N$ has components $N^i, i=1,2,3$. But using the Minkowski metric, $(*)$ reads $$\partial_t N^0=\nabla\cdot \vec N$$ Why is there not a minus sign on the RHS? Thank you.


title: Velocity field induced by vortex points along ellipse

text: I'm investigating the velocity field induced by a continuous distribution of 2D vortex points distributed along an ellipse $\\{a\cos\theta,b\sin\theta\\}$. I'm interested in the field inside the ellipse, and I need some help to prove whether this field is zero or not. The intensity of each vortex point is proportional to $d\theta$ and not to the length along the ellipse. A vortex point located at a point $\boldsymbol{x'}$ induces a velocity field $\boldsymbol{u}(\boldsymbol{x})=\frac{d\theta}{2\pi |\boldsymbol{x}-\boldsymbol{x'}|} \boldsymbol{e}_\perp$ where ${e}_\perp$ is the unitary vector orthogonal to $(\boldsymbol{x}-\boldsymbol{x'})$ which is in 2D: $\boldsymbol{e}_z\times(\boldsymbol{x}-\boldsymbol{x'})/|\boldsymbol{x}-\boldsymbol{x'}|$. The total velocity field at a point $(x,y)$ inside the ellipse is obtained by integration over $\theta$. Numerical experiments seem to show that the field is zero inside the ellipse, but I cannot prove it. Dropping the factor $2\pi$, the field is in cartesian coordinates: $$\boldsymbol{u}=\int_0^{2\pi} \left\\{\frac{-y + b \sin\theta}{(x - a \cos\theta)^2 + (y - b \sin\theta)^2}, \frac{x - a \cos\theta}{(x - a \cos\theta)^2 + (y - b \sin\theta)^2}\right\\} d\theta \stackrel{?}{=}\\{0,0\\}$$ Is the field really zero? Maybe there is no need for the integrals to prove it. Maybe complex analysis is of help? I should mention that the following property is true in this problem: For any closed contour inside the ellipse the circulation is zero: $$\oint \boldsymbol{u}\cdot\boldsymbol{dl} =0$$ Since we are in a simply connected region, the velocity is the gradient of a potential which is single valued. But then is this potential constant?... I'm able to prove that the velocity is zero on both the $x$ and the $y$ axis. Also $u_x$ is an even function of $x$ and an odd function of $y$. The opposite applies for $u_y$. I'm able to prove the result for a circle $a=b$. Can somebody help me for the ellipse? Any idea?


title: Reconstruction of "wavefunction" phases from $|\psi(x)|$ and $|\tilde \psi(p)|$

text: Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$ Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$. Have we enough information to reconstruct the "wavefunction" $\psi(x)$, that is, obtain the phase of the "wavefunction" for all $x$ (up to a global phase, we are only interesting to the relative phases between the $x$)?


title: Why was PACER abandoned?

text: The PACER project is described in this question: How much of the energy from 1 megaton H Bomb explosion could we capture to do useful work? Why was it abandoned? It seems that it is the only readily economical and engineeringwise useful path to fusion power, and it seems that its breeder possibilities can easily let it pay for itself for generating fissile elements and helium (which is getting to be rare too nowadays!) Was it political or technical limitations that killed it? Is there hope for a renewed interest in this in todays energy conscious politics?


title: The speed of tachyons

text: The other day I was wondering: When a tachyon is coming towards you faster than the speed of light, will you see it before it hits you? Then I thought of course not, since the light waves aren't traveling faster than the tachyon then how could you see it before it hits you? Now I thought today, if an tachyon is traveling away from you faster than the speed of light, would you see it? If you fire a ball at an initial velocity of 20mph south out of a car that is going 50mph north, the final velocity of the ball would be 30mph north, is this also how light acts when the initial velocity of the object it is reflecting off is not equal to 0? So in my case, if the speed of light were 100mph (dummy math) and a tachyon was traveling at 110mph north that means the light reflecting off the tachyon would be traveling at 10mph north, so then really would you be able to see it? More generally, how does relativistic addition of velocities work for tachyons? update: This question is a hypothetical question: IF tachyons exist, then what would happen? After a few hours of research I see why a usual massive object CAN'T travel faster than (or even reach) the speed of light, but this question is about tachyons.


title: Dielectric in Parallel Plate Capacitor

text: Given a parallel plate capacitor of width $w$, length $l$, with a dielectric moving along the length $l$. Let the dielectric be from $x$ onwards. The capacitance will be $\frac{w \epsilon_0}{d} (\epsilon_r l - \chi_e x)$. Griffiths (p. 195) says that the total charge $Q$ in the $C=\frac{Q}{V}$ expression is constant as the dielectric moves. But $Q$ here refers to the free charge, and the free charge definitely increases as you move the dielectric in increasing $x$. What am I misunderstanding?


title: Classical car collision

text: I have a very confusing discussion with a friend of mine. 2 cars ($car_a$ and $car_b$) of the same mass $m$ are on a collision course. Both cars travel at $50_\frac{km}{h}$ towards each other. They collide. Ignoring any shreds and collateral damage, what is the speed of collision that the driver of $car_a$ felt? What I mean is, if $car_a$ were to be driven into an infinite mass wall, what would the velocity be to replicate the damage caused by the initial collision?


title: Is a collision with an immovable object equal to a head-on collision?

text: **Scenario 1** Two identical and bilaterally symmetrical cars, driven by identical drivers in the exact center of the car and in the same body position, each traveling the same speed, collide exactly head on. (Essentially, eliminate all variables that would cause rotation or interpenetration during the collision). **Scenario 2** One of the above cars hits an immovable wall instead of an oncoming car. **Question** Will each car in these scenarios experience the same forces? For clarity, the cars in scenario 1 have twice the closing speed of the car and wall in scenario 2.


title: Is two cars colliding at 50mph the same as one car colliding into a wall at 100 mph?

text: I was watching a youtube video the other day where an economist said that he challenged his physics professor on this question back when he was in school. His professor said each scenario is the same, while he said that they are different, and he said he supplied a proof showing otherwise. He didn't say whether or not the cars are the same mass, but I assumed they were. To state it more clearly, in the first instance each car is traveling at 50mph in the opposite direction and they collide with each other. In the second scenario, a car travels at 100 mph and crashes into a brick wall. Which one is "worse"? When I first heard it, I thought, "of course they're the same!" But then I took a step back and thought about it again. It seems like in the first scenario the total energy of the system is the KE of the two cars, or $\frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2$. In the second scenario, it's the KE of the car plus wall, which is $\frac{1}{2}m(2v)^2 + 0 = 2mv^2$. So the car crashing into the wall has to absorb (and dissipate via heat) twice as much energy, so crashing into the wall is in fact worse. Is this correct? To clarify, I'm not concerned with the difference between a wall and a car, and I don't think that's what the question is getting at. Imagine instead that in the second scenario, a car is crashing at 100mph into the same car sitting there at 0mph (with it's brakes on of course). First scenario is the same, two of the same cars going 50mph in opposite directions collide. Are those two situations identical?


title: If two objects collide, will the collision force be twice as big as the original force?

text: Say, if we have two trucks travelling towards eachother at 60m/s, and those two trucks carry the same payload, and are the same model (aka they're exactly the same), will the collision force be as if just one truck crashed into a wall at 120m/s? Theoretically the forces will balance each other out, but it won't always be this way.


title: At what point can we assume the tip of a fin is adiabatic?

text: Let's say there is a fin that is 1mm thick, extends 8mm from the surface, and is 10 mm wide. The fin is exposed to a moving fluid. Can we assume the adiabatic tip condition and use the characteristic, "Corrected", length for calculations?


title: Russian Doll Teller Ulam?

text: Can you trigger a thermonuclear explosion from a smaller thermonuclear explosion in a scaling way, so that starting from a small laser ignited fusion within a small fissile container, using the X-rays from the first explosion to implode a tiny adjacent Li-d wire, which is then used to implode a bigger wire, and so on to a big explosion after a few cycles? Or does the Teller-Ulam design not scale to miniature explosions? If there is a russian-doll design, can you then trigger the first explosion using a free-electron x-ray laser available today? Any other laser trigger? I am asking because from the answer to this question: How much of the energy from 1 megaton H Bomb explosion could we capture to do useful work? , the PACER power-plant design presumably relies on a fission triggered bombs, and fission resources are limited and non-renewable. Further, the PACER folks suggested that future thermonuclear explosions can be laser-triggered, and this was the only way I could think of doing this. Is this really possible? (I should add that it would also help with regards to proliferation and terrorism safety if the trigger was an expensive bulky free electron laser, as opposed to a standalone explosive nuclear bomb.)


title: What is electric/magnetic field in a medium?

text: One method of introducing electric field is based on the measurement of the force acting on moving charged particle. By equating F to qE we determine the electric field E if the electric charge q is known. Similarly, one can determine magnetic field. This approach is fine for vacuum but it generally fails for fields in a medium, especially for dispersive medium since one needs first to distigush Lorentz force from other forces that act on a particle there, e.g., from friction force. Once I met the above cited description somewhere but I forgot where it was written. Could some one direct me to an approriate reference? I also cannot remember tha name of scientist who pointed out on the above cited difficulty in determining the field. Perhaps, it was Rosenfeld or something like that. And finally, is there a better definition of the electric / magnetic field in a medium?


title: Liquid with freezing point above 0 Celsius that could be use at ice rinks

text: Please help me with an answer to my dilemma: Is there a liquid that could be used to fill an ice rink (non-explosive, non- poisonous, etc), and have the freezing point above 0 Celsius?