title: Classical Viewpoint on Electromagnetism

text: **Note:** This question may be difficult or impossible to answer within the rules of these forums due to its philosophical nature. I will delete the question if I am violating the rules. Onto the question! Recently I have been curious whether classical electromagnetism is fully solved (up to the divergences). Specifically, can we completely mathematically describe and then interpret the classical world of charges and currents and them the associated fields. Let us assume that our world is classical and that electromagnetism is a complete theory even though there are certain inconsistencies (self-interactions, infinite energy of point charges, etc). A common description of $\textbf{E}(\textbf{r}, t)$ and $\textbf{B}(\textbf{r}, t)$ among physics textbooks is that changing electric fields induce magnetic fields and vice-versa. This is assuming that there are no external fields or sources of field present. In symbols, $$ \partial_t\textbf{E} \neq \textbf{0} \implies \exists \textbf{B} \quad (1) $$ $$ \partial_t\textbf{B} \neq \textbf{0} \implies \exists \textbf{E} \quad (2) $$ Many physicists come to this conclusion from Maxwell's equations. Specifically, they argue that Faraday's law, $$ \nabla \times \textbf{E}(\textbf{r}, t) = -\partial_t\textbf{B}(\textbf{r},t), $$ implies (1), and that Ampere's law (with Maxwell's correction term and no currents), $$ \nabla \times \textbf{B}(\textbf{r}, t) = \partial_t \textbf{E}(\textbf{r},t), $$ implies (2). Note that we are using natural units with $c = 1$. However, these equations do not have any obvious causal connection. While we may like to pretend that _right_ implies _left_ , this is purely notational convention. Who is to say from these equations alone that one field having a nonzero curl doesn't produce a changing dual field? One attempt at reconciling this problem seems to be in Jefimenko's equations. I will state the equations without derivation, but the fields can be solved for completely in terms of the source charges and currents (I'm lazy and the following equations are in mks units from Wikipedia): $$ \textbf{E}(\textbf{r}, t) = \frac{1}{4 \pi \epsilon_0}\int [\frac{\rho(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^3} + \frac{1}{c}\frac{\partial_t \rho(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^2}] (\textbf{r} - \textbf{r}') - \frac{1}{c^2}\frac{\partial_t \textbf{J}(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^2} d^3\textbf{r}', $$ $$ \textbf{B}(\textbf{r}, t) = \frac{\mu_0}{4 \pi}\int [\frac{\textbf{J}(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^3} + \frac{1}{c}\frac{\partial_t \textbf{J}(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^2}] \times (\textbf{r} - \textbf{r}' )d^3\textbf{r}' , $$ where $t_r = t - |\textbf{r} - \textbf{r}'|/c$ is the retarded time. These equations seem to imply that neither of the fields "causes" the other. Instead, Jefimenko's equations imply that only the source charges and currents generate the fields (without the presence of external charges, currents, or fields). My question is related to this approach. Is it valid? What are the arguments for and against? Is the matter settled in the classical context of electromagnetism, or are there certain subtleties I've skipped over? As an extra question, is it instead better to consider $F_{\mu \nu}$, and treat it as one object arising solely from $J^\mu = (\rho, \textbf{J})$, instead of looking at the individual fields? Thanks in advance for any and all answers!


title: What are the differences between Comet versus Asteroid versus Meteoroid versus Meteor versus Meteorite?

text: What are the differences between comet versus asteroid versus meteoroid versus meteor versus meteorite in practical terms?


title: Does the particle vibrate in 3D space?

text: As far as I know, particles vibrate with a frequency and wavelength determined by their energy level. Is this vibration in 3D space?


title: Is angular resolution important when we want the spectra of an Earth-like exoplanet?

text: Right now, our resolution + light gathering power are still far too low to take direct images of exoplanets, so we're limited to subtracting the planet spectra from the parent star spectra when the planet undergoes a transit (and this isn't going to be possible for decades, according to Jim Kasting's latest book). So in this case, it seems that light gathering power is more important. So, Is angular resolution more important when we want to measure the spectra of an Earth-like Exo?


title: Does the term "dark matter" apply to nonluminescent bodies which still interact electromagnetically?

text: On the new Astronomy.SE site, I was having a short discussion on one of my answers. The basic discrepancy was; can MACHOs like black holes/brown dwarfs/neutron stars be termed "dark matter"? My reasoning is that these objects do not radiate EM radiation on their own but they do gravitate, and thus constitute a small part of the total dark matter in the universe. I agree that there is a lot of dark matter which doesn't In other words, can the term "dark matter" be applied to nonradiating (or faintly radiating) bodies which still participate in the electromagnetic interaction (baryonic or otherwise)? Or is it necessary for all dark matter to not interact electromagnetically?


title: Cosmological models other than FRW

text: The FRW is a nice isotropic and symmetric metric but I think its assumptions are too many. I was wondering about alternative models. Specifically are there any prominent alternatives that have more fleshed out local detail? For example, are there any models that can ascribe a more local interpretation to data which was considered as crucial support for the Big Bang.


title: Is deep-space radiation greater in the Sun, or in the shadow of Earth?

text: http://space.stackexchange.com/questions/604/is-it-possible-to-get-pregnant- through-natural-means-in-space indicates cosmonauts are at risk of irradiation whilst in space. What little I know about space is that planets, and inhabitants are largely protected from deep cosmic rays by \- Sun's magnetic sheath \- Solar Wind \- Planet's own magnetic sheath When in orbit, a cosmonaut and satellites are periodically in the Sun. At other times, they are in the dark. On the sunward side, any body is directly in the Solar wind, whereas in shadow the planet/body probably acts as a shield against the Solar Wind; I may be wrong!   1. Where is the deep-space radiation higher - on the Sunward side of Earth, or on the side in Earth's shadow? I guess this may also be rephrased to read - what fraction of shielding from deep cosmic rays within the Solar System is provided by the Solar wind?   2. Is a cosmonaut at greater health risk on the Sunward Side, or in Earth's shadow? ( _This second part may fit better on Biology SE but since it is contingent on the former I post it here_ )


title: Why do most metals appear silver in color with gold being an exception from a scattering and EM viewpoint?

text: Related: Why are most metals gray/silver? After reading Johannes’ impressive answer to Ali Abbasinasab question of why do most metals appear silver in color with the exception of gold (and copper), my lack of experience with quantum mechanics (I’m an undergraduate and I will take graduate quantum mechanics this coming quarter) is not enough for me to appreciate the relativistic quantum solution. What struck me most was Stefan Bischof’s statement: “Here physics has to explain more than just “is there a d valence electron”.” I took this as a challenge to see if I could explain the golden color of gold (and copper). Here is my humble attempt to answer this very question using scattering and EM approaches.


title: Why is there an absolute entropy?

text: Why is there an absolute entropy? Given any non-discrete probability distribution, we don't really have an absolute entropy because the entropy depends on the parametrization of the distribution (e.g. Beta vs. Beta-prime) which was arbitrarily chosen. Another way to put it is that instead of entropies we only have Kullback–Leibler (KL) divergences aka relative entropies. So, why isn't there a physics analogue to KL-divergence? Just as we have relativistic velocity, which has some properties, why don't we also have relative entropies, which have some properties? Instead of saying the absolute entropy of the universe increases, why don't we say that the relative entropy given our prior belief of the universe increases? Alternatively, what is the relation between entropy and number of microstates when the physical system is continuous, and how do we "count microstates"?


title: Paper airplane physics

text: I am working on a game involving flying and steering a paper airplane for WP7. I want the plane to fly just like how normal paper airplanes fly (see this game for an example http://armorgames.com/play/7598/flight) but I can't seem to find an equation for how paper airplanes fly. Anyone have any experience with this? In my game now, it just follows the usual motion for an object in a vacuum, which makes for some flight, but it doesn't feel perfect, and traveling at a slight downward angle makes you lose speed, which isn't right. Thoughts?


title: How are Euler's laws of motion applied to gyroscopes?

text: Euler's laws of motion for a distributed mass are: $$F = \frac{d}{dt} MV_{cm},\ N = \frac{d}{dt} L$$ $F$ are the sum of the external forces, $M$ the total mass, $V_{cm}$ the velocity of the centre of mass. $N$ are the sum of the moments of the external forces about some given point, L the total angular momentum about the same point. If a gyroscope is supported at its base with its axis horizontal, it precesses at a constant angular velocity. Using the above equations, how does one show this?


title: Why do clouds fly?

text: I know it's probably the most stupid question there is, but why do they fly are the clouds lighter than air? What's keeping those tiny ice structures floating miles about the ground? I've been looking all over the internet and I can't find acceptable answer to this basic physical question. Can you please help me? This question was marked as answered - but I don't believe it has been explained satisfactory. Please bare with me on this one.


title: why does gravitational force bring the clouds down?

text: > **Possible Duplicate:**   >  The Density of Clouds according to definition of cloud : A visible mass of condensed water vapor floating in the atmosphere, typically high above the ground as there's an enormous amount of gravitational force on earth why can't the cloud particles cant get attracted towards earth , my general question is if we threw an object which weighs even 1milligram to ground it gets attracted and falls down on earth, As per the gravity definition:               an attractive force that affects and is produced by all mass and other forms of energy as well as pressure and stress      so why can't the cloud's particle can't get attracted by gravitational force as the cloud is an form when they have tons of water particles in them?


title: Why don't clouds fall?

text: Well I do know that they sometimes fall as rain, but my question is why don't the droplets fall as soon as they condense from steam to cloud. Clouds are white by the process of Mie scattering so the particles are on the order of light wavelength ~$0.5 \mu m$, although small would a particle of this size not slowly fall to the ground? When a cloud does rain many of the rain drops fall at the same time what is physically happening to the droplets to encourage them to join into large drops that fall?


title: How does Fraunhofer diffraction depend on the orientation of the sides of a lens?

text: Matt in his answer on What does a hexagonal sun tell us about the camera lens/sensor? mentions > Incidentally the number of (distinct) points to the star is equal to double > the total number of unique orientations* in the sides of the aperture shape > i.e. three blades would be six points, six blades would also be six points > as a hexagon has only three unique orientations in its sides. > > * a hexagonal aperture has six sides but only three unique orientations as > there are three pairs of parallel sides. I think It makes sense to have three pointed star in case of three blades. But how does a six blade aperture also result in a three pointed star?


title: Mapping a given density matrix to the generalized 2-qubit state

text: The generalized 2-qubit state is given as: $$ \rho = \frac{1}{4}[ I\otimes I + (m_x\sigma_x + m_y\sigma_y + m_z\sigma_z)\otimes I + I \otimes (n_x\sigma_x + n_y\sigma_y + n_z\sigma_z) + \sum_{ij}t_{ij}\sigma_i\otimes\sigma_j] $$ Then, is there a method to map a given density matrix: $$ \rho_g = \pmatrix{a & b & c & d \\\ e &f &g &h \\\ i & j & k & l \\\ m &n &o &p } $$ to the generalized state in terms of relations between the coefficients, without having to expand and compare terms?


title: Evidences against Supersymmetry

text: Recently, some experiments show that the supersymmetry is not realised by Nature according to the simple models that we currently have. Nevertheless, it is far from saying that the "game is over" as SUSY might be realised at an unknown energy. I would like to ask what are the experiments that could give us a clue whether SUSY assumption is wrong. Will the LHC / VLHC be able to rule it out in the future?


title: How much water from ice and after how much time?

text: I need two formulas in dealing with ice and water. First:   If I have 2 x 2 x 2 (cm) ice cube, after it melts, how much water will there be ? Second:   If I have 2 x 2 x 2 (cm) ice cube, after how much time will it melt ? I know that there are many variables: temperature, density and so on.   So you can give and more complicated formulas. Also, for second question I suspect that outer area of ice is also effecting speed of melting. Thanks in advance


title: 1 Mohm Resistivity at cryogenic temperature for doped silicon

text: I would like to pick the correct doping of silicon to get 1 Mohm cm resistivity at liquid helium temperature of 4.2 K. The metal insulator transition implies that I need a very accurate doping. Are there any other effects and is there a way to calculate the correct doping?


title: Do photons actually generate a slight kinetic force?

text: My question is even though photons have no (rest) mass, do they emit a external force due to EM radiation causing electrons to be excited and jump to higher energy shells which electrons have mass thus photons can emit a kinetic force? I am new so I would like to get the record straight on this issue.


title: Quantum Entanglement - What's the big deal?

text: Bearing in mind I am a layman - with no background in physics - please could someone explain what the "big deal" is with quantum entanglement? I used to think I understood it - that 2 particles, say a light-year apart spatially, could affect each other physically, instantly. Here I would understand the "big deal". On further reading I've come to understand (maybe incorrectly) that the spatially separated particles may not affect each other, but in knowing one's properties you can infer the other's. If that it the case, I don't see what the big deal is... 2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties...? What am I missing? Is it that the particles properties are in an "un-set" state, and only when measured do they get set? (i.e. the wave-function collapses). If this is true - why do we think this instead of the more intuitive thought that the properties were set at an earlier time?


title: Entanglement, really?

text: If I have two " _entangled_ " particles and I know the spin state of every one of them. Then, I change the spin state of one of the particles, will it affect the spin state of the other particle even if it is far away? Also, it turns out that entanglement is actually real and can be used for many things, is this just the result of a false interpretation and understanding of Quantum Mechanics by those who claim it or it can be really done? And if it is real, will it just change our interpretation of GR?


title: magnetic moment of proton

text: I just tried to calculate the magnetic moment of a proton. I took the proton g-Factor of $g=5.585694$   nuclear magneton of $\mu_k = 5.050783 * 10^{−27}$ J/T   proton spin of $I=1/2$   At first I calculated the norm of the proton spin $|\vec{I}|=\hbar \sqrt{I*(I+1)}=\hbar \frac{1}{2}\sqrt{3}$ And then I put everything together in $\mu=g\mu_k\frac{|\vec{I}|}{\hbar}$ and obtain 2,44134228 × 10^-26 instead of 1.410606 × 10^-26 ... Interesting enough, I obtain the correct value if I devide by $\sqrt{3}$. But I see no reason to do this... It would be great if you could help me. Thanks in advance ftiaronsem


title: What creates the differences between the two channels of a stereophonic signal?

text: Given two identical microphones arranged in an ideal XY pattern, recording a single sound source at equal distance from both capsules, the two signals obtained are equal in amplitude, perfectly in phase with each other, and distorted identically by microphone frequency response. Therefore, other than small differences that result from noise, what exactly causes the difference between the left and right channel, such that when heard, a stereo field is perceived? I ask because I am developing an audio synthesis application, and need a way to simulate real stereophonic sound given a single input signal.


title: Quantum computers: are they possible or impossible?

text: I know quantum computers are very complicated and my question is is there any way in "Principle" to create one? Are there already quantum computers being created?


title: Is energy always proportional to frequency?

text: Google has no results found for "energy not proportional to frequency" and many results for E=hf. Is there an example of an energy that is not proportional to frequency?


title: Toric Code and Random Bond Ising Model

text: It was established by Dennis, Kitaev et al. that the 2D Toric Code can be mapped to a 2D Random Bond Ising Model. The original derivation was given in the paper "Topological quantum memory" which can be found in J. Math. Phys. 43, 4452 (2002); doi: 10.1063/1.1499754 or online at http://dx.doi.org/10.1063/1.1499754 . A free arXiv preprint version is available online here. The derivation that shows that the 2D Toric Code can actually mapped to a 2D Random Bond Ising Model is given in section IV - D "Derivation of the model" (Starting page 4469). The point where I'm getting lost is when he is trying to set up a function that for a fixed chain E outputs the probability of a "homotopically equivalent" chain $E'$ (which he calls $p(E'|E)$). To obtain this function he first calculates the probability of a link being occupied that lies on $E'$ but not on $E$. He obtains (Eq. 13) that the probability is equal to \begin{equation} \left(\frac{p}{1-p}\right)^{n_{C}(\ell)} \end{equation} up to an overall normalization and with $C$ being the cycle corresponding to $E'$. I don't really see how he arrives at that results or what the overall normalization factor actually is (after all "normalization" is crucial when talking about probabilities - what is the meaning of probability 35 if I don't know the normalization?) Then he obtains in Eq. (15) an explicit exponential- like form for $p(E'|E)$ which isn't clear to me neither. He also seems to omit a lot of explanations at least at this stage. Could anyone explain to me in a bit more details how the derivation is carried out in more detail? This would really help me a lot.


title: wind versus compressed air

text: What would the calculation look like when computing the wind pressure of a moving vehilce versus compressed air pressure pushing against the the moving vehicle, how much compressed air pressure would be needed to combat the wind of the moving vehicle.?


title: How do magnets work?

text: I've read a classbook on the field theory (including EM): it perfectly describes quantitive patterns in EM-theory, but I have no luck understanding how and why it works. I mean, magnetic substances are described mainly by magnetic moments of electrons, but all explanations of the phenomenon I've found are rather of deep high-level focus on fields exclusively, than on explanation why it works and what underlying mechanisms bring all those ideas to life (including explanation of what field is, except that it is an abstraction). So, the question is: may anyone try to give (or point to) a popular and thorough explanation of magnets and on low-level mechanisms, which unifies and explains how this long-distance interaction really works (i.e., not only modelled and described mathematically)? p.s.: also, I'd like to see some papers on computing magnetic properties of bodies (iron ball, for example, finite plane, NeoCube's ball chains, etc). Thanks in advance!


title: Using 'Euler's Homogeneous Function Theorem' to Justify Thermodynamic Derivations

text: I've been working through the derivation of quantities like Gibb's free energy and internal energy, and I realised that I couldn't easily justify one of the final steps in the derivation. For instance, in deriving the formula for Gibb's Free Energy, we first found the differential equation: $$dG = dH - TdS$$ which has the property that, for spontaneous processes, $dG \leq 0$. We then went from there to defining the state function: $$G = H - TS$$ and claimed that this had the analagous property that $\Delta G\leq0$ for all spontaneous processes. Apparently we can reason this way because the second equation can be obtained from the first by integration. But I'm not entirely sure of this. For instance, temperature is not necessarily independent of entropy, so I'm not convinced that $TS$ must be the integral of $TdS$. Physically I'm not convinced because the derivative refers to small changes at constant temperature, while the state function applies at all temperatures. Wikipedia's Gibbs free energy page said that this part of the derivation is justified by 'Euler's Homogenous Function Theorem'. Now, I've done some work with ODE's before, but I've never seen this theorem, and I've been having trouble seeing how it applies to the derivation at hand. If anyone can shed any light on the matter or recommend some further reading I'd appreciate it. Thanks


title: A question about definition of Fermi energy

text: Wikipedia states the definition of Fermi energy as for "a system of non- interacting fermions". If we have to assume free electrons in a solid behave this way before we are able to calculate Fermi energy, how can Pauli exclusion be justified (because electrons are non-interacting)? Can Fermi energy be similarly defined for electrons confined to a single atom?


title: Helmholtz's "Ueber die Bewegungsgleichungen der Elektricität für ruhende leitende Körper" in English?

text: Is Helmholtz's 1870 paper "Ueber die Bewegungsgleichungen der Elektricität für ruhende leitende Körper," _Journal für die reine und angewandte Mathematik_ **72** :57-129 translated into English anywhere? thanks


title: A question about the numerical simulation on classical three-body gravitional system

text: Related to this post Just out of curiosity, I have a question about the current status of classical (Newtonian) three-body gravitional numerical simulation. I found wikipedia is relatively limited on this topic. Given three stellas, how accurate can we predict their trajectories using numerical methods based on current compuational techniques and computer capacity? Is there any comparison with astrophysics observation? (Presumably we can update our observed initial conditions during calculation)


title: interior surface of furnaces

text: For industrial furnaces, in order to improve radiation heat transfer and save energy, some people say applying a high emissivity coating on to the interior surface of a furnace will do, while some people say applying a high reflectivity coating will do. we all know that at given wavelength and for an opaque object surface, emissivity + reflectivity =1, anyone can explain to me please? With best regards Sean


title: Normal ordering of the identity operator

text: I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator):   * looking at it from the "Fock space operators POV",the prescription is to move all the creation operators to the left and the annihilators to the right, but the identity by definition as non of those, so should be left invariant. But I run into several contractions:   * normal ordering of any operator should have a vanishing vev, so $$<\, : \mathbb{I} :\, > = 0 \ne\, < \mathbb{I}>$$   * using the commutator between the $a$ and $a^{\dagger}$: $$:aa^{\dagger}: \, = \,a^{\dagger}a =\, :a^{\dagger}a + \mathbb{I}: = \,a^{\dagger}a + :\mathbb{I}:$$ where I used the commuator to get from the first to the third and the normal ordering otherwise. Identification of the second and fourth gives me again $\,:\mathbb{I}:\,\, = 0$.   * last the vev of the exponential of any operator should be zero ($<:e^{A}:>\, = 0$), expanding the exponential gives me again $<\,:\mathbb{I}:\,>\, = 0$. Is this the final result or are there more subtleties?


title: Field Strength Renorm in Peskin&Schroeder

text: On page 237 in PS we have (the unnumbered equation after eq. 7.58) $$\mathcal{P} \sim \frac{iZ}{p^2-m^2-iZ\,\mathrm{Im}M^2(p^2)}$$ but after deriving it myself I obtained $$\mathcal{P} \sim \frac{iZ}{p^2-m^2-iZ\,\mathrm{Im}M^2(p^2)-iZ\frac{\mathrm{d}\,\mathrm{Im}\, M^2}{\mathrm{d}\,p^2}\cdot(p^2-m^2)+\dots}$$ why do they omit the derivative term? Why is it considered small? **Note: My mistake was that I also expanded the imaginary part of $M^2$...please see answer below for solution.**


title: Electric field in a cylinder

text: We have electric charge density $\rho(r) = kr$ in a cylinder of infinite height and radius $a$. I'm asked to find the electric field. I'm doing it using two methods and I don't undesrtand why then don't yield the same result **Method 1** Gauss' theorem applied to a cylindrical surface; $$E(r) 2\pi rh = \frac{Q}{\epsilon_0}$$ $Q = h\int_A \rho = h\int_A kr$, where A is the unit circle $\Rightarrow Q = h\pi k r^2$ So I find $$E(r) = \frac{kr}{2\epsilon_0}$$ **Method 2** Divergence in polar coordinates: $$\nabla \cdot E (r) = \frac{\rho}{\epsilon_0}$$ $$\nabla \cdot E(r) = \frac{1}{r} \frac{\partial rE(r)}{\partial r} = \frac{1}{r} (\frac{\partial E(r)}{\partial r} + r\frac{\partial E(r)}{\partial r}) = \frac{\partial E(r)}{\partial r} \frac{r+1}{r} = \frac{\rho}{\epsilon_0}$$ $$\frac{\partial E(r)}{\partial r} = \frac{k}{\epsilon_0} \frac{r^2}{1+r}$$ $$\Rightarrow E(r) = \frac{k}{\epsilon_0} (\frac{r^2}{2} + r + \ln{(1+r)})$$ What's wrong with that?


What is the cause of centripetal/centrifugal force? When an object of mass $m$ is moved in a circular orbit, it experiences a centrifugal force radially away from the center. What is the cause of this centrifugal force? Is these related to the four fundamental forces (gravity,electromagnetic,weak and strong forces)? This force is equivalent to a force experienced while stopping a mass in motion (Inertia). But is this inertia caused by some force? or what causes inertia? A photon particle does not have inertia of rest.


title: Why in $F = iLB$, $L$ is a vector but $i$ is not?

text: I learned $F = iLB$ recently. However, I don't understand why $L$ is marked as a vector but $i$ is not.   For a normal rod, how should I define the direction of length vector $L$? And if I reverse the current in it, the force exerted on it by the magnetic field would reverse direction, correct?   So I think in this formula, $i$ should be the vector but not $L$. Am I right? I'm using the Physics II by _Halliday Resnick and Krane_


title: Do you use the magnitude equation to get speed from an accelerometer?

text: A guy suggested to me that getting speed from an accelerometer required the use of this equation: $\text{speed} = \sqrt{x^2 + y^2 + z^2}$ This does not make any sense to me, all that you would get from this equation would be the magnitude in $m/s^2$ of the acceleration in the $x, y$ and $z$ axes. Am I correct? Or is my reasoning flawed?


title: What is the meaning of inflection points in the dispersion relation inside the first Brillouin zone?

text: I have a question regarding the $E$ vs $k$ curve in the first Brillouin zone. Why does the curve have an inflection point at some value of $k$ in the curve? How does it physically support it? ![E vs k](http://i.stack.imgur.com/Y4DGl.jpg)


title: How do you explain Kepler's third law in general terms without complex math?

text: I understand the first law-elliptical orbits, and the second-equal area in same time, but I need help with the third one. Note that I am not in an AP course or taking calculus at the moment so simple quadratics/cubics/CP level explanations would suffice.


title: What's the meaning of the age of the universe?

text: I'm not asking about how we worked backward from an expanding universe to the age of the big bang, but rather what is the meaning of time in a near infinitely dense point in the context of general relativity? Wouldn't time flow infinitely slowly for a theoretical (though physically impossible) observer?


title: What latitude is needed before you can reliably see Omega Centauri?

text: What latitude is needed before you can reliably see the globular cluster Omega Centauri, say it reaches 20 degrees above the horizon? What about if you are up on a hill looking down, what's the theoretical highest latitude where you could have any chance of seeing it?


title: Are retrograde capture orbits "easier" than prograde capture orbits?

text: After reading up on irregular moons in the solar system - moons that are thought to be captured, most seem to be in retrograde orbit around their parent body. That led me to wonder if retrograde orbits are easier to capture objects than prograde orbits - say prograde orbits are more likely to gravitationally slingshot the object away from the parent body before capture, whereas retrograde orbits would be more likely to capture before flinging the object away. When viewing the capture from the perspective of the parent body, an object that is moving retrograde past the body appears to slow down as it interacts with the gravity well of the body, whereas an object moving prograde past the body appears to accelerate in the same frame of reference. Is there any validity to this, or is that just a flaw in reasoning?


title: What causes millisecond pulsars to speed up?

text: Millisecond pulsars are supposed to be old neutron stars. However, they are spinning even more rapidly than newly formed pulsars. Since pulsars slow down as they age, something must have caused these older pulsars to "spin up" and be rotating as fast as they are. What is the mechanism for doing so?


title: Where can I find public domain astronomical pictures?

text: Where can I find public domain astronomical pictures of nebulae, stars, etc. that can be freely used?


title: Measuring energy when no work is done -- comparing push-ups to planks

text: **tl;dr** Is there any meaningful (physical) way to compare the energy expended in the exercise of doing $x$ pushups in $t_1$ seconds, vs the exercise of doing the plank for $t_2$ seconds? * * * I'm confused about the concepts of energy, work, and power, in the case where no distance is traversed. For instance, take a push-up. I can approximate the work I've done by using the change in potential energy $\Delta U = mgh$, ie. $$W = mg \Delta h$$ $$= m [kg] * 9.8 [m/s^2] * h[m]$$ where $m$ is some proportion of my body mass, and $h$ is a bit less that the length of my arm. The result has the units of $Newton-meters$, or $kg m^2/s^2$, aka $Joules$. I can also measure the average _power_ of my push-up using $\Delta W/\Delta t$, which has the units of $J / s$, or $kg m^2/s^3$, aka $Watts$. Fine so far -- but now take a plank (holding the push-up position for some time). Obviously when you do a plank, you are expending some energy, even though you are not doing any work (since there is no change in potential energy). What are you doing then? You are counteracting the force of gravity on your body mass (that is, $9.8 [m/s^2] * mass [kg]$) for a fixed time. This would produce units of $kg m/s^3$ -- these units are not named, that I can recall. So my question is twofold:   1. Is there an actual unit that describes the energy I've expended in doing a plank, beyond $(mass * g * time) kg m/s^3$   2. Is there any meaningful way to _compare_ the energy I've expended by doing a plank, with the energy I would expend in doing $x$ pushups? (Beyond _work_ or _power_ , since those values are always _zero_ for the plank, which is not helpful.)


title: How much detail can telescopes actually provide?

text: For example, could the numbers / letters on a postage stamp in a randomly specified location be clearly visible from space. This is to settle a discussion with a friend that piqued my curiosity.


title: What determines a progenitor's fate as a spiral or elliptical?

text: I was thinking about my answer to Are the inner planets on planar orbits because there was more dust in the inner solar system (early on in planetary accretion)? \- when it occurred to me that maybe I was reversing cause and effect. Specifically, that perhaps spiral galaxies simply arise from high- angular-momentum progenitors, while elliptical arise from low angular-momentum progenitors. In this scenario, the spiral wave-pattern would merely be how that excess angular momentum organizes itself, and the matter can't help but stir itself. For a warm-up question and consistency checker, do spiral galaxies in fact have substantially greater total angular momentum than elliptical?


title: In astronomy what phenomena have theory predicted before observations?

text: As far as I know, astronomy is generally an observational science. We see something and then try to explain why it is happening. The one exception that I know of is black holes: first it was thought of, then it was found. Einstein's relativity is middle ground to me, he thought of light beams at the speed of light but obviously could observe gravity's effects. Anyway, I guess my question is, what are the biggest discoveries that were thought of before they were seen in the sky?


title: Temperature of glowing materials

text: As I understand it, Stars emit visible light, OBAFGKMRNS, in the range of $10^3 - 10^4 K$. Yet materials such as steel emit similar frequencies at much lower temps; red is around 800K. Why the difference? I thought black body radiation applies to all materials and environments. I am an interested amateur.


title: North and south of magnetic field

text: ![enter image description here](http://i.stack.imgur.com/uCrT3.jpg) The current I is flowing upward in the wire in this figure. The direction of the magnetic filed due to the current can be determined by the right hand rule. Can we determine the north and the south of the magnetic field produced by the current I by using a hand rule?


title: What is the quantum / Berry-Pancharatnam phase for a spin-j state with z-component m?

text: Quantum phase arises when a spin-j state is sent through a sequence of transitions that return it to its original position. For example with spin-1/2, a state picks up a complex phase of $\pi/4$ when it goes through three perpendicular (not orthogonal!) transitions: $\frac{<+z|+y><+y|+x><+x|+z>}{|<+z|+y><+y|+x><+x|+z>|} = e^{2i\pi/8}$ where $|+y>$ means the state with spin-1/2 in the +y direction. Since the above path covers an octant, or one eighth of the surface of a sphere, the total quantum phase for the whole sphere in the spin-1/2 case is $2i\pi$. More generally, for a spin state $|j,m,+z>$, we will have $\frac{<j,m,+z|j,m,+y><j,m,+y|j,m,+x><j,m,+x|j,m,+z>}{|<j,m,+z|j,m,+y><j,m,+y|j,m,+x><j,m,+x|j,m,+z>|} = e^{2i n_{jm} \pi/8}$ where $n_{jm}$ is an integer. The same phase is picked up for a spin-1 state with m=1, but a spin-1 state with m=0 picks up no phase at all. (In fact the m=0 states do not depend on the orientation.) At the moment, I believe that the j=1,m=-1 state will pick up n=-1 (in analogy with the spin-1/2 case), but I haven't bothered to actually run the computation. * * * Yes, Dr. Motl found it. For a journal reference see "Geometric Phases" by Péter Lévay. The appropriate equation is (36) on page 19. Note that our $m$ is his $r-J$. http://arxiv.org/abs/math-ph/0509064v1 Also, the method of looking at an infinitesimal rotation works because Berry- Pancharatnam phase doesn't depend on stuff like the arbitrary complex phases of spinors and of course there's no preferred region on the sphere of possible orientations for the spin axis. As an aside, I was convinced that the j=1, m=+-1 case had the same coefficient as the j=1/2,m=+-1/2 case. This was because the geometric phase for light is the same as that for spin-1/2. But now it's clear that this is mixing apples and oranges.


title: Is relativity necessary for the existence of life?

text: If the universe didn't have the relativity principle, would it be able to support life? Life consists of very complicated organisms. The operation of these organisms depends on the laws of physics. If the laws of physics depended on absolute velocity, then it seems that life would have a more difficult task; organisms would have to adapt their biochemistry to the different absolute speeds of the planet as it moves with or against the motion of the sun around the galaxy. If the laws of physics depended on the absolute gravitational potential, or on acceleration, then the biochemistry of life would have to adapt to the different accelerations / gravitational potential as life colonized higher altitudes. In addition, there would be a seasonal effect as the earth moves closer and farther away from the sun. * * * I think there's a way this question could be answered quantitatively. Begin with a modification of general relativity such as the post Newtonian parameters. See wikipedia article:   http://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism   Now analyze an important biological molecule whose shape is extremely important to life such as hemoglobin. Find out what range of post-Newtonian parameters are compatible with the operation of that molecule. Unfortunately, I suspect that this is a research problem. If someone solves it, I presume they will publish it.


title: Are multi-megaton antimatter induced fusion explosions theoretically possible?

text: I just finished reading  this research about antimatter induced fusion and thermonuclear reactions. And one conclusion I could make is that very little mass of antimatter (in range of micrograms) is needed to initiate a fusion reaction in lithium-deuteride fuel. Also, in page 14 in this PDF file, there is a theoretical design of a 1-kiloton antimatter induced fusion bomb. Now, I actually have 2 questions. Firstly, 100 grams of lithium-deuteride is used in this theoretical 1 kt design. But according to this, 100 grams of lithium-deuteride should yield 6.4 kilotons not only 1 kiloton, so is there any explanation of this ? Secondly, since I seem to find only low yield designs of antimatter induced/catalyzed fusion bombs, a doubt about the feasibility of larger yield came to my mind. So, if a single kiloton fusion reaction is feasible with a certain amount of antimatter, then should we consider fusing more fuel with even more antimatter feasible too ? Note : I completely understand the difficulty of making, handling and storing antimatter, and I am not saying this thing is going to be made any time soon. I am just curious about the physics part behind it.


title: Entropy and Information

text: Several posts and my classes in thermodynamics equate increase in entropy with loss of information. Shannon clearly showed that the information content of a message is zero when its entropy is zero and that its information content increases with increasing entropy. So entropy increase leads to more information, which is consistent with the evolution of the universe from a disordered plasma to one that contains lots of order. Why does physics continue to get the relationship between entropy and information backwards?


title: Electric Power $P$

text: In my textbook there are 2 formulas for electric power: $$\begin{array}{cccr} P &=& E/t &\hspace{10pt} (1) \\\ P &=& VI. &\hspace{10pt} (2) \end{array}$$ What is the difference between the 2 formulas? Do they both calculate electrical power? If so, how do you know which formula to use in a given situation?


title: Which force makes a wheel roll down the hill? What causes friction?

text: ![enter image description here](http://i.stack.imgur.com/kUyg1.png) A wheel rolling down a hill has two axis of rotation. One is where the center or mass is and the other is the point of contact with the surface which acts as a fulcrum. I was trying to understand how this happens, how it rotates down the hill. What causes friction and torque? Please mind me over this simple issue, but I just want to further understand how things work. My ideas are as follow:   * **F1** , the component of gravity, pulls the wheel from the center of mass ( **CM** ). This pull will create friction **f**. Friction then creates a torque about its center of mass. The object is at the same time rotating around the point where it touches the surface, like a fulcrum, the torque is created with lever-arm **L** and force **W**.  I don't know if this is right or not. Maybe one should not consider the component of gravity at all acting on the wheel. If so, what is rotation originated? Is it because the position of the **CM**? Maybe someone can explain better. Thanks you.


title: Curve Fitting and Multiple Experiments

text: Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the average of all the experiments and then do fit a single curve to that?


title: Light Apparent Brightness

text: Let's say I have a triangular light source. From that light source I want to calculate a pyramidal frustum (tetrahedron with no apex). How would I calculate the maximum bottom area where light would be visible to the naked eye? And how would I calculate the apparent brightness?


title: space acrobatics

text: In the newly released movie "Gravity" the main character is left alone floating in space in spacesuit, in a fast spin. Assuming there are no external forces (no tether, jet-pack etc) the system is closed and the initial angular momentum has to be preserved. However, can the astronaut change the axis of rotation? Say, initially the rotation is around the principal axis going perpendicular to the body in the plane of the shoulders (Fig. A). Can the astronaut, using only his/her body, transition to rotation around the principal axis going along the body (Fig. B)? The body shape is the same in the initial and final position, only the orientation is different.![enter image description here](http://i.stack.imgur.com/iG1UZ.jpg)1


title: How deep is the region near an event horizon where Hawking radiation is generated?

text: In other words, how strong does gravity have to be to cause Hawking radiation to occur?


title: Do mirrors have a "resolution" quality to them?

text: I would like to know if mirrors have a quality of "resolution" to them like a regular photograph might, or like a JPEG does. For example, if you looked to closely, or magnified a photograph, you would soon come to a point where the image blurs, or that which was too far away when photographed is not discernible. If a high quality microscope was applied to a mirror, would it be possible to observe far-away objects reflected by the mirror, taking only the quality and scope of the microscope into account? Are there different quality mirrors, all other things ( _e.g._ shape) being the same? Could you also talk about any loss of luminescence if possible? Also, do all plane mirror have the same converging/diverging power i.e, the image size will be same on all plane mirrors ?


title: Measurement problem in the orthodox interpretation

text: Let's look at the measurement problem in the orthodox interpretation of quantum mechanics as an inconsistency between inner and outer treatment of the measurement apparatus. You can always push your boundaries of treating the evolution of your system as unitary further and further. You can say OK, the universe as a whole is evolving unitarily (let's not worry about information loss in a blackhole). So it's up to me to consider the boundary to see the evolution of my system and apparatus together or just my system. And I should be able to work out the reduced density matrix of my system equally in every treatment unambiguously! However, If you treat the apparatus externally, the evolution of the system would be: $$a|\uparrow\rangle + b|\downarrow\rangle \to |\uparrow\rangle$$ with probability $|a|^2$ or $$a|\uparrow\rangle + b|\downarrow\rangle \to |\downarrow\rangle$$ with probability $|b|^2$. Whereas, an internal treatment of the apparatus would give $$|\uparrow\rangle\otimes|\text{ready}\rangle\to U\bigl(|\uparrow\rangle\otimes|\text{ready}\rangle\bigr) = |\uparrow\rangle\otimes|\text{up}\rangle$$ and $$|\downarrow\rangle\otimes|\text{ready}\rangle\to U\bigl(|\downarrow\rangle\otimes|\text{ready}\rangle\bigr) = |\uparrow\rangle\otimes|\text{down}\rangle$$ with $U$ a linear operator, $U(a|\psi\rangle + b|\phi\rangle) = aU|\psi\rangle + bU|\phi\rangle$, which evolves $$\bigl(a|\uparrow\rangle + b|\downarrow\rangle\bigr)\otimes|\text{ready}\rangle$$ to $$U\bigl[a|\uparrow\rangle\otimes|\text{ready}\rangle + b|\downarrow\rangle\otimes|\text{ready}\rangle\bigr] =a|\uparrow\rangle\otimes|\text{up}\rangle + b|\downarrow\rangle\otimes|\text{down}\rangle$$ However, pushing the boundary after the measuring apparatus gives a difference physics. This could be viewed as a problem with measurement in orthodox quantum mechanics (as opposed to realist or operational strategies to solve the measurement problem) But I was thinking it's not really an inconsistency within a theory. It's just an inconsistency between two different choices of the internal-external boundaries! I'm not asking about the role of decoherence theory. It sounds to me like the measurement problem wasn't really a problem in the first place! Am I right about that? **update:** It has been pointed out that the question is not clear enough yet. Here is my last attempt: It's believed that for an adequate postulates for quantum mechanics, the inner and outer treatment of measuring apparatus shouldn't affect the physics of the system. Which for the orthodox interpretation of quantum mechanics does. For instance in the Bohm's model this has been resolved by denial of representational completeness. And in Operational interpretation it's bypassed by avoiding talking about physical state of the system. Here the question is **Are we really allowed to change the boundaries?** Because if you don't believe you can, the problem will never appear in the first place. I hope that explains what I'm asking. Because I don't think I can make it more clear :-)


title: Mnemonics for remembering physical data

text: Anyone have good mnemonics for remembering standard packets of data in physics? Any field within physics would be welcomed. Examples of such "packets":   * data in the standard model of particle physics   * charges/masses of common subatomic particles and light atoms   * cosmological data -- numbers, masses, luminosities (I recall "Oh Be A Fine Girl/Guy Kiss Me Right Now Smack" for star types)   * thermodynamics   * material properties (I note also Mnemonics to remember various properties of materials)


title: Do non-metal objects reduce the signal strength of a computer wireless network device?

text: Would an object like a wooden bed interfere or block the signal coming from a 802.11 wireless router?


title: Why does salty water heat up quicker than pure water?

text: > **Possible Duplicate:**   >  Why does adding solutes to pure water lower the the specific heat? Why do higher concentration salt solutions heat up more quickly?


title: Can the Hubble constant be measured locally?

text: The Hubble constant, which roughly gauges the extent to which space is being stretched, can be determined from astronomical measurements of galactic velocities (via redshifts) and positions (via standard candles) relative to us. Recently a value of 67.80 ± 0.77 (km/s)/Mpc was published. On the scale of 1 A.U. the value is small, but not infinitesimal by any means (I did the calculation a few months ago, and I think it came out to about 10 meters / year / A.U.). So, can you conceive of a measurement of the Hubble constant that does not rely on any extra-galactic observations? I ask because, whatever the nature of the expansion described by the Hubble constant, it seems to be completely absent from sub-galactic scales. It is as though the energy of gravitational binding (planets), or for that matter electromagnetic binding (atoms) makes matter completely immune from the expansion of space. The basis for this claim is that if space were also pulling atoms apart, I would naively assume we should be able to measure this effect through modern spectroscopy. Given that we are told the majority of the universe is dark energy, responsible for accelerating the expansion, I wonder, how does this expansion manifest itself locally? Any thoughts would be appreciated.


title: Why are there two ways to solve for energy of a spring?

text: I can find the energy of a spring using $F = -kx$, or by using the formula $e = 1/2mv^2 + 1/2I\omega^2 + mgh + 1/2kx^2$. The first way, I get $mg/k = x$, but the second way, I get $2mg/k = x$. Which one is correct to use and why? This is assuming that there is no angular velocity or linear velocity.


title: What's wrong with this derivation that $i\hbar = 0$?

text: Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\hat{p} |\psi_p\rangle = p |\psi_p\rangle,$$ where $p$ is the eigenvalue of $\hat{p}$. Then, we have $$ [\hat{x},\hat{p}] = \hat{x} \hat{p} - \hat{p} \hat{x} = i \hbar.$$ From the above equation, denoting by $\langle\cdot\rangle$ an expectation value, we get, on the one hand $$\langle i\hbar\rangle = \langle\psi_p| i \hbar | \psi_p\rangle = i \hbar \langle \psi_p | \psi_p \rangle = i \hbar$$ and, on the other $$\langle [\hat{x},\hat{p}] \rangle = \langle\psi_p| (\hat{x}\hat{p} - \hat{p}\hat{x}) |\psi_p\rangle = \langle\psi_p|\hat{x} |\psi_p\rangle p - p\langle\psi_p|\hat{x} |\psi_p\rangle = 0$$ This suggests that $i \hbar = 0$. What went wrong? **EDIT:** This seemingly little curiosity, as I perceived it initially, evolved into a major problem, as can be seen from the discussion below. Quantum mechanics indeed leads to non-physical results in my opinion and not just to resolvable paradoxes, at that regarding one of its basic claims, the one involving commutators. It seems most of what could be said on this issue has already been addressed below and probably there is no need to continue the exchange here. I would suggest that we discuss it further in the chat.


title: The notion of bounded states in quantum mechanics and their characterization with operators

text: > Is there any case of potential $V$, such that the continuity of the operator > $H=c\ \Delta+V$ is not spoiled? And I don't know any non-differnetial operator examples for continous spectra. I guess I don't know the full spectrum of operators (ba dum tss!). My question comes about as I wonder about the justification of the term "bounded state", especially in relatoion to the fairly straight forward classical concept.


title: "Time" by epistemic subdivision of a closed system

text: There is the idea that there is no time in a completely closed (thus unobservable) system. Within such a system, a subsystem may be imagined to be split off by some virtual boundary. However, one wants to continue to consider the remaining system as if it was the complete system (which of course is not correct). In that case the description of the effects of the subsystem on the remaining system requires "time". Does anyone know more about this idea?


title: Multi-component mixture phase equilibrium criteria

text: For a pure species, the equlibrium between liquid phase and vapour phase is given by the equality of molar Gibbs energy in both phase: $$\underline{G}^l=\underline{G}^v$$ Where $\underline{G}$ with an underline represents molar Gibbs energy. In a multicomponent system, the equilibrium criterion is the equality of the **partial** molar Gibbs energy (aka chemical potential) for each component between each phases: $$\overline{G}^l_i=\overline{G}^v_i$$ Now, because of the relation between molar quantities and partial molar quantities, $\underline{G}=\Sigma x_i\overline{G}_i$, the second criterion contains the first by setting $x_i=1$. My question is : is the first criterion still valid between phases? If yes this would seem to imply that $x_i^l=x_i^v$, which is clearly wrong.


title: stress work of uniformly deforming continuum

text: I have a volume which is deforming (using explicit time-integration scheme) uniformly with velocity gradient $L$ and stress tensor $\sigma$. I would like to determine work done by the volume deformation during one timestep $\Delta t$, knowing both current and previous values of $L$, $\sigma$ and volume $V$. I've seen somewhere the formula $\Delta W={\mathrm tr}(L\sigma)V\Delta t$, but I don't know if it is correct and how to derive it. $L\sigma$ should be energy density, but why are its non-diagonal terms discarded? _Remark:_ the integration scheme actually is leap-frog, but I ignored mid- step/on-step business for now and supposed everything are on-step values. The formula above would correctly compute the increment mid-step, reading $\Delta W(t-\Delta t/2)={\mathrm tr}\left(L(t-\Delta t/2)\frac{\sigma(t-\Delta t)+\sigma(t)}{2}\right)\frac{V(t-\Delta t)+V(t)}{2}\Delta t$


title: Can microwaves affect wifi - and can they harm us?

text: I listen to the radio via my iPad with wifi. When I switch the microwave oven on, the radio cuts out. When the microwave oven is finished, the radio comes back on. (This is 100% reproducible!) So - is it (as I suspect) the microwave oven affecting the wifi? If so, how can that happen (I thought microwaves could not escape the oven)? And (most importantly for me), is it harmful? **Update:** when I stand between the microwave and the iPad, the radio comes back on! :S


title: Can we obtain non-Lorentzian metric from Lorentzian metric, through renormalization methods?

text: Since low-energy, non-relativistic thermal field theories are defined in Euclidean spacetime, while high-energy relativistic theories are define in Minkowski spacetime, I was wondering if there are renormalization methods that can show such a change in metric signature.


title: How did they measure the speed of light observing Jupiter's moons, centuries ago?

text: I am interested in the pratical method and I like to discover if it is cheap enough to be done as an experiment in a high school. Thank you.


title: Terminologies for moment of inertia

text: Perhaps someone can suggest the right terms for the following mathematical objects related to moment of inertia?   1. A _inertia tensor_ $I$. $$I \equiv \begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} \\\ I_{2,1} & I_{2,2} & I_{2,3} \\\ I_{3,1} & I_{3,2} & I_{3,3} \\\\\end{bmatrix}$$   2. A _product of inertia_ is an off-diagonal entry in the tensor: $I_{1,2} = I_{2,1}$, $I_{1,3} = I_{3,1}$, or $I_{2,3} = I_{3,2}$.   3. A _principal moment of inertia_ is a diagonal entry in the tensor: $I_{1,1}$, $I_{2,2}$, or $I_{3,3}$. This is the semantic of moment of inertia discussed in elementary treatment of Physics.   4. What is the term for $I_2$ and $I_3$ in the last line below? $$\begin{align*} I_{1,1} &= \sum_{j} m_j\;\left(r^2_{j,2} + r^2_{j,3}\right) \\\ &= \sum_{j} m_j\,r^2_{j,2} + \sum_{j} m_j\,r^2_{j,3} \\\ I_{1,1} &= I_2 + I_3 \end{align*}$$


title: Adiabatic approximation and time-dependent problems

text: I am an undergraduate physics student. I have a question in approximation methods for time-dependent problems in quantum mechanics. I read the proof of the adiabatic theorem but I didn't understand it. This theorem states: > _If the Hamiltonian $H(t)$ changes very slowly with time, a system which is > initially in a discrete non-degenerate state $u(0)$ with energy $E(0)$ is > very likely to go over to the corresponding state $u(t)$ with energy $E(t)$ > at time $t$, without making any transition._ Could you present a comprehensive and simple proof for it?


title: Linear Superposition Principle - Origin of interference term

text: ![enter image description here](http://i.stack.imgur.com/Z1Sko.jpg) How do you from 2.8 to 2.10 ? I'm confused :/


title: How does a thin film of soap help defog bathroom mirrors?

text: We often find bathroom mirrors get fogged while hot water is used. Looking this up on the internet, we find several easy solutions:   * use a hot air blower   * use a heater behind the mirror   * vinegar + water mixture   * thin film shaving cream/foam lather   * thin film of regular(any) bath soap lather I would like to know: **How does a thin film of soap lather prevent condensation on the mirror surface during a hot shower?** A thread on DIY(StackExchange): technique to make a shower mirror fog-free.


title: Everything is a Photon

text: If we accept $E=\hbar \omega$, $p=\hbar k$, and $E^2-(pc)^2=(m_0 c^2)^2$, then we find $$(m_0 c^2)^2 = (\hbar \omega)^2-(\hbar kc)^2 = \hbar (\omega^2 - (kc)^2)=0$$i.e. the rest mass of any particle is $0$. Therefore every particle is a photon. What is wrong with this?


title: WHy is SUSY QM important?

text: why is SUSY QM important ? i mean for each one DImensional Hamiltonian , can we write 'H' as $ H= A.A^{+}+C$ (or similar constant) here $ A= \frac{d}{dt}+A(x)$ and $ A^{+}= -\frac{d}{dt}+A(x)$ 1) can we always express 'x' and 'p' as a combination of A and its adjoint? 2) does SUSY QM makes easier to solve the Hamiltonian 3) can we apply second quantizatio formalism with operators $ A,A^{+}$ in the same way we did for the Harmonic oscillator 4) let be $ Ay(x)=\lambda _{n} y(x) $ an eigenfunction of the anhinilation operator, then is it true that $ E_{n} =C+ |\lambda _{n}|^{2} $ ¿what happens for the eigenfunctions ??


title: Why don't black holes form from forces other than gravity?

text: Gravity is the weakest of the fundamental forces, so what is so special about gravity that it can form an inescapably strong field while a force like the EM force cannot? It seems to me that if there were some extremely strong electric field, that it would only have an influence on particles that carry a charge, and so it would make sense that a black hole could not from from the EM force because it could not affect particles like neutrons. However, the same logic cannot be applied to black holes caused by gravity, because things like photons, which have no mass, still cannot escape the intense gravitational field. Why does gravity influence things that have no "gravitational charge" while the EM force cannot influence things that have no "electric charge." Because otherwise it would follow that a strong enough EM field should be fundamentally inescapable in the same way a black hole is inescapable. I know E=mc^2, so obviously a strong enough EM field would eventually act like a gravitational black hole, but why wouldn't it be easier for an EM black hole to form than a gravitational one since the EM force is so much stronger?


title: Speed of light as a universal speed limit

text: It follows from special relativity that nothing can travel faster than light. Einstein believed this would have to hold so generally that he assumed the Einstein-Podolsky-Rosen paradox to indicate a contradiction in quantum physics. Nowadays we know we have to be a bit more specific, and maybe it would be safe to say that nothing that can carry information can travel faster than the speed of light. If information can be transferred faster than light, it would be possible to change the past. If I understand correctly it is the contradictions arising from the hypothetical possibility to influence the past that lead to the assertion that nothing can travel faster than light. My questions:   1. Is this indeed the reason why it is said to follow from special relativity that nothing can travel faster than the speed of light?   2. Aren't the contradictions associated to influencing the past more of a philosophical nature than of a purely logical (or at least physical) nature? **EDIT** I realize now that instantaneous transfer (action at a distance) is actually physically in contradiction with special relativity: what is instantaneous/simultaneous in one inertial frame is not in the other, so the laws of physics would not be the same. I suspect that for finite velocities faster than that of light there must be a similar argument, I would appreciate if anyone could elaborate on that.


title: entropy relation between ads$_2$ black hole and near extremal ads RN black brane

text: It seems that the entropy for AdS$_2$ black hole is independent of the temperature $s=s_0$. While for near extremal AdS RN black brane, $s=s_0+ s(T/\mu)$. Should not these two entropies be the same since the AdS$_2$ black hole can be thought of a near horizon geometry of near extremal AdS RN black brane?


title: Mass density of dark matter in solar system near us

text: What is the approximate mass density of dark matter in our solar system at the radius of the Earth's orbit? I would like some idea of the mass of dark matter going through each cubic meter of material on earth.


title: Ground state of spin in magnetic field

text: I am trying to solve a time dependent perturbation theory problem, and it involves the Hamiltonian $$H=-\mu B\sigma_z$$ And a perturbation $$V=-\mu B_1\sigma\cdot(\cos(\omega t)\hat x-\sin(\omega t)\hat y)$$ I want to find the probability that the spin will go from the Ground state of $H_0$ at t=0 to the excited state at time t. I understand the perturbation theory, I just need to figure out the initial and final states. I thought there were two ground states (either up or down). So what are the excited state and the final state?


title: Can you clean the air in a small room using a vacuum cleaner?

text: You can use a vacuum cleaner to vacuum the floor but if you tried to vacuum the the air in a small room would this work or will the laws of physics stop this from working in any way? In what way would attempting to vacuum clean the air be different from using an air scrubber(in terms of physics). Thanks


title: Using Ampere's Law for a Solenoid

text: ![solenoid](http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/imgmag/sol.gif) To calculate the magnetic field, a rectangle amperian loop was drawn, and since the sides of the rectangle are perpendicular to the magnetic field, and the top is too far away to have any field lines cross it, only BL makes a contribution. ![equation](http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/imgmag/sol2.gif) I don't understand why this magnetic field is only contained within the solenoid. Shouldn't the magnetic field at any point on the amperian loop equal μnI, according to Ampere's law?


title: Is this webcomic accurate?

text: I was considering this xkcd comic from 5/10/14, with the alt-text "Trains rotate the Earth around various axes while elevators shift its position in space." I'm wondering about its accuracy. ![http://xkcd.com/1366/](http://i.stack.imgur.com/EvUsH.png) I first considered the elevator scenario. As the elevator accelerates (to start or stop its motion), the occupants feel a fictitious force, demonstrating conclusively that it is the elevator and not the planet which is in motion. By this argument, an elevator is fundamentally different from a device which physically displaces the planet. Similarly, I suspect that a train doesn't "rotate the Earth," since such a rotation would be felt by occupants on Earth as an angular or centripetal acceleration. Any observer can conclude definitively that it is the train which is in motion around the Earth, not the Earth rotating under the train. For this reason, is the comic is not strictly correct? (Note that there is a minimal motion of the Earth resulting from the action- reaction force pair between the train and the planet, but it's safe to say that this motion is negligible relative to the distance traveled by the train, since $I_{earth} \gg I_{train}$.)


title: Twin Paradox: Still a Paradox?

text: Alright, so David Griffiths in his "Introduction to Electrodynamics" states that the Twin Paradox is not a paradox at all since the traveling twin returns to Earth. By returning to Earth, the twin had to reverse direction, thus undergoes acceleration, and therefore cannot claim to be a stationary observer. However, what if the traveling twin simply Skypes the twin that is on Earth. The twin on earth will still appear older, which would make no sense since in that case the rocket can be seen as the stationary frame of reference while the Earth "travels" at a speed close to the speed of light. No acceleration is undergone, yet the paradox remains. Is Griffiths just completely glossing over important nuance again?


title: Is it possible to hear the past?

text: From this Stack Exchange Physics Post, I am certain that it is possible to view the past. But then this interesting question came to me. Is it possible to hear the past? Ok, you might say, "Well, we are hearing the past, aren't we?" and go onto immense details about the speed of sound against the difference and stuff. But what if I wanted to hear the sound of a place extremely close by days, weeks, months or even years back? Would that be possible?


title: Different colors in LED/LCD display when viewed from different angle?

text: I want to know why different colours appear when viewed with a different angle. Can anyone tell me why?


title: Origin of Electromagnetic Interactions between Molecules

text: What is the origin of electromagnetic interaction between molecules? Anyway, it should have some relation with atoms. Also, These electromagnetic interactions are playing a major role in different properties of matter including the transition between solid-liquid-gas. Hence, what would be the **source** of these interactions..? If these interactions originate from atoms, then some other questions come into **focus** :   * Are they related to the transition of electrons between various energy levels and emission of photons from an atom..?   * Are they related to cohesive forces between molecules in solids & liquids?


title: Folded and/or compacted dimensions in M-theory?

text: I've on many occasions that there are various numbers of 'extra' dimensions above the 4th. However, I've heard that they are 'compacted' or 'folded' tightly and unimaginably small. Now, as I understand, we live in 3 dimensional space coupled with time. So, by extension, those four dimensions should be present all around us and in our universe. Now, if these hypothetical extra dimensions are folded and compacted, obviously smaller than our universe, doesn't that mean that the extra dimensions are 'somewhere' in the universe and contained within it rather than enveloping it?


title: Maxim Raykin's "solution to the measurement problem" using infinitely many derivatives

text: Recently I was made aware of the following arXiv preprint by Maxim Raykin: Analytical Quantum Dynamics in Infinite Phase Space. As far as I understand it, Raykin's idea is to reinterpret quantum mechanics as a theory of ordinary classical particles moving around in R3 \-- except with the twist that now the differential equation governing the particles takes as input not only their positions and the first derivatives of the positions, but all the higher derivatives as well. Once you decide to look for this sort of "evolution rule," Raykin says that you can find a not-even-very-complicated one that perfectly reproduces all the predictions of QM, at least in the (special?) case that the wavefunctions are analytic. Indeed, the differential equation that you get, describing the particle trajectories, ends up being identical to the guiding equation in Bohmian mechanics. However, the key difference from Bohmian mechanics is that here one makes no explicit reference to any "wavefunction" or "guiding field" -- only to all the higher derivatives of the "actual" particle positions. On this account, Raykin wants to explain "quantum nonlocality" as arising from the fact that an analytic function is determined by its collection of higher derivatives at any point. The idea, I suppose, is that you can't specify all the higher derivatives of the positions of the particles in Alice's lab, without also knowing what's going on in Bob's lab. While Raykin _doesn't_ discuss this, I imagine one could also explain the exponentiality inherent in (say) quantum computing, in terms of the fact that given n particle positions x1,...,xn, the number of tuples (k1,...,kn) for which one can form the combined derivative $\frac{d^{k_1}}{dx_1^{k_1}} ... \frac{d^{k_n}}{dx_n^{k_n}}$ increases exponentially with n, even after a finite cutoff is imposed on max{k1,...,kn}. In any case, there has to be _some_ explanation, since (just like in Bohmian mechanics) this account is specifically constructed to give exactly the same predictions as standard QM. Look: if you follow the quant-ph arXiv, you see another revolutionary solution to the conceptual problems of QM trumpeted every week. Most of those solutions can safely be rejected on the ground that they contain no new idea, _not even a terrible idea._ But I was unable to reject the present idea on that ground: this way of thinking about QM seems bizarre to me, but it's not one that I've seen before, nor would it ever have occurred to me. I can envision at least five possible reactions to Raykin's idea: (0) There's some error that prevents it from working. (1) There's no error, but it's a completely uninteresting triviality. _Because_ an analytic function is completely determined by its higher derivatives at any point, Raykin's "differential equation" is really just an elaborate way of stating the tautology that a particle's future trajectory is completely determined by its future trajectory. Obviously, that would be true _regardless_ of what the trajectories were (provided only that they're analytic): nothing specific about QM is being used here. Differential equations that depend on infinitely many derivatives simply lack predictive power. And as for the use of analytic functions' global nature to "explain quantum nonlocality" --- it's no better an explanation than Gerard 't Hooft's "superdeterminism." In both cases, the cosmic conspiracy one needs to posit, with every particle's trajectory mysteriously constrained by every other's since the beginning of the universe, is astronomically worse than the relatively-benign nonlocality (e.g., Bell inequality violations) that one was trying to explain in the first place. (2) Sure, the fact that QM can be formulated in this way is a tautology, but it's a kind of _cute_ tautology, one that could conceivably provide useful insights. (3) This is a genuinely new reformulation of quantum mechanics, in the same sense that (say) Bohmian mechanics was -- so it's important in a similar way, regardless of whether you like it philosophically. (4) [Raykin's own view, apparently] This is the true, unique solution to the measurement problem, which resolves all known conceptual problems with QM, is a necessary stepping-stone to quantum gravity, etc. etc. Of the five views above, the only one that I feel extremely confident in rejecting is (4). Since Raykin's paper appears to have gotten zero (public) reaction so far, my question is the following: **Does anyone else have thoughts or observations that would support or rule out reactions (0), (1), (2), or (3)?**


title: Difference between using displacement and current configuration as unknown?

text: We could use either the current configuration $x$ or the displacement $u$ as unknown while solving for the deformation, for example, of a solid object. I want to know what's the difference between them? Is it that there is no difference, or one is numerically better than the other?


title: Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid. Why?

text: I need a full explantation for this concept. Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid.


title: single-particle wavepackets in QFT and position measurement

text: Consider a scalar field $\phi$ described by the Klein-Gordon Lagrangian density $L = \frac{1}{2}\partial_\mu \phi^\ast\partial^\mu \phi - \frac{1}{2} m^2 \phi^\ast\phi$. As written in every graduate QM textbook, the corresponding conserved 4-current $j^\mu = \phi^\ast i \overset{\leftrightarrow}{\partial^\mu} \phi$ gives non-positive-definite $\rho=j^0$. If we are to interpret $\phi$ as a wave function of a relativistic particle, this is a big problem because we would want to interpret $\rho$ as a probability density to find the particle. The standard argument to save KG equation is that KG equation describes both particle and its antiparticle: $j^\mu$ is actually the charge current rather than the particle current, and negative value of $\rho$ just expresses the presence of antiparticle. However, it seems that this negative probability density problem appears in QFT as well. After quantization, we get a (free) quantum field theory describing charged spin 0 particles. We normalize one particle states $\left|k\right>=a_k^\dagger\left|0\right>$ relativistically: $$ \langle k\left|p\right>=(2\pi)^3 2E_k \delta^3(\vec{p}-\vec{k}), E_k=\sqrt{m^2+\vec{k}^2} $$ Antiparticle states $\left|\bar{k}\right>=b_k^\dagger \left|0\right>$ are similarly normalized. Consider a localized wave packet of one **particle** $\left| \psi \right>=\int{\frac{d^3 k}{(2\pi)^3 2E_k} f(k) \left| k \right>}$, which is assumed to be normalized. The associated wave function is given by $$ \psi(x) = \langle 0|\phi(x)\left|\psi\right> = \int{\frac{d^3 k}{(2\pi)^3 2E_k} f(k) e^{-ik\cdot x}} $$ $$ 1 = \langle\psi\left|\psi\right> = \int{\frac{d^3 k}{(2\pi)^3 2E_k} |f(k)|^2 } = \int{d^3x \psi^\ast (x) i \overset{\leftrightarrow}{\partial^0} \psi (x)}$$. I want to get the probability distribution over space. The two possible choices are: 1) $\rho(x) = |\psi(x)|^2$ : this does not have desired Lorentz-covariant properties and is not compatible with the normalization condition above either. 2) $\rho(x) = \psi^\ast (x) i \overset{\leftrightarrow}{\partial^0} \psi(x)$ : In non-relativistic limit, This reduces to 1) apart from the normalization factor. However, in general, **this might be negative at some point x, even if we have only a particle from the outset, excluding antiparticles.** How should I interpret this result? Is it related to the fact that we cannot localize a particle with the length scale smaller than Compton wavelength ~ $1/m$ ? (Even so, I believe that, to reduce QFT into QM in some suitable limit, there should be something that reduces to the probability distribution over space when we average it over the length $1/m$ ... )


title: Could dark matter be a kind of Goldstone boson?

text: The common argument for vanishing Goldstone boson is like, if there is some massless particle generated from the spontaneous symmetry breaking, it should be detected. Since we never saw that, it is not possible. However, if there is some particle which has not been found, it could be (a) very heavy; (b) very weakly interacting. The Goldstone particle was rejected based on reason (a). Could (b) somehow be true? e.g. as a candidate of dark matter? Or since the Higgs mechanism was confirmed, this possibility (very weakly interacting) is already ruled out?


title: Lagrangian - Particle wave interaction

text: Assume that we got a Lagrangian of a classical wave $\phi(x,t)$ and a classical particle $q(t)$. The interaction term in the lagrangian is $$ L_{int}=\frac g 2 \dot{\phi}(x=q(t),t)$$ It is very likely that this interaction has already been investigated, but I find it hard to search for such a lagrangian on the web. Does someone know an efficient method to search the web for a lagrangian? Or maybe does someone recognize it? This interaction might be similar to a setup where one catches an atom in a dipole field. Then the field would correspond to the electric/magnetic field. Another similar setup might be catching a dielectric sphere in a standing wave.


title: Do really, two oppositely charged bodies (of equal charge in magnitude) attract each other?

text: It is well known fundamental behaviour that, _oppositely charged bodies attract each other_ (I don't know whether it applies also for charges of equal magnitude or not), and identical charges repel each other. It is also well known that, a system of oppositely charged bodies with equal charge in magnitude, has zero net charge. If a system has oppositely charged bodies, with equal charge in magnitude, it would imply that, there will be zero net charge, and the vector field's had got cancelled each other, like around a current carrying conductor, there is no electric field because, charge on current carrying conductor is zero, as one electron enters the conductor, the other will be leaving the conductor. Now, if there is no field, how could the oppositely charged bodies (of equal charge in magnitude) attract each other, they should not feel any force, isn't? Taking into account all the above statements, would it imply that, two oppositely charged bodies (of equal charge in magnitude) attract each other? _[All statements made, are up to my view. Any correction advisory is welcome]_ _LINKS_   * What are the fields produced around a current carrying conductor? (Provides an idea of whether, there exists a electric field around a neutral body or not, like the current carrying conductor)


title: What forces are exerted on a clothespin in space?

text: Let's say a clothespin is modeled as a simple torsion spring as follows. **Given:**   * $p_1,\ p_2,\ p_3$: point-like objects of equal mass in 2-D space.   * All objects float in space, i.e. the center of mass will not change.   * At time $t_0$ a torsion spring is inserted at $p_1$, such that it exerts torque on $p_2$ and $p_3$, with:      * $\theta$: the angle of twist from its equilibrium position in radians     * $\kappa$: the spring's torsion coefficient     * $\tau = -\kappa \theta$ is the torque exerted by the spring **Question:** what are the resulting forces on $p_1$, $p_2$ and $p_3$? **My answer:** Because all objects have equal mass, we can leave mass out of the equation. F2 is a force perpendicular to $p_1,\ p_2$ of magnitude $\dfrac{\tau }{ |p_1-p_2|}$ . By Newton’s 3rd law, F2’ is a force of equal magnitude and opposite direction as F2 Similar for F3 and F3’ ![Torsion spring at p1 exerting torque on p2 and p3](http://i.stack.imgur.com/seUSe.png)


title: Charged particle between two parallel likely charged plates, is it affected by the plates?

text: Imagine two parallel conductive plates. Charge up both to have the same amount of positive charge. Then put positive test particle between the two. The Coulomb's law is an inverse square law, so one might think the positive test particle is repelled from the nearby plane and accelerated towards the middle between the plates making it do an oscillating motion (until it radiates away it's energy due to the acceleration and stops in the middle). On the other hand, since the two plates have the same charge, there is no voltage between the plates that would do work on the electric charge. So it won't move at all. I'm a confused here. Is Coulomb's law just a special case for 2 point charges?


title: equivalent spring constant for a drum floating in water

text: If I have a drum floating in water, how do I find the equivalent spring constant? I know that the water has density $\rho$, the drum has diameter $d$ and height $\ell$, and the positive direction is in the downward direction. Therefore, the buoyancy pushing up on the barrel is my spring. I have been reading the section of this topic in Mechanical Vibrations by Rao but there isn't any information that helps me figure this out.


> Quantum mechanics (QM; also known as quantum physics, or quantum theory) is > a branch of physics which deals with physical phenomena at nanoscopic scales > where the action is on the order of the Planck constant. -wiki Time is much less finite than matter. Matter can be split to the point where it is so small that it gains the classification of nanoscopic and subatomic. However the Planck constant is a measurement of mass and time so I find it difficult to separate the measurements. What measurement of time is so small that it qualifies as quantum? Could someone clear up the misconception?


title: How to model the form of a surface water wave?

text: Normal surface water waves, as generated by wind, do not have sine form but wave peak is higher and shorter than wave trough with different wave steepness. What parameters characterize such a surface water wave and how can one predict amplitude of water for given waves as function of time?


title: How to prove that generalized coordinates are independent?

text: I know that the generalized coordinates obtained after taking into account the constraint forces are independent. How can I prove this?


title: Are branes in 4D-spacetime moving, or are they static?

text: Given that a worldline, worldsheet, worldvolume, are representation in a 4D-spacetime of a point particle, a string or a brane, respectively, I was wondering if those objects necessarily have to be static in spacetime, or if they are allowed to move. If they can move, what dimension is being used to measure their changes in position?


title: How should semiconductors be understood?

text: In the chemistry setting, a doped semiconductor is a crystal lattice with holes/extra electrons in it. In the band/quantum mechanical picture, these holes/extra electrons can be seen as the amount of missing/excess charges from a full band of energy states. Therefore, in a pure silicon crystal, with full covalent bonds and thus no holes or extra electrons, the valence band is completely full, and the conducting band is completely empty. Which means that Silicon is an insulator, since the Fermi level lies above the valence and below the conducting band. What are the holes in my reasoning?


title: At what fraction of the speed of light have people traveled?

text: I'm guessing that, this would be someone in a rocket or something... When they hit their top speed, at what fraction of $c$ are they traveling?


title: Mathematical Physics SUSY QM Resource Recommendation

text: I want to study SUSY QM. I found some excellent physically motivated articles on Arxiv. Despite, I am especially interested in the mathematical structure behind SUSY QM. Does anybody know whether there is a particularly good reference to start studying SUSY QM from a rigorous mathematical point of view. The reference should cover Spectral analysis and a wide range of all the algebraic properties concerned with the topic.


title: phase-space volumes or cells for N particle system

text: For N non interacting spinless particles in a volume, we have 3N degrees of freedom and we can divide the phase space into 6N dimensional cells of volume h raised to power 3N. And each cell represents a state of the system. That means the system can be in any of those cells. How can a cell be visualized? Is it combination of many volumes of the individual particles where the particles might be in? Fixing one particle in its one of the individual states (i.e., in one cell when the system is only one spinless particle) and all the possible combination of the other particles in their cells, does this represent one cell for the N particle system?


title: Why is the triple product nτT used instead of the individual three criteria?

text: Why is the criteria for fusion represented as a triple product nτT instead of the n,τ,T individually? i.e saying the density has to be above this value, temperature above this value? What are the advantage we gain from representing it this way?


title: The BICEP2 data are evidence of gravitational waves and of inflation. Are they also the first observation that requires quantum gravity?

text: It strikes me that the recent announcement of data from BICEP2 contains two really Big Deals:   * the first evidence of gravitational waves   * the first evidence of inflation. Is there also a third?   * the first observation that will require quantum gravity to explain? Will the BICEP2 data, or data from any other experiment, require quantum gravity to explain? (I'm thinking here of other unexplained phenomena, like dark matter, etc ) **Update, two days after OP:** Today Lawrence Krauss, Alan Guth, and John Kovac appeared on NPR's Science Friday. Krauss called the event a triple discovery, and a trifecta, providing first direct evidence of all three of these things. Guth said that he agreed. But I understand all the comments and answers; it seems what constitutes "direct evidence" is in the eye of the beholder.


title: What is spin on a lattice site is it electrons or atom as a whole?

text: Hi I wanted to know what is spin half in lattice site means? Is it electron or atom or total spins half of electrons in a atomic 1d chain or 2d?


title: What are the general solutions to a hard sphere collision?

text: Surely someone has found the solutions to the hard sphere collisions (in $n$ dimensions) of two bodies of mass $m_1$ and $m_2$, respectively--that is the resultant velocities (or momenta) of the two bodies after a hard sphere collision given $\vec{p}_1$ and $\vec{p}_2$. I have a solution (both for elastic and inelastic collisions), but I wonder whether or not it's correct. FYI, here are my solutions for elastic, hard-sphere collisions: $$ \vec{p}_{1f} = \vec{p}_1 + 2\frac{\left(m_1\vec{p}_2 - m_2\vec{p}_1\right)\circ \hat{r}}{m_1 + m_2}\hat{r} \\\ \vec{p}_{2f} = \vec{p}_2 - 2\frac{\left(m_1\vec{p}_2 - m_2\vec{p}_1\right)\circ \hat{r}}{m_1 + m_2}\hat{r} $$ wher $\hat{r}$ is in the direction of the "hard sphere" (i.e. the unit vector connecting the two radii of the spheres). You can see that my formulation trivially conserved momentum--the question is whether or not it conserves energy as well.


title: Why is kinetic energy only "often $1/2mv^2$"?

text: I am reading the first few pages of Nakahara and refreshing my memory on physics I learned a while ago as a physics math undergrad. Nakahara defines a field $F$ to be conservative if it's the gradient of some potential $V(x)$. (standard) Then he defines the total energy as $1/2mv^2+V(x)$. He then states that $E$ is "often" the kinetic+potential energy, hence deserving the name "total energy." I was taught that kinetic is defined as $1/2mv^2$. So what does he mean here? **Addum:** The exact quote of Ref. 1 is: > _The function $E$, which is often the sum of the kinetic energy and the > potential energy, is called the **energy**._ References:   1. M. Nakahara, _Geometry, Topology & Physics,_ 2nd edition, 2003, Section 1.1.1.


title: Experimental relationship between linear dependence and superposition

text: From Griffith's _Introduction to Electrodynamics_ > The principle of superposition may seem obvious to you, but it did not have > to be so simple: if the electromagnetic force were proportional to the > _square_ of the total source charge, for instance, the principle of > superposition would not hold, since $\left(q_1+q_2\right)^2\ne q_1^2 + > q_2^2$ (there would be cross terms to consider). Superposition is not a > logical necessity, but an experimental fact. I am confused as to why it is important that electromagnetic force must depend linearly on charge for the superposition principle to hold. Suppose it _was_ the case that electromagnetic force depended on the square of charge. Then, defining, a new quantity "squarecharge" $k = q^2$, electromagnetic force would depend linearly on "squarecharge", and therefore does superposition not hold? Put another way, every object with charge $q$ has an associated quantity $m = q^{1/3}$. Suppose that $m$ is more naturally measured than $q$, therefore it is used in place of $q$ in all formulas. Then, even though electromagnetic forces are proportional to the cube of $m$, superposition still holds.


title: Does light loses its energy when it passes through denser medium

text: I know it does not because it emerges out of denser medium at 300,000 KM per second, but according to E=MC square and given that speed of light decreases inside denser medium with refractive index greater than 1, does not it suggest that energy of light inside denser medium is less?


title: Symmetry breaking in Bose-Hubbard model

text: According to Landau's symmetry breaking theory, there is a symmetry breaking when phase transition occurs.   1. What is the symmetry breaking of superfluid-Mott insulator transition in Bose-Hubbard model?    2. Why metallic state to Mott insulator state transition in Fermi-Hubbard model is not a phase transition, but a crossover.


title: What's the physical meaning of the imaginary component of impedance?

text: As you know, impedance is defined as a complex number. Ideal capacitors: $$ \frac {1} {j \omega C} \hspace{0.5 pc} \mathrm{or} \hspace{0.5 pc} \frac {1} {sC} $$ Ideal inductors: $$ j \omega L \hspace{0.5 pc} \mathrm{or} \hspace{0.5 pc} sL $$ I know that the reason why they 'invented' the concept of impedance is because it makes it easy to work with circuits in the frequency domain (or complex frequency domain). However, since in real-life circuits both voltages and currents are real numbers, I'm wondering if there is any actual physical meaning behind the imaginary component of impedance.


title: CMBR temperature over time?

text: How has CMBR temperature dropped as function of time? A graph would be nice, but I'd be happy with times (age of universe) when it cooled enough to not be visible to human eye, became room temperature equivalent, or reached some interesting temperatures regarding matter in the universe. If there is a nice formula giving the temperature as function of time, that would be great too.


title: How to measure the curvature of the space-time?

text: I know G.R. change our vision of space and time as a unique surface than can bend. We can associate the curvature of the space-time as the gravity created by the mass of planets, stars... But how can we measure the curvature of the space-time for calculate, as an example, the position of a star?


title: Is there an upper limit to temperature in thermodynamics or statistical mechanics

text: In many presentations of statistical mechanics where we have a system of particles having mass, such as the molecules of an ideal gas, the temperature is often equated to the average relative velocity of the particles. However no such velocity can exceed that of light. Is there therefore an absolute upper bound to temperature as well as an absolute zero?


title: Is plant photosynthesis more efficient than solar panels?

text: Is photosynthesis more efficient than solar panels? If so, by how much?


title: Motion in a central field and angular momentum

text: Is it correct that for a motion in a central force field, e.g. a gravitational field, the absolute value of the total angular momentum of the particle and the component of the perpendicular to the plane of motion are conserved, but the other components of the angular momentum in general not?


title: Why are bicycle pedal threads' handedness left on the left and right on the right?

text: I understand the reason that bicycle pedals are oppositely threaded on either side. What I don't understand is why it works because I'm missing something. Take the right pedal for example. It's threads are right-handed, so it would be tightened by turning it clockwise (if the pedal were in between you and the bike). Now imagining this from the same perspective, a person pedaling makes the right pedal constantly turn counter-clockwise, relative to the crank. If there's any friction between the pedal and the bolt attaching it to the crank, I would imagine that it would create a force that would try to turn the bolt counter-clockwise along with the pedal. This is the opposite of what happens, and I've seen the bolts unthread after pedaling backwards a lot myself. What am I missing?


title: Why does the video recorded by a falling, rotating camera seem to stabilize?

text: Here is a video of a GoPro falling from a plane: http://www.youtube.com/watch?v=QrxPuk0JefA Any idea what is happening when the image "stabilizes" around 0:35? I think it is because the camera's tangential velocity approaches the speed of the GoPro's rolling shutter. Follow-up question:   * Why is the image skewed with the left side skewed "up" and the right side skewed "down?" A challenge:   * Find the axis of rotation (with supporting diagram and explanation of how it was computed).


title: What if LHC finds SUSY?

text: Here and on many other forums and blogs people ask the question "What if LHC does not find SUSY?". I would like to ask the opposite. What if it finds it? What would the implications be? Is it going to just confirm something understood and expected or is it gong to bring something new? Will there be implications for string theory? String theorists will take it as a good sign, but by itself it is not a confirmation of string theory. I do realize this is more that one question, but they are on the same topic. p.s If this question is not suitable for this site, please delete/close


title: A telescope with a bunch of small mirrors

text: Together with colleagues we got this question. Imagine to take small mirrors, the size for example of a dentist mirror, and stick them to a wooden frame with a parabolic shape. Each mirror is flat, and once set of the frame it can be aligned properly with screws so that it redirects its reflection towards the focus of the parabola. All the parabolic surface is covered with these small mirrors. Now, if you build a Newtonian telescope out of this system, would you get a decent telescope? I expect it to be affected by various aberrations, but would it work ?


title: What is the SI Unit for intercept of a graph

text: I have a graph that i drew and i got all the point and a question came up that what is SI Unit for intercept of a graph. I have no idea what the answer is. (I'm a total newbie in Physics)


title: Inertial and non inertial frames of references

text: I've heard from a physics professor that there's no stationary platform to observe and analyze a body in motion.Why did he mention that? Is it because even seemingly stationary objects like a parked car on a road moves along with the earth's rotation? As per my intuition there is a stationary platform.If you or your car is at rest then that becomes a stationary platform and you can study the kinematics of other objects.Am i right?? Also what is a frame of reference? Which one is intertial and non inertial? (I might have not framed the question correctly.appologize for that)


title: How does a change in temperature affect relative humidity

text: Assume that the air pressure and the amount of water in the air stay constant. How can I figure out how much a change in temperature affects the relative humidity?


title: Product of ρ and φ

text: Consider a charge distribution $ρ_1(r)$ and the potential $φ_1(r)$ due to it. Consider another charge distribution $ρ_2(r)$ and $φ_2(r)$ due to it. Both distributions have finite extent, but are otherwise arbitrary and need not have anything to do with each other. Show that the $$∫ρ_1(r) φ_2(r) dv = ∫ρ_2(r) φ_1(r) \; dv$$ where the integrals are taken over all space. Solve this in two different ways as follows. a) Consider the two collections of charges to be rigid objects that can be moved around. Start with them initially located very far apart, and then bring them together. How much work does this require. Imagine moving collection 1 toward collection 2, and then the other way around. b) Consider the integral of $∫E_1 E_2 \; dv$ over all space, where $E_1$ and $E_2$ are electric fields due to the two distributions. By using the vector identity $∇ \cdot (E_1φ_2) = (∇ \cdot E_1)φ_2 + E_1 \cdot ∇φ_2$ (and similarly with the $1$'s and $2$'s switched), rewrite the integral $∫E_1 E_2 \; dv$ in two different ways. This is a homework question, and I have no idea where to start. If someone could show me how to set this equation up to solve, then I would deeply be thankful.


title: Center of mass of a car

text: This might be little of the track but my question is more on possibility of application of principle of center of mass.   The probability of a vehicle overturning depends more on the level of center of mass of the body above the ground, the higher it is and smaller is the base to support it, the easier it is to tumble it.   In case of vehicles on the road, how much is the possibility that lives can be saved to some extent if the driver or some intelligent system in car can let go the tires to bring the center of the mass of body closer to the ground and preventing it from deviating in the traffic to a great extent.   I can fully understand the momentum of the body but it might be helpful in preventing it from crossing over the divider to the other side of the road.   This question is amateur but seeing these horrific videos of road accident, this thought came to me.


title: Can a SQUID perform real time measurements of a slow varying magnetic flux?

text: It is well known that a SQUID can detect an extremely small magnetic flux. What if the flux is time-dependent? E.g. a slow varying sinusoidal magnetic flux $$\Phi=\Phi_0\sin(\omega t).$$ Can I still perform a real time measurement of the slow varying flux considering the induced EMF? By "slow" I mean $\omega$ is smaller than Josephson frequency (AC Josephson effect due to the EMF).


title: Solution to Klein-Gordon equation always valid?

text: We know that there is a relativistic version of Schrodinger equation called Klein-Gordon equation. However, it has some problems and due to these problems, there is Dirac equation that handles these problems. So, the question is, if there is a solution that is allowed by Klein-Gordon, but not by Dirac, can this solution be considered valid? Also, can Dirac equation be used for spin-0 particles?