title: Why do we have an elementary charge but no elementary mass?

text: Why do we have an elementary charge $e$ in physics but no elementary mass? Is an elementary mass ruled out by experiment or is an elementary mass forbidden by some theoretical reason?


title: Reconciling refraction with particle theory and wave theory

text: I have searched the web for good answers to why refraction occurs when light moves from one medium to another with different density. I have limited background in physics and want to know if there is an easy way to understand this without having to go back to school for years. Most explanations come in some variant of the "Marching line of soldiers" analogy, where a straight line of soldiers change pace when they hit an boundary. I have no problem understanding the analogy, my problem is understanding how this is even relevant to light.   * The analogy does not work for light as a particle. The particle does not "know" of particles around it, and should follow a straight line.    * How does a single ray of light (wave), as part of a "wave front" know what other rays are doing?


title: scattering theory quantization condition

text: where and how is found the scattering quatniztion condition $ det (1-zS)= 0 $ so the energies or bounded states of scattering come out from this here $ S=S(k)=e^{2i\delta (k)} $ where could i find some info about it ?? wouldn't be easier to impose the condtion on the functional determinant of the Schroedinguer operator $ det(E-H)=0 $


title: Why is the relationship between atomic number and density not linear?

text: What are the factors that affect the density of an atom?


title: Polarization of the gluon

text: I think that, by now, it's understood that the gluon propagator in QCD has a dynamically generated mass. Ok, so my question is the following: where does the extra polarization degree of freedom come from? Or, asking in another way: suppose you try to define an S matrix for QCD, apart from the usual problems for doing so, would it be unitary? How? In the case of a Higgs mechanism, it is clear that the extra degree of freedom comes from "eating" the Higgs, as they say. But and where does is come from in theories where the mass is dynamically generated?


Most textbooks on quantum field theory never mention dispersion relations at all. Where can one learn about dispersion relations for S-matrices?


title: What is the notion of a spatial angle in general relativity?

text: Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror in such a way, that I measure a right angle between the incoming and outgoing photon. How can I now calculate the wordlline of the outgoing photon? Of course I cheated a bit since I just used the old euclidean notion of an angle when I say "I measure a right angle between the incoming and outgoing photon". But I hope that this is allowed locally.


title: Why does this equation refer to magnetic tension?

text: The force density in a plasma, according to MHD, is $$ \mathbf{F} = -\nabla P -\nabla(\frac{B^2}{2\mu_0}) + \frac{1}{\mu_0}(\mathbf{B}\cdot\nabla)\mathbf{B} $$ where first term is the thermal pressure gradient, the second term is the magnetic pressure gradient and the third term is the **magnetic tension** The magnetic tension tends to straighten out the field lines thus reducing their curvature. How do I see this from the form of the equation $\frac{1}{\mu_0}(\mathbf{B}\cdot\nabla)\mathbf{B}$?


title: Change in volume of air held at the top of a tube with upper end closed and holding some liquid

text: Consider a tube both end open.dip one end in a container with water..close the top with your thumb..(make sure some air remains between your thumb and the liquid in the tube)..now take the tube out of water.. Now Does the volume of air inside the tube expands..I mean does the volume of the air held at the top increases?(top end tightly close)


title: Different hamiltonians for quantum harmonic oscillator?

text: The Hamiltonian for a _classical_ simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = \frac{1}{\sqrt{2m\omega\hbar}}(m\omega\hat{x} +i\hat{p}) , \ \ \ \ \ a^{\dagger} = \frac{1}{\sqrt{2m\omega\hbar}}(m\omega\hat{x} -i\hat{p})$$ we get the quantum Hamiltonian $ \hat{H} = (a^{\dagger}a + \frac{1}{2})\hbar\omega$ so that $ E_n = \hbar\omega(n+\frac{1}{2})$. **BUT** If I write the classical Hamiltonian $$H= \frac{1}{2m}(p+im\omega x)(p-im\omega x)$$ and replace $a$ and $a^{\dagger}$, we get $\hat{H} = \hbar\omega aa^{\dagger}$ and $E_n = \hbar\omega(n+1)$... **SO** On the plus side the energy spacing is the same, but the zero-point energy is different. Now I guess that since we only detect energy differences and not absolute energies, this does not change the Physics. Is this right? How come we get two answers though? Mathematically I mean. Are there any other quantum Hamiltonians that can be obtained in a similar way?


title: Question on Equipartition at Reheating?

text: Apologies in advance if this is a silly question. I am fimiliar with the idea of Reheating in cosmology, as discussed briefly here. I have been reading several papers which make the assumption of 'Equipartition at Reheating', however I have not come across a definition/ description of it. So, in Reheating we say that at the end of inflation the Inflaton transfers it's energy to the Standard Model (SM) particles, or some as of yet unknown particles, through some coupling. As regards the Equipartition then, is it making the assumption that all particles of all particle species have equal energy after the end of inflation? For this to be the case, would not we need either the inflaton to couple directly to each SM particle species? Alternatively, we need it to couple to some as of yet unknown particle, perhaps a WIMP, which would then in turn need to be having many interactions with the SM particles in order to 'equipartition' (can we use this as a verb?) the energy out between all particle species. If this is the case, it seems like a big assumption to make. That is of course, unless I am missing something, which is highly likely!


title: Minus Sign in Feynman Diagram

text: I've been reading these notes and I can't figure out the why on P.120, it is said that > The fermionic statistics mean that the ﬁrst diagram has an extra minus sign > relative to the ψψ scattering of Figure 25. Would someone mind explaining it to me? I see that the calculations below, illustrating this, but is there an intuitive way of seeing that the signs of the respective first diagrams in Figures 25, 27 should be opposite? Thank you.


title: How to relate wingtips vortices with the energy loss it engenders?

text: On planes, wingtips vortices engender a loss in terms of lift, and an incresed of the drag, so an energy loss. Is there any possibilty to find a mathematical model relating both ? Thanks for the help.


title: Moduli Stabilization in 6D Einstein-Maxwell theory - Fluxes and O3 planes

text: I'd like to do the maths for the moduli stabilization of 6D Einstein-Maxwell Gravity $$ S= \int d^6X \sqrt{-G_6}(M_6^4R_6[G_6]-M_6^2|F_2|^2), $$ where the 6D metric is specified by $$ ds^2 = g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+R^2(x)\hat{g}_{mn}(y)dy^mdy^n, $$ and $\hat{g}_{mn}$ is the metric of a compact 2D manifold with unit volume. The setup of this model can be found in Denef, Douglas, Kachru, starting on page 10. Now I want to perform the dimensional reduction of the theory and thereby obtain the correct effective potential $V(R) = \frac{\chi}{R^4}-\frac{n^2}{R^6}$. I could manage to derive the first part of the potential by rewriting $$ ds^2= R^2(x)\\{\frac{g_{\mu\nu}(x)}{R^2(x)}dx^{\mu}dx^{\nu}+\hat{g}_{mn}(y)dy^mdy^n\\} $$ which allows me to rescale the 6D action by $G_{MN}=R^2\tilde{G}_{MN}$. Then we are left with a product space, i.e. we can write $R_6[\tilde{G}_6]=R_4[\tilde{g}]+R_2[\hat{g}]$, where $\tilde{g}_{\mu\nu}=\frac{1}{R^2}g_{\mu\nu}$. Further rescaling brings a factor $R^{-4}$ for the term $\chi$, while a factor $R^2$ has been absorbed in order to write $M_4^2=M_6^4R^2$. However, I'm not sure how to obtain the term $\frac{n^2}{R^6}$. I'd appreciate any help. My idea is as follows: $$ \int d^6X \sqrt{-G_6}M_6^2 |F_2|^2 = \int d^6X R^6 \sqrt{-\tilde{G}_6}M_6^2 |F_2|^2 = \\\ = \int d^4x \sqrt{-\tilde{g}}\int d^2y\sqrt{\hat{g}} R^6M_6^2 |F_2|^2 = \int d^4x \sqrt{-g}\int d^2y\sqrt{\hat{g}} R^2M_6^2 |F_2|^2 \sim \\\ \sim M_6^2 \int d^4x \sqrt{-g}R^2 (\frac{1}{R^2})^2 \cdot (\frac{n}{R^2})^2 = M_6^2 \int d^4x \sqrt{-g} \frac{n^2}{R^6} $$ In the last line, we obtain a factor $R^{-4}$ because of $\gamma^{m\hat{m}}\gamma^{n\hat{n}}F_{mn}F_{\hat{m}\hat{n}}$ and the second factor comes from $F_2 \sim \frac{n}{R^2}$, since $$ \int_{M_2} F_2 =n. $$ However, I'm puzzled because the factor of $M_6^2$ remains. If I want to rewrite the action s.t. we have a factor $M_4^2$ in front, then the term I obtain reads $\frac{1}{M_4R} \frac{n^2}{R^6}$ (reminder: $M_4^2 =M_6^4R^2$). So, where is my mistake? Furthermore, it is claimed that O3 planes give rise to an additional term $\frac{m}{R^4}$ (after Weyl rescaling), but I've no idea how it arises. I would start with a CS term $\int C_4$ for the action... It would be great if anyone could help me with these questions. Thanks in advance!


title: What would be the minimal size of an aerial vehicle capable of sustained suspersonic flight?

text: I wonder, what's the smallest possible size for a UAV capable of sustained supersonic flight at the current technology level? Let's say 10 minutes of flight at 1.1 M.


title: A change of sign in the electron-hole second quantization form

text: It is common to see people do a change of sign in the so called electron-hole representation, namely, $$ b^{\dagger}_{-k}=a_{v,k} $$ similar argument also seen in 1992 mattuck's book "guide to feynman's diagrams" pp.138, eq.7.78b. Can somebody explain what is the physical meaning of changing the sign of momentum, since it make a difference in specific calculations, clearly it is not just a convention, meanwhile it seems to change momentum conservation law in a way that is unambiguous in the electron-electron representation..


title: Is a billion volt electrical transmission line possible?

text: Because $$ P_{\,\displaystyle\rm loss}=\frac{P^2R}{V^2} $$ in an electrical line where $R$ is the total resistance Why not use a $1$ billion volt line instead of a $800\,\rm kv$ line? Is there a maximum voltage that copper can carry?


title: Superconducting loop as a particle detector?

text: Imagine that one has induced a persistent current in a superconducting loop. If a particle interacted with an electron in one of the Cooper pairs then would some decoherence and dissipation occur? I guess if that happened then the current would be diminished somewhat leading to a detectable change in the magnetic field around the loop. Could one use such an effect to detect particles (perhaps even dark matter particles)?


title: Query on Polarization

text: Describe the state of polarization represented by (here w = omega and TT = pi) a) $E_x = E\sin(kz - \omega t)\,\& \,E_y = E\cos(kz - \omega t)$ b) $E_x = E\cos(kz - \omega t) \,\&\, E_y = E\cos(kz - \omega t + \pi/4) $ c) $E_x = E\sin(kz - \omega t) \,\&\, E_y = E\sin(kz - \omega t)$


title: Is it easier to move or rotate an object?

text: This is a problem I'm facing on designing a moving bookcase for my home, I just don't have enough physics background to tackle the problem. Which of the following requires more force/effort?   * A bookcase with a pivot point in one of the back corners that is rotated 180° to its new position (therefore we now face the back of the bookcase).   * A bookcase that is pushed/pulled to the same relative position but with a diagonal push (guided by one rail at the top). In both cases the bookcase will be supported by equivalent fixed caster wheels aligned with the direction of movement or rotation, with the pivot and the rail acting only as guide. I don't think it should matter, but the bookcase will be about 1 ft deep and 3 ft wide, I don't have any good source, but I think it would be around 200 kg if fully loaded.


title: Sum total distance of electrons on a spherical surface

text: What is the sum total distance between every possible pair of point charges when there are n point charges on a spherical surface? All point charges can only and are located on the infinitesimal spherical surface. Basically, we're going to have a bunch of points staying as far apart as possible on the surface of a sphere. Is there a general equation for this? And whats the sum total distance between every pair of points? (Both the distances by travelling the shortest distance on the spherical surface and the shortest distance travelling through the interior volume of the sphere. I'm confident that the relationship between those two values require a relatively simple equation. Is there a similar mathematical model/problem which can be used to solve this problem?


title: Explaining to a five year old - why don't bubbles run like water

text: My five-year old daughter was asking about astronauts the other day and why they float in space. After me showing her a few bits on the kids section on the NASA web site I started explaining about the planets and how the sun was a ball of gas and you couldn't stand on it. I then had to explain to her what a gas was, using ice, water and steam as examples for the three main states. A few hours later she asked me if the bubbles from the bubble-bath in the bath water were a liquid and if they were why didn't the gravity make them run off her hands like water. That had me stumped. I'm not a scientist but I've been doing my best to try and figure this out and explain it to her but I'm not fully sure, either. Firstly, are bubbles from bubble-bath a liquid or have I made a bad assumption? I thought it might be down to friction but I assume bubble-bath is a detergent and shouldn't encounter much friction. I then thought it might be because the bubbles had a large surface area relative to their weight and would suffer wind resistance as a result. My best guess now is that it's because the bubbles were more viscous than the water. Are any of these right, if not can someone explain it to me in simple terms so I can tell her, please? **UPDATE** Wow, thanks for all the responses! I'm usually on Stackoverflow and I don't think that in two and half years on there I've asked anything that's generated this level of response. Thank you to everyone who took the time to answer or comment - I've read all that's been written here. I explained this to my daughter last night saying that the bubble bath liquid was sticky and as a result would stick to things and slow down gravity's pull. I drew a bubble and explained about the liquid stopping the air in the middle getting out and by standing opposite and holding hands both I explained how we were like the liquid of the bubble squeezing the air in the middle which she understood. I expect that if she's asking questions like this now I'll be on here again soon asking something else. I've certainly learnt a lot! Thanks again.


title: What is the meaning of $G/cm^2$?

text: I'll prefix this question by stating that I am not a physicist, I am a programmer. I am working on updating a (very old) program that calculates the performance of a magnet. This whole program is in CGS units and it is required of me to change it to SI units. A particular equation in this program gives the Gauss/square centimeter. I understand this as the gauss going through a 1 square centimeter area. Is there a proper name for the unit of $\frac{G}{cm^2}$?, the closest I can find in CGS terms in a Maxwell unit which is $G \cdot cm^2$ It's less about the name of the unit and more about being able to check whether this empirically derived equation is correct. Unfortunately I cannot post this equation as I do not have the necessary permissons. * * * In the comments it has now been shown that $$\frac{G}{cm^2} = \frac{T}{m^2}$$ So the question now becomes, what does this mean? The field strength in an area?


title: What is the relation between photoelectric current and frequency of incident light?

text: I googled it a bit and found that photoelectric current is independent of frequency(of incident light).Some further look revealed that actually "saturation current" is independent of frequency.I could not find about the instantaneous current(current other than saturation current). > Speculation 1: If saturation current is not reached, then radiation with > higher frequency will give greater photoelectric current. > > Reason 1: Greater frequency means greater velocity of electrons, which will > help them to counteract "space charge" and more photons can reach the anode. * * * > Problem 1:Let's say the intensity (W/m²) of the radiation remains the same, > and only the frequency increases. The intensity multiplied by the Area of > the plate results in the total energy that arrives the plate in each second. > So IA = E/Δt, where E = nhf (n photons of frequency f) Let's say each photon > is able to pull out one electron from the plate, so the current i = ne/Δt, > where e is the charge of the electron. That gives n/Δt = i/e, and IAe=hfi -> > i = IAe/hf, so when we increase the frequency, if the intensity of the > radiation remains the same, the current decreases. Is this right? Are my speculation and reason correct? And please help me to resolve my problem ?


title: XCrysDen Structure file

text: Does anyone know how to directly convert a .cif file to a Xcryden structure file(.xsf) ? I know how to extract the lattice vectors and the atom positions from a .cif file, but don't whether the .cif file contain the conventional lattice vectors that need to generate a .xsf file. Thanks.


title: Gravitation - why is it so confusing?

text: First, does gravity emit from all directions around a star or planet equally. Secondly, how is it that gravity pulls a planet closer, then it goes further out, without the planet changing orbit permanently?


title: Conservation of momentum and energy in an explosion

text: One simple problem is physics is to determine the mechanical energy difference after an explosion. To do this, you must assume that momentum is conserved because in a explosion you have internal forces, so, using Newton's third law, you get that the total momentum is conserved. How true is that? I mean, when a explosion happens you have LIGHT and SOUND being propagated. And they carry energy and momentum. It's clear to me that if you only consider the momentum of the parts of the body + the momentum of the sound wave + the momentum of the light wave this should be equal to zero. However, it's not so clear if the momentum of the body should be equal of the sum of the momentum of its fragments. I have some conjectures, but I didn't find nothing related to that: **I)** The waves propagate symmetrically, so, because momentum is a vector, it should be zero. However, the explosion may not be total symmetric, so I don't know if this argument is valid. **II)** The momentum is not conserved for the body, but the momentum of the wave is negligible. These are the two possibilities I considered, so I don't know if they are valid.


title: Derivative of a constant tensor field along a path

text: I understand that the time derivative of a tensor filed $\mathbf T$ along a curve $\gamma\left(t\right)$ can be shown to be, $$ \frac { d\mathbf{T}}{ dt } = \left(\frac { dT^{ i } }{ dt } + V^{ k }\Gamma ^{ i }_{ km }T^{ m }\right)\mathbf e_i = T^i\mathbf e_i$$ where, $\mathbf e_i$ is the covarient basis and $V^k$ is the velocity component of the position vector w.r.t covarient basis ($\mathbf V = V^k\mathbf e_k$).$$$$ My question is how to prove that $\frac{d\mathbf T}{dt}$ becomes zero when the tensor field $\mathbf T$ along the curve $\gamma\left(t\right)$ is constant? Any hint on this will also be very important.


title: Momentum operator of a particle in an electromagnetic field

text: In quantum mechanics, to all observables correspond some self-adjoint operators. In the absence of an electromagnetic field the momentum operator is clearly $\vec{P}:=\frac{\hbar}{i}\vec{\nabla}$. However in the presence of an electromagnetic field (which in QM is introduced by means of a gauge dependent potential) it is difficult to interpret $\frac{\hbar}{i}\vec{\nabla}$ as the momentum operator since it is a gauge dependent object. The natural choice seems to be $\vec{\Pi}:=\frac{\hbar}{i}\vec{\nabla}+e \vec{A}$. Is it true that measured values of the components of momentum of a particle moving in an EM field are the eigenvalues of $\vec{\Pi}$ not $\vec{P}$?


title: Richter scale equation

text: According to Wikipedia, the equation for finding magnitude according to the Richter scale is $M_\mathrm{L} = \log_{10} A - \log_{10} A_0(\delta)$. However, it is rather ambiguous as to the effect of epicentral distance ($\delta$) on the equation. Other sources seem to suggest that the distance from the epicentre is taken into account with the function of $A_0$, but do not specify how. Is it just basic multiplication, or is it actually more complex?


title: What are "finely subdivided" substances?

text: As in the title, what does it mean for a substance to be _finely subdivided_ , or _finely grained_? For example: > It is only in the cases for which the ratio area/volume is very large (for > example, a _finely grained_ substance) that the surface must also be > considered. or: > In the case of a system composed by two homogeneous substances, it will be > possible to express the internal energy of the system as the sum of the > energies of the two substances only if we can neglect the surface energy of > the two substances where they are in contact. The surface energy can > generally be neglected only if the two substances are not very _finely > subdivided_ ; otherwise, it can play an important role.


title: Modular invariance of CFT

text: I am looking at the Cardy formula for entropy in CFT, and in the article 'Kerr/CFT correspondence and its Extensions' there is a sentence: > In any unitary and modular invariant CFT, the asymptotic growth of states... I've looked a bit, but couldn't find any simple explanation of the term modular invariant CFT. All I find is that modular invariance is the invariance under the group $SL(2,\mathbb{Z})$. But what does that mean, what's the physical explanation of it and what's it's effect?


title: How can a transversality condition be invoked to reduce the Euler-Lagrange equation?

text: I asked this question regarding the Euler-Lagrange equation at MSE and have gotten no response. I will ask it here too. I think I might have more luck here since the E-L equation is at the core of just about everything a physicist thinks about. The question is basically this: Under what transversality condition can the E-L equation $\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0$ be reduced to: $$L-\dot{x}\frac{\partial L}{\partial \dot{x}}=0$$ In a seminal 1931 paper the economist H. Hotelling justified this reduction by invoking the condition that the action be at a maximum. I do not understand how such a condition justifies the reduction of the E-L equation.


title: What is the name of the temperature limit beyond which an explosion will form a mushroom cloud?

text: Many months ago I saw a picture that was taken many years ago of an explosion, possibly in the Atlanta area. If I recall the explosion was caused by fuel in railroad cars. However, the explosion formed a distinctly mushroom shaped cloud (caused by Rayleigh-Taylor instability) and I remember reading that any explosion (specifically meaning it does not have to be a nuclear explosion) can form a mushroom cloud if the temperatures involved are above a certain limit. However I forgot what the name of this limit is and I have been unable to find it through searching. I seem to recall it had a rather Russian sounding name. Can anybody tell me what it is?


title: Deduction of relativistic mass formula in 1+1 dimensions

text: I have read the explanation/calculation of relativistic momentum and relativistic mass in the Feynman lectures, see chapter 16.4 here: http://www.feynmanlectures.caltech.edu/I_16.html. (I guess this explanation goes back to Lewis/Tolman.) The argument seems to use two spatial dimensions in order to make the situation of the elastic collision of two objects with the same rest mass not too symmetric. My question is whether there is a similar argument that already works in one spatial dimension. Please note that I am _not_ looking for a more sophisticated explanation using four-momentum (or rather two-momentum in 1+1 dimensions), although it will be much more elegant mathematically.


title: How can a biochemist become a physicist?

text: I am currently 22 years old, I graduated in Biochemistry and am now finishing my Master's in Bioengineering and Nanosystems, both with high grades. I've been thinking of leaving life sciences since I've rediscovered my passion for Physics (and Philosophy). Having some background on classical mechanics, physical-chemistry and electronics, how can I start a career in Physics? I'm thinking about applying abroad for a 1/2 year Master's in Physics, for which I need some background in electricity and magnetism, classical mechanics and introduction to QM; after the Master's, I would be apt to apply for a phD.   * Is there any other way to shorten this path? I've heard for instance that to apply for phD in USA I only needed to be successful in GRE tests.   * Given economic limitations I was thinking of preparing the background needed for the MSc independently. Are there ways to refer this study such that it'd be recognized by an educational institution (e.g. online courses)?


title: About memory effect in E&M when E and B field has dependence in the history of E and B fields

text: How should one define the memory effect in E&M formally? What is the physical way to visualize the memory effect? How should one understand it intuitively?


title: Why does inflation (the inflaton field) push Omega down closer to zero (flatten the universe)?

text: I know that in our particular case the inflaton field expanded the volume of the universe while simultaneously maintaining a mass-energy density close to the critical density all the while, thus the flatness. But why that number? Why didn't the inflaton field maintain a density which would have caused some other kind of curvature instead? I thought inflation was supposed to solve the fine tuning problem but I still run into the problem. Basically, what makes inflation "choose" which energy density to maintain? And if it always picks the critical density (for flat curvature), why is that?


title: What is the relative acceleration of a projectile fired at in a low gravity vacuum?

text: This question is to end an old argument. Given a (space) vehicle with an acceleration of X, and projectile with an acceleration of 2X (rocket, not bullet), what would the relative acceleration of the projectile be if fired from the vehicle along its axis of motion? My assertion is that, relative to the vehicle, the rocket would accelerate at 2X, and to an outside observer, the rocket would accelerate at 3X (the acceleration of the launch platform and the projectile itself). My friend’s assertion is that relative to the vehicle, the rocket would accelerate at X, and an outside observer would see the rocket accelerating at 2X.


title: Transfer of angular momentum during oblique impact with(out) adhesion

text: I am solving a problem of 2 identical spherical particles colliding. The impact is oblique. I consider both normal and tangential contact forces. Tangential force is a combination of elastic force and friction. I seems reasonable to me that after collision angular momentum should be the same for both particles, i.e. they start spinning in the same direction (particle B acting on particle A creates positive torque, particle A acting on particle B creates opposite force, but the sign of torque is the same). This also agrees with some experimental and theoretical results. The problem starts when adhesion force is added. What happens with the momentum transfer once particles stick together and therefore cannot spin relative to each other? It seems to me that they must start spinning as a whole. However I cannot physically reason it quite well. Any suggestions? UPDATE: thanks for your replies. This is indeed a complex problem. To clarify some details: I am trying to follow a paper by Thornton & Yin, Powder Technology, 65 (1991), "Impact of elastic spheres with and without adhesion". I am timestep integrating all of the forces involved, i.e. collision is not instantaneous (~100-300 timesteps). When the particles stick due to adhesion there is the effect called "peeling", and this occurs until critical value of tangential force is reached. Then sliding starts (either a rigid body sliding, i.e. pure friction or micro-slip, which is a combination of elastic force and friction, as by Mindlin and Deresiewicz (1953), ASME J. Appl. Mech,20, p327). I have my simulation match the results of Thornton and Yin except the case of adhesion. The rebound angles (of particles' centres) match, so the model must be almost working, but the contact patch reflection angle does not (this one depends on spin of the particles caused by collision). This makes me think I am not implementing the transfer of angular momentum right in the case of sticking particles. I will try to work out how to modify the angular momentum of the adhered pair from you replies and the paper you attached.


title: Is there an upper frequency limit to ultrasound?

text: Wikipedia has this diagram of the acoustic frequency spectrum: ![enter image description here](http://i.stack.imgur.com/N8klp.png) Is there an upper limit to the frequencies you can transmit through the air? Are they absorbed more and more at higher frequencies, resulting in shorter travel distances? Are there "windows" of absorption and transmission like there are for radio waves? ![enter image description here](http://i.stack.imgur.com/9x3VJ.jpg) **Update:** This says > An additional problem is created by the strong absorption of ultrasound in > air, at least for frequencies higher than 250 kHz. As a consequence, > ultrasound of e.g. 1 or 2 MHz can propagate in air over a distance of not > more than a few centimeter Also found some references, but I can't access them: http://scholar.google.com/scholar?q=Atmospheric+absorption+of+sound


title: Does all information in the universe come from the observer?

text: In absence of the observer any system undergoes unitary evolution, that is reversible evolution without entropy change. It is believed that the initial state of the universe had very low entropy, possibly, zero. So the entropy of the universe grows only in interaction with the observer, i.e. in the process of wave function collapse. But the total amount of entropy in the universe characterizes the amount of information stored there (and needed to describe the state). As such it turns out that the wole information stored in the universe comes from the interaction with the observer. The observer serves as the channel which swells the universe with its content from outside. Am I correct?


title: Walter Lewin's lecture on Faraday's law

text: In Walter Lewin's lecture on Faraday's law, he does a demonstration with a solenoid where he shows that current is induced in a loop that surrounds the solenoid as the magnetic flux is changing. He claims that the induced current is the same regardless of the size of the loop of wire that he uses (since the magnetic flux is the same in both cases). But it seems to me that the smaller loop should have a higher induced current. Faraday's law says that: $$\oint_C{\mathbf{E} \cdot d\mathbf{l}} = - \frac{d}{dt}\int_S\mathbf{B} \cdot d\mathbf{S}$$ If we imagine a solenoid with radius $R$ producing a changing magnetic field, and surrounding it is a wire of radius $r_1 > R$ forming contour $C_1$ and a larger wire with radius $r_2 > r_1$ forming a contour $C_2$ (all concentric), we can see that for the flat surfaces $S_1$ and $S_2$ which enclose the contours $C_1$ and $C_2$ respectively the change in magnetic flux must be the same: $$\int_{S_1}\mathbf{B} \cdot d\mathbf{S} = \int_{S_2}\mathbf{B} \cdot d\mathbf{S}$$ Then for any given time $t$, $\oint_{C_1}{\mathbf{E_1} \cdot d\mathbf{l}} = \oint_{C_2}{\mathbf{E_2} \cdot d\mathbf{l}}$. By symmetry, the electric fields will be uniform along their contours so $E_1(2\pi r_1) = E_2(2\pi r_2)$. Since $r_2 > r_1 \implies E_1 > E_2$. So this means that the electric field would be larger in the smaller loop and so the current should also be higher. It seems to me that if you pay close attention to the video, the current reading for the smaller loop is actually higher. That said, it's much more probable that Walter Lewin is correct and I made a wrong assumption somewhere. Is my understanding of Faraday's law incorrect?


title: Compatibility between solutions of explicit Maxwell equations vs. wave equation?

text: When trying to solve for the allowed propagation frequencies in a cylindrical waveguide, I approached the problem by solving the wave equation for all three components of $\bar{E}$, and subsequently apply boundary conditions - $E_z(R) = E_\phi(R) = 0$ and $\nabla\cdot\bar{E}=0$. This yields the modes: $$E_i=A_ij_m(k'\rho)e^{\pm im\phi}e^{i(k_zz-\omega t)},\ k'^2\equiv\frac{\omega^2n^2}{c^2}-k_z^2, \ i=x,y,z$$ Where the boundary conditions set the degrees of freedom (the discrete values for $k'$, setting the $A_i$'s, etc.) On the other hand, reading through Jackson (3rd ed. chap. 8), solving directly Maxwell's equations, he states that only $E_z$ should satisfy a wave equation, and the other (traverse) components are given by $E_z$'s derivatives. This obviously leads to Bessel derivatives $j'_m(k'\rho)$, as he demonstrates later in that chapter, which are absent from the modes I found above, if we ignore the relation between $j'_m$ and $j_{m\pm1}$. Where does this discrepancy come from? I would imagine that the transverse elements can easily be found by   $$E_\rho=\cos(\phi)E_x + \sin(\phi)E_y, \ E_\phi = -\sin(\phi)E_x+\cos(\phi)E_y$$ If one wishes to avoid solving the wave equation directly in polar coordinates, as then the components are mixed together.


title: Question about gravity in 1 dimension

text: Consider 2 pointmasses in 1D space. More specific ; object $A$ with mass $m_a$ and object $B$ with mass $m_b$. $A$ and $B$ are a distance $d$ away from eachother. Let these pointmasses $A,B$ have no initial speed or rotation. Let $d(t,m_a,m_b)$ be the distance between $A$ and $B$ after time $t$. At collision $d(t,m_a,m_b)=0$ and at starting position ($t=0$): $d(t,m_a,m_b)=d$. What is the closed form or integral equation or differential equation for $d(t,m_a,m_b)$ according to the different theories of gravity, such as,   * Newtonian?    * special relativity(*)?    * general relativity? (* I mean Newtonian + the velocity addition from special relativity ). I also wonder about the analogues for the Laplacian for each gravity theory.


title: The Alcubierre drive and closed timelike curves

text: Under what conditions would it be possible to create closed timelike curves, assuming an Alcubierre drive could exist? Would it be possible to have the latter without the former? See here for information on the chronology protection conjecture.


title: Are black holes stationary?

text: If light/energy with zero mass hits the speed limit, are black holes with infinite density at a universal standstill with everything moving relative to them? Am I barking up the wrong tree as they would move relative to each other?


title: Double rainbows

text: In my garden, when I'm watering the plants I sometimes see a rainbow or two. How did two rainbows appear? Why can't I see three rainbows then, or how can I see three rainbows?


title: Does a 27 hp engine output the same amount of energy as lifting a 1 ton stoneblock almost 3 meters per second?

text: I'm trying to get a sense of how much energy a $27$ horsepower engine outputs. $$27 \mathrm{hp} = 20133\mathrm{W}$$ Potential energy can be calculated as $$E = mgh$$ where $g \sim9.8\mathrm{ms}^{-2}$ on earth. $$\frac{20133\mathrm{W}}{9.8\mathrm{ms}^{-2}} \sim 3000\mathrm{kgs}^{-1}$$ So for example if you use a $27\mathrm{hp}$ engine to pump hydraulic fluid. Would it output as much fluid as a $3000 \mathrm{kg}$ stone block pushing down a piston at $1\mathrm{ms}^{-1}$?


title: Using $A = {\lambda}N$ to find when a the amount of a radioactive source becomes constant

text: The question and mark scheme I will write in bold and my own thoughts in normal sized text. I'm told that: **When a $\bf{_{92}^{235}U} $ nucleus is exposed to free neutrons it can absorb a neutron. The resulting nucleus decays, first to $\bf{_{93}^{239}Np}$ and then to $\bf{_{94}^{239}Pu}$.** I'm then asked a few questions this, one of which I don't understand the answer to. They say that: **The number of $\bf{_{93}^{239}Np}$ nuclei present eventually becomes constant, calculate this constant number of $\bf{_{93}^{239}Np}$ nuclei, given that the half life of $\bf{_{93}^{239}Np} = 2.04 \times 10^{5} s$ and that the number of $\bf{_{93}^{239}Np}$ nuclei produced is at a constant rate of $ \bf{1.80 \times 10^{7} s^{-1}} $** The mark scheme uses the equation: $ \bf{A = \lambda N}$ and rearranges this to say that: $\bf{N=\dfrac{A}{\lambda}}$ **and we also know that** $\bf{\lambda = \dfrac{0.693}{t_\frac{1}{2}}}$ $\bf{\therefore \lambda \approx 3.397 \times 10^{-6}}$ I understand all of this and I understand the maths of the next step, I just don't understand why it gives you they value of the number of nuclei of $_{93}^{239}Np$ when the rate of decay of $_{93}^{239}Np$ equals the rate of formation of $_{93}^{239}Np$. **Substituting this value of lambda into our equation for $\bf{N}$ gives us:** $\bf{N = \dfrac{1.8 \times 10^{7}}{3.397 \times 10^{-6}} \approx 5.3 \times 10^{12}}$ **$\bf{{\therefore}}$ the number of $\bf{_{93}^{239}Np}$ nuclei when this number becomes constant is $5.3 \times 10^{12}$**


title: Radar as range detector between cars-static sender vs dynamic sender

text: Say I have a radar transmitter on a car,used to detect its distance from a car in front of it by measuring the doppler shift on the reflected wave. i.e Car 1 is the source, its wave gets reflected by Car 2 in front of it. My book splits the problem of calculating the Doppler shift by computing the frequencies w.r.t relative velocity. 1.When the source emits(Car1),take it as stationary by considering the receiver to move at relative velocity. 2.When the receiver reflects(Car2),take it as stationary by considering the source to move at relative velocity. The book goes on to give the following statements: 1.If the sender moves relative to a static receiver,the wavelength changes.The received frequency now will be: $$f_{receiver}=\frac{c_{sent}}{\lambda_{receiver}}=\frac{{c_{sent}}}{\lambda-\frac{v_{rel}}{f}}$$ 2.If the receiver moves relative to a static sender,the propagation velocity changes* _(but not the wavelength??)_ *.The received frequency now will be: $$f_{receiver}=\frac{c_{receiver}}{\lambda}=\frac{c_{sent}+v_{relative}}{\lambda}$$ Q* **Shouldn't the wavelength also change when the waves get reflected back by the moving receiver?*** The above equation no.2 says it doesn't. I can't attach images because I just joined.


title: Problem on apparent height of postage stamp underneath paperweight

text: > A postage stamp is placed on a surface and a glass cube of refractive index > $1.5$ is placed over it. When observed through the cube, the stamp appears > at a height of $1.5cm$ from the bottom of the table. Another glass cube made > of different material and same thickness is then placed over the first glass > cube. This time when observed, the stamp appears at a height of $4cm$ from > the bottom of the table. What is the refractive index of the second cube? I started off by trying to figure out the height of the first cube with the following figure: ![enter image description here](http://i.stack.imgur.com/btvat.png) $EF$ is the postage stamp. I sampled two rays from the centre of the stamp at an angle of $60$ with the normal (ignore the $30$ in the figure). Clearly, $\angle NIG = 30$. Using Snell's Law, $n_{cube}\sin\angle NIG = n_{air}\sin\alpha$ $\implies \frac{3\sin30}{2} = \sin\alpha \implies \alpha = \sin^{-1}\left(\frac{3\sin30}{2}\right)$ Producing the emergent rays backward they intersect at a point $M$. It can be seen that $\tan\angle NMI = \frac{IN}{NM} \space (1)$ Let height of cube equal $h$. $MG = 1.5cm \implies NM = h - 1.5$ $\angle NMI = \alpha, \tan\angle NGI = \frac{IN}{d} \implies IN = d\tan60$ Putting all these together into $(1)$ we get, $\tan\alpha = \frac{d\tan60}{d - 1.5} \implies d - 1.5 = \frac{d\tan 60}{\tan\alpha} \implies d - 1.5 = \frac{d\tan60}{\tan\left(\sin^{-1}\left(\frac{3\sin30}{2}\right)\right)}$ But, $\frac{\tan60}{\tan\left(\sin^{-1}\left(\frac{3\sin30}{2}\right)\right)} = 1.52752523165$, which would mean that $d - 1.5 = 1.5d$, which makes no sense. I'll include the image with two blocks too, in case it aids in answering: ![enter image description here](http://i.stack.imgur.com/nGTUh.png)


title: Is Joule heating only between charged particles?

text: The Wikipedia page for Joule heating explains _"It is now known that **Joule heating is caused by** interactions between the moving particles that form the current (usually, but not always, electrons) and the atomic ions that make up the body of the conductor. **Charged particles** in an electric circuit are accelerated by an electric field but give up some of their kinetic energy **each time they collide with an ion**. The increase in the kinetic or vibrational energy of the ions manifests itself as heat and a rise in the temperature of the conductor. Hence energy is transferred from the electrical power supply to the conductor and any materials with which it is in thermal contact."_ However, I wonder why they only talk about ions. Of course, electrons can easier interact with particles, which also have a long range field, but what about the non-charged particles which are also all around? > Is Joule heating only between electrons and ions?


title: Can D and H form an 'in materials' version of the electromagnetic tensor?

text: In analogy to the electromagnetic tensor, with the components defined as the electric field $E$ and magnetic field $B$ as such: $F^{ab} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\\ E_x/c & 0 & -B_z & B_y \\\ E_y/c & B_z & 0 & -B_x \\\ E_z/c & -B_y & B_x & 0 \end{bmatrix}$ Can one form a relativistic tensor from the electric displacement field $D$ and magnetizing field $H$ like so: $\bar{F}^{ab} = \begin{bmatrix} 0 & -D_x/c & -D_y/c & -D_z/c \\\ D_x/c & 0 & -H_z & H_y \\\ D_y/c & H_z & 0 & -H_x \\\ D_z/c & -H_y & H_x & 0 \end{bmatrix}$ and thus obtain an "electromagnetic tensor" that can be easily used to handle the effects of fields in materials? Since $D = \epsilon_0 E + P$ and $H = \frac{1}{\mu_0}B - M$, this is equivalent to asking if $P$ and $M$ can be used to form a relativistic tensor.


title: Quaternions and 4-vectors

text: I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity: $(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$ Is it useful? Is it used? Does it bring anything? Or is it just funny?


title: Computer simulation of Schrödinger equation

text: I am looking for a computer program which simulates the Schrödinger equation (say for a single particle) in two dimensions and for potentials and initial states specified by the user. Typical situations I have in mind are the double slit experiment, harmonic oscillator, Aharonov–Bohm, tunneling effect etc. Unfortunately I was only able to find the one dimensional case on the web. So are there some 2d programs available? PS: I don't think my question is a duplicate of this one. I only ask about a single particle, not about thousands. On the other hand I want freedom in my choice of potential. Also I want a numeric solution of a PDE, while the other question is about (as far as I understand, which is not ver far) a Monte Carlo simulation. Anyway these are completely different requirements.


title: Moment calculation

text: Consider image below. The weight of the fire-fighter is 840 N. **What is the torque of the fire-fighter's weight about P and what is the value of the force C which cancels out the torque?** ![enter image description here](http://i.stack.imgur.com/xs1QM.png)


title: Calculating the derivative of the average number of particles by the chemical potential

text: This should be a trivial calculation but somehow I have managed to get myself confused about this. The grand partition function is: $\mathcal Z = \sum_{N=1}^\infty \sum_{r(N)} {\text e}^{-\beta E_r +\beta N \mu}$ And the average number of particles: $\left \langle N \right \rangle = \frac{1}{\mathcal Z} \sum_{N} \sum_{r(N)} N{\text e}^{-\beta E_r + \beta N \mu } = \frac{1}{\beta} \frac{\partial}{\partial \mu} \ln \mathcal Z$ How can I take the derivative of that by the chemical potential? I thought about taking the inverse of the derivative of the chemical potential by the number of particles by I suspect that I'll be in the same position.


title: black hole escape speed

text: $$S_e = \sqrt{\frac{2GM}{R} } > c $$ Once an object hits the event horizon it should be pulled at the speed of light. Or because of the mass of the object, while light has zero mass, it may have to get closer than just the speed of light. The equation above shows how much speed is needed to escape the bh's pull beyond the event horizon but, my question is, how do I figure it for an object with the mass of 2,030 tons (1,841,585.02 kg). Also the bh is hypothetical so it can be any size needed to pull the mass. Honestly, as I'm completely new to physics, don't know where to start in the equation. Now again I apologize if any of my question comes off as unintelligent or uninformed but I'm just beginning to learn physics so plz take it easy on my guys and gals. Ok, sorry, the bh has a mass equal to the object...we're just going to say it's 1.8*10^6 so here's my dilemma... 2GM/R √((2×(6.67×10^(−11))×(1.8×10^(6)))÷(14×10^(−4))) =0.4141428325 What's wrong with this? -edit for Kyle- With the Rs I got smthg like 1 nanometer which I don't think is right but it could be Rs=2GM/c^2 where M is 1.8*10^6 n c is 300,000,000 meters So... (2×(6.67×10^(−11))×(1.8×10^(6)))÷((3×10^(8))^(2)) =2.66800000E−21 meters=1.12227082 nanometers So a black hole with the mass of a space shuttle would only be a nanometer? Just doesn't seem right.


title: Bragg diffraction and lattice planes

text: Crystalline substances show, for certain sharply defined wavelength and incident directions, very sharp peaks of scattered X-ray radiation. From the illustration below we see that we get constructive interference when the path-length difference is a multiple of the wavelength $\lambda$. ![Bragg diffraction](http://i.stack.imgur.com/mxwmj.png) In real crystalline materials we have a large amount of closely packed lattice planes. This large amount accounts for the sharp peaks for certain $\theta$. I do not understand how this follows from the Bragg reflection formula $$ n\lambda = 2d \sin \theta , $$ since $d$ is not constant anymore. I understand the model for two lattice planes as in the illustration. Is it true that $d$ can only take on values of the seperation of lattice planes, so $d$ is defined to be the seperation of points in the reciprocal lattice, or in others words, is $d$ constrained to be the absolute values of some reciprocal lattice vector? How does the Bragg condition account for very sharp peaks when we let $d$ run through all such absolute values?


title: White hole and Schwarzschild solution

text: What is the relation between white hole and the Schwarzschild solution commonly found in textbooks of physics and interpreted usually as black hole?


title: Concrete example of a two-dimensional harmonic oscillator

text: I am a student of mathematics and some time ago I showed in general that for a two-dimensional harmonic oscillator one can apply the recurrence theorem. So far so good.. now I would like to have a concrete example of a two-dimensional harmonic oscillator. Can you giuve me one?


title: How to model tea diffusion/osmosis?

text: How should I model the tea concentration as a function of time after a tea bag has been submerged? Is there a simple way of measuring the tea concentration?


title: Weighted average whenever a variance is 0?

text: So I have data for the number of times a certain event happens. Each column is a different event and each row is one trial. A sample of data would look like \begin{bmatrix} 1& 4& 7& 6& 5\\\ 1& 3& 9& 4& 5\\\ 1& 3& 7& 8& 8\\\ \end{bmatrix} I then collapse this data down into one row, and each number is just the average of the column and has its own variance. I want to take a weighted average of this row and obtain the associated variance as well, but in this example the first number has a variance of 0, and thus the weight would be 1/0. What does one do in a situation like this?


title: Why are all-Sky images drawn as a filled ellipse?

text: There is some convention? how is this 3d to 2d mapping done? here an example ![Fermi_Data](http://i.stack.imgur.com/rHRCq.jpg)


title: Gamma Ray Bubble at the center of our galaxy seen by Fermi Telescope

text: How could we measure high energy photons, whithout measuring them ? I can't understand how we can "see" those Gamma Ray Bubbles if they are not reaching here In this graph from Nasa you can see those "bubbles" are not reaching solar system: ![Art about bubles](http://i.stack.imgur.com/PEM2t.jpg) Then how could be measure that Gamma Ray without the Gamma Rays Thanks for any answer!


title: Question about the Killing form in Georgi

text: I'm trying to understand the Killing form described on page 49 in the book by Howard Georgi. He starts by saying that one defines the inner product between two generators $T_a$ and $T_b$ in the adjoint representation as follows: \begin{equation} \mathrm{Tr} (T_a T_b) \end{equation} and then subsequently he says that this is a real symmetric matrix. I'm not sure why this is the case, because the trace would just be a number? First, I thought this might just be a small mistake. However, he derives that a linear transformation on the generators $X_a$ (in an arbitrary representation): \begin{equation} X_a \rightarrow X_a' = L_{ab}X_b \end{equation} results in the following transformation: \begin{equation} \mathrm{Tr} (T_a T_b) \rightarrow \mathrm{Tr} (T_a' T_b') = L_{ac}L_{bd} \mathrm{Tr} (T_c T_d) \end{equation} And then he states _we can diagonalize the trace by choosing an appropriate $L$_ such that we can write (after dropping the primes): \begin{equation} \mathrm{Tr} (T_a T_b) = k^a \delta_{a b} \end{equation} I really don't understand where this equation comes from. I've never heard of diagonalizing the trace (because this is a number, not a matrix) and I couldn't find anything useful on Google. Any help with my problem would be much appreciated. Best regards,


title: Intuitive understanding of the definition of entropy

text: In Wikipedia, the definition of entropy goes like this: $ d S = \dfrac{\delta q_{\rm }}{T}$. The literal interpretation of this equation is that some amount of heat transferred into a system, if the temperature of the system is high, the entropy is low. Wikipedia also said that the entropy stands for the "waste energy" which means the amount of thermal energy that cannot be used to do work. So entropy is a function of the state of a system. My question is, why does the equation above mean "waste energy"? There are a few similar questions I found here (like The definition of entropy), but I don't think they address this "waste energy" concept.


title: Is ocean color affected by Rayleigh Scattering?

text: We know that sky color is due primarily to Rayleigh scattering. Is Rayleigh scattering the dominant effect for ocean color too?


title: Trouble understanding phase matching equation

text: consider a 2nd order non linear optical material, i.e. a material in which it holds that $ P = \epsilon_0 E + 2dE^2$. In the Born approximation, the non linear contribution to the polarization density $P$ can be approximated simply by calculating $2dE^2$ where $E$ is electric field of the incident light. Looking at only the time dependent part of an incident wave that is a superposition of two plane waves with frequencies $\omega_1$ and $\omega_2$, gives a nonlinear contribution at frequencies of $\omega_1 \pm \omega_2$, $2\omega_1$, $2\omega_2$ and a time constant electric field. For the output of the material to really radiate at these frequencies the initial plane waves also need to interfere constructively inside the nonlinear medium, giving the phase matching condition: $\omega_3 = \omega_1 + \omega_2$ and $\vec{k_1} + \vec{k_2} = \vec{k_3}$ My question now is how I should interpret the subscript 3. Are $\omega_3$ and $\vec{k_3}$ the wavevector and frequency of all the single non-linear oscillators inside the non-linear medium? Or what do these two quantities belong to? Please help me clear up my confusion and please correct any mistakes or wrong things I said in this post. Cheers and thanks in advance!


title: How to understand the definition of vector and tensor?

text: Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having difficulty understanding this kind of definition. To me, it goes like this:   1. First we have a collection of bases or coordinate systems (do these represent reference frames?) and the transformations between them.   2. A vector/tensor is an assignment of an array of numbers to each basis, and these arrays are related to each other by coordinate transformations. I am wondering how coordinate systems and transformations are specified.   1. Do we need all possible coordinate systems and transformations to define a vector space or just a few of them?   2. How to define the concept of basis and coordinate transformation without completing the very notion of vector?


title: AM Receiver and sideband frequencies

text: How sorry for "dummy" question. I have difficulty to understand how the AM receiver. Being a child I was taught that AM receiver resonates only with the radio frequencies it is tuned to, and "passes" only this frequencies to "other" circuits. Recently I have read in articles that somehow the AM frequencies have the component of carrier frequency and component of side band frequencies. I even can more or less understand the formula derivation of it, which is quite simple. What I have difficulty to understand is: If AM receiver tuned to receive only one frequency, e.g the carrier frequencies, how all this side band frequencies are not getting filtered out by the receiver? Please be patient all you radio gurus.


title: Time variation of the distance between two falling objects

text: I have solved this question below from my textbook. My answer differs from the one given in the book. I have tried my best to see where I am wrong. I need your help to find the truth of this question. **Q.** Two stones are thrown up simultaneously from the edge of a $200\,\mathrm{m}$ high cliff with initial speeds of $15\,\mathrm{m.s^{-1}}$ and $30\,\mathrm{m.s^{-1}}$. The graph representing the time variation of the relative position of the second stone with respect to the first stone would be linear for 8 seconds. Is it true or false? I think it is false and the graph would be linear for $1.5$ seconds only when the first stone reaches its peak but the second stone continues upward journey. The answer in the text book is $8\,\mathrm{s}$.


title: Why does lambertian surface reflection result in uniform radiance measured from every direction?

text: I'm reading through the light measurement handbook and page 28 is confusing me. I knew previously that lambertian diffuse reflections distribute the reflection of incoming intensity such that the surface looks equally bright from every direction from which it is measured (only dependent on incoming light direction and surface normal). Specifically, the measured radiance stays constant. However, the page shows that the actual reflected energy (the text says energy, I'm assuming it's talking either about radiant flux or intensity) isn't actually uniformly distributed, but forms essentially a cosine lobe centered about the surface normal: ![figure from page 28](http://i.stack.imgur.com/AaJ0u.png) I'm assuming that the "uniform radiance" results somehow from the measured area in relation to Lambert's cosine law cancelling out this cosine distribution to somehow result in a uniform distribution. How can this be shown mathematically? Also, how does this influence the light received by surfaces receiving this reflected energy? Are they lit such that every surface placed at equal distances to the surface receives the same amount of light, regardless of its orientation w.r.t. the surface normal, or do they essentially receive this cosine distribution?


title: Is the Lagrangian "math" or "science"?

text: I've seen in class that we can get from Lagrangian to derive equations of motion (I know its used elsewhere in physics, but I haven't seen it yet). It's not clear to me whether the Lagrangian itself follows from the equations of motion, or whether it represents a fundamentally different approach - whether it's a different "model", or whether it's largely a mathematical observation (it may illuminate/integrate what came before, but for all intents and purposes it is equivalent). I'm just curious about how these relationships are generally understood. I ask because it's not at all obvious to me that the principle of least action should be true. I'd also be curious whether the answer to this question would be the same across different fields of physics, though if the answer's "no" much detail would probably go over my head.


title: Tried to do the double slit experiment, failed. Why?

text: I tried, and failed, to do the double-slit experiment. I thought I had a good experiment setup, but obviously I was wrong. Can anyone offer some insight in what I might have done wrong? Most of this was done using some vague memory of experiments way back in High School, so there might be a better approach. I did search through past questions. This question was the closest to what I am shooting for, but there are significant differences and did not enlighten my particular situation. **Setup:** I used a variety of laser pointers as the light source, both red and green. One laser pointer was home built using a red laser diode (200-250 mW) from a DVD burner going through an adjustable lens. The slits were made by depositing soot from a candle onto a microscope slide, then "cutting" slits in the soot using razor blades. Two blades were held together and the slits "cut" at the same time with a single swipe of the razor blades. Initially I used old fashioned double-edged razor blades but the blades were too thin and only a single slit was made in the soot. I switched to X-Acto blades, which made two slits approximately 0.5 to 0.8 mm apart. I didn't measure this, I just eyeballed it. I did both a single slit and a double slit. I varied the distance from the laser to the slit, from inches to 20 feet. The distance from the slide to the wall was varied, inches to 8 feet. **Results:** I never clearly saw an interference pattern. I saw things that were probably some kind of interference pattern, but there was no consistency. One experiment showed a pattern of lines that was about 2 cm apart, while just changing the "quality" of the slit changed the pattern to about 0.25 cm apart (not changing the laser/slit/wall distances)-- this just seemed wrong to me and implied some fundamental experiment setup problem. I also expected a difference between the single and double slit, but that was not the case. The variation of patterns seen for a single slit was similar to what I saw with the double slit. When the laser-slit distance was 20 feet, and the slit-wall distance was inches then I could clearly see an impression of the slits on the wall with no hint of an interference pattern. **Other experiment notes:** The soot was surprisingly difficult to put slit into. If the soot was made thick enough to block all light then the edges of the slit were ragged and the width of each slit was not consistent. If the soot was thinner then the slit was smoother and consistent, but there was some light leakage in the areas that should have been solid black. **Questions:** Is the "soot on a microscope slide" a valid method for making the slits? Is the laser pointer with adjustable lens an appropriate light source? Should I expect to see a diffraction pattern with only the double slit, and not with a single slit? Did you see any other improvements I could make to this experiment?


title: Is the G2 Lie algebra useful for anything?

text: Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. I know only of one false alarm in the 1960s or 1970s before SU(3) quark theory was understood, some physicists tried to fit mesons into a G2 representation.


title: Kaluza-Klein Christoffel Symbols

text: I have a question regarding the connection coefficients as they pertain to the following paper: http://www.weylmann.com/kaluza.pdf . When I try to calculate the 4D Christoffel symbols from the 4D part of the 5D metric $\tilde{g}_{\mu \nu}=g_{\mu \nu}+kA_\mu A_\nu$ I get an extra term that shouldn't be there. I get: $$\tilde{\Gamma}^\lambda_{\mu \nu }=\frac{1}{2}\tilde{g}^{\lambda \sigma}(\partial_\mu \tilde{g}_{\nu \sigma}+\partial_\nu \tilde{g}_{\mu \sigma}-\partial_\sigma \tilde{g}_{\mu \nu})$$ $$=\Gamma^\lambda_{\mu \nu }+\frac{k}{2} g^{\lambda \sigma}(\partial_\mu (A_\nu A_\sigma)+\partial_\nu (A_\mu A_\sigma)-\partial_\sigma (A_\mu A_\nu))$$ $$=\Gamma^\lambda_{\mu \nu }+\frac{k}{2} g^{\lambda \sigma}(A_\mu (\partial_\nu A_\sigma -\partial_\sigma A_\nu)+A_\nu (\partial_\mu A_\sigma -\partial_\sigma A_\mu)+A_\sigma (\partial_\mu A_\nu +\partial_\nu A_\mu ))$$ $$=\Gamma^\lambda_{\mu \nu }+\frac{k}{2} g^{\lambda \sigma}(A_\mu F_{\nu \sigma }+A_\nu F_{\mu \sigma}+A_\sigma (\partial_\mu A_\nu +\partial_\nu A_\mu ))$$ According to the paper the $A_\sigma$ term shouldn't be there, but I can't figure out how to make it go away. Anybody have any ideas?


title: Explanation of the experiment with Mach–Zehnder interferometer (self-interference of a single photon)

text: Here is the description of experiment which uses Mach–Zehnder interferometer http://en.wikipedia.org/wiki/Mach-Zehnder or http://departments.colgate.edu/physics/research/Photon/root/ajpbs02.pdf (see section "C. The experiments: The photon interferes with itself") However I don't think I can fully understand the nature of what's happening in this experiment Could possibly someone give an explanation preferably in simple terms or point to another source?


title: Newton 2nd Law: Does vertical force (mass) affect the horizontal acceleration?

text: I learnt before that if 2 forces are perpendicular to each other, they should not affect each other. However in a recent experiment setup (asked in another question): ![](http://i.stack.imgur.com/xW1il.png) I believe the theoratical equation by newton 2nd law is $$\begin{aligned} F_{horizontal} &= F_{vertical} \\\ m_{cart}a_{cart} &= mg \\\ a_{cart} &= \frac{mg}{m_{cart}} \\\ \end{aligned}$$ Am I right so far? If so, this seem to imply that $m_{cart}$ (vertical force) is somehow affecting acceleration (horizontal)? Why is that?


title: $N=4$ supersymmetric yang-mills theory and S-duality

text: [Question suggested by @Lawrence B. Crowell's answer to the question on T-duality] There are three parts to my question: A. What is the action for $N=4$ SUSY Yang-Mills and what is the physics of the various terms in the action? B. Give a simple explanation for the origin of Montonen-Olive duality in this theory. This duality maps the physics at gauge coupling $g$ to the physics at $1/g$, i.e. this is a duality between the strong and weak coupling regimes of the theory. C. **Auxiliary Question** : Explain how one can decompose a SUSY gauge field into its bosonic and fermionic parts. [ **Background** : I am familiar with non-abelian gauge theory and have had a class on SUSY some years back. So I pick up on the jargon but am missing a fuller picture.]


title: Green functions in Quantum Mechanics

text: How the Green's functions and the Quantum Mechanics are related? Do they can be used to solve the Schrödinger equation of an particle subjet to some potential that is not a Dirac's delta? And the proprieties of some Green's functions that are symmetrical, i.e. $ G(x|\xi) = G(\xi|x)^{\ast} $, has some relation with the propriety of the inner product $ \langle \alpha \vert \beta \rangle = \langle \beta \vert \alpha \rangle^{\ast} $?


title: How should 500ml water bottles be stored in the fridge so they will be coldest?

text: Should they be taken out of the wrapping and stored as single units, or should I leave them in the package? EDIT to add I mean, the package is 24 bottles bundled together. Should I just leave them in the crate, or disperse them around the fridge?


title: Truck moving uphill, object suspended from ceiling, determine acceleration

text: > A truck is moving with constant acceleration $a$ up a hill that makes and > angle $\phi$ with the horizontal. An object of mass $m$ is suspended from > the ceiling of the truck by a light cord. If the pendulum makes a constant > angle $\theta$ with the perpendituclar to the ceiling, what is $a$? By vector addition, the tension in the rope becomes $$mg+ma$$ Right? It will make the angle $\theta$ with the perpendicular of the ceiling. The triangle consisting of these vectors will have one side, $m$ times $a$, with angle ($\theta-\phi$) on the opposite side. Therefore I tried using the law of cosines, but it did not work out. Is there a better way?


title: Coulomb Exhaustion

text: In thinking about a perturbation model for Coulomb's law, one could imagine photons spewing off of a charge in all directions. The chance of interaction with a near by charge being proportional to the photon density at a given radius (a la, $\frac{1}{4\pi r^2}$). But, one could also imagine there are a finite number of photons being spewed. Imagine an experiment with a central charge of $q_1$, around the charge exists a perfectly spherical conducting shell a distance $r$ from the center. A sensor is created that can detect the outward force on the shell. (Perhaps the shell is a set of free moving conducting scales connected with wires where the outward force on each scale can be measured) Gradually a charge $q_2$ is continually added to the shell. According to Coulomb's law the outward force should be proportional to the charge of the shell. However, if there are a finite number of photons being spewed from the central charge, after the point where all of its photons are interacting with the charge of the shell, adding additional charge to the shell would not further increase the outward force. Have any such experiments been performed or such results observed? Is there any theoretical reason why such a phenomena would not exist? How might we guess what the ratio of $q_2/q_1$ would be before such full exhaustion might occur?


title: Weight held by jet stream of water

text: > _Consider a bucket of weight $W$, held in air by a jet stream of water from > the ground (moving vertically upwards) at height $h$. The water is fired > from the ground at the rate of $\mu$ (units kg/s) with an initial velocity > of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve > the same._ (To observers, this is a problem from Kleppner & Kollenkow: _An Introduction to Mechanics_ ). **What I did:** Consider a differential mass element $dm$ near the bucket before hitting it. Its velocity is $v_f = \sqrt{v_0^2 - 2gh}$, so its momentum is $v_f dm$. Suppose after hitting the bucket it had a velocity of $v'$, and the time of contact was $\Delta t$. Since we must exert a force of $W$ upwards on the bucket, the rate-of-change of momentum of $dm$ is $W$. Clearly $dm(v' + v_f) = W\Delta t$, which in the limit $\Delta t \to 0$, becomes $W = \mu (v' + v_f)$. Thus $2gh = v_0^2 - \left(\frac{W}{\mu} - v'\right)^2 $. This raises a question. Setting $v' = \frac{W}{\mu}$, we can attain $h = v_0^2 / 2g$, but at that height, the water would have no velocity, so no momentum change. What are my errors, and how to attack such problems (with continuous masses)? ( Please excuse me for any silly mistakes. I'm only a beginner :( ). A detailed solution which I can study would be extremely helpful.


title: How uncertainties affect values

text: If I calculate the equivalent resistance of a circuit, for example: $$1/R_{eq} = 1/R_1 + 1/R_2 + 1/R_3 = 1/1472 + 1/3260 + 1/5580 \Rightarrow R_{eq} = 858.22\,\Omega$$ And then calculate its uncertainty: $$\Delta R_{eq} = |\delta 1/\delta R_1| \times \Delta R_1 + |\delta 1/\delta R_2| \times \Delta R_2 + |\delta 1/\delta R_3| \times \Delta R_3 = 0.001\,\Omega$$ Will the uncertainty affect the value $(858.22)$, given that it only has 2 decimal numbers?


title: Does light or observation collapse wave functions?

text: I hear that observation is what causes the wave function collapse, but that doesn't make sense considering that an eye or camera is just a physical system with no particularly special properties. In explanations it say that light causes the collapse if that's the case why do you need an observer? If you shined a light while not in the room of the double silt experiment would it still act as a particle?


title: Why doesn't a car slide down a banked road when there is no friction?

text: In case of banked roads without friction, there is an additional $mg \sin(\theta)$ which is unbalanced. Why isn't this taken into account because it is responsible for making the vehicle slide down the inclined plane?


title: Absorption from a classical to quantum

text: Today, I learned that using rubidium atoms at very low temperature in a Magneto-optical trap, one can experimentally show that the Lorentz classical derivation of absorption using dipole is valid. The atoms were laser cooled to show this. Why is this valid only at low temperature and not at room temperature?


title: Can we make a camera that fits in one of the bands of the double slit experiment?

text: So, in Young's double slit experiment, we see where the particle lands on the back wall. What if we were standing at the back wall, looking toward the slits? Could we gain more information not by simply observing the light bouncing at the back wall, but looking at it our selves? Of course, for electrons or lasers, that ain't a great idea. And the slits are too small for our eyes to really make out, I'd gather. Frogs can possibly detect a single photon. What if we trained a frog to lick its left eye if it detects a photon from the left slit, and its right eye if it detects a photon from the right slit. Or better yet, build a tiny robot with a camera. Would that robot see both slits illuminate as it detects the photon? Seems testable in theory.


title: Tool for Born-level Cross Section Generation with Color Decomposition

text: I'm looking to find a tool, like CalcHEP, that calculates basic cross sections and allows for symbolic output which is decomposed by color. I understand that MADGRAPH 5 will decompose amplitudes by color in the calculation, but I'm not sure if I can extract those individual amplitudes in a reasonably easy way. Being unfamiliar with MADGRAPH, I'm also not sure how that output would look...or if it's symbolic in nature. I am also beginning to look at Comix* but I have not made much progress through the manual yet. I would like to be able to specify my basis, but I suppose that's a luxury. Any recommendations?


title: Why is normal ordering a valid operation?

text: Why is normal ordering even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that? Is its definition motivated by the relation between the normal ordering and the time-ordered product, which is basically the content of Wick's theorem?


title: What is "localisation" of instantons?

text: I often encountered the term "localization" in the context of instantons, as for example in the work of Nekrasov on extensions of Seiberg-Witten theory to N=1 gauge theories. Could someone give a "intuitive" explanation of the concept of localization as well as a "simple" concrete realisation of it?


title: Number of dimensions in string theory and possible link with number theory

text: This question has led me to ask somewhat a more specific question. I have read somewhere about a coincidence. Numbers of the form $8k + 2$ appears to be relevant for string theory. For k = 0 one gets 2 dimensional string world sheet, For k = 1 one gets 10 spacetime dimensions and for k = 3 one gets 26 dimensions of bosonic string theory. For k = 2 we get 18. I don't know whether it has any relevance or not in ST. Also the number 24, which can be thought of as number of dimensions perpendicular to 2 dimensional string world sheet in bosonic ST, is the largest number for which the sum of squares up to 24 is itself a square. $(1^2 + 2^2 + ..+24^2 = 70^2)$ My question is, is it a mere coincidence or something deeper than that?


title: Why does weak equivalence principle say gravity is equivalent to acceleration?

text: I am told that the weak equivalent principle, that $m_i=m_g$ (inertial and gravitational masses are equivalent) is equivalent to the statement that in a small system you can't tell whether you are in a uniform gravitational field, or in an equivalent accelerating frame. My question is about light: Special relativity doesn't say anything about how light falls in a gravitational field, so if you are in a rocket ship accelerating at $g$ why should it be true that when you shine a light in the rocket ship, you see the same thing happening as if you shined a light on earth? (to clarify, if you shot a bullet in the rocket ship, I understand why the trajectory of the bullet would look the same as if the bullet were shot on Earth, but this is because special relativity (or classical mechanics) tells us what the trajectory of a bullet looks like on Earth).


title: What does unfolding of attractor mean?

text: What does unfolding of attractor mean? Effect of time scales on the unfolding of neural attractors paper talks about Takens embedding theorum. It says that the embedding dimension should be large enough to unfold the attractor. What does the term unfolding of the attractor mean in non technical terminology? What is the significance, why to map the time series to higher dimension when we can work with single time series as higher dimension adds more complexity? An attractor consists of the coordinates of the variables, say Lorenz system. SO the attractor will have 3 coordinates. Then what does unfolding mean?


title: Is it really to solve problem below by using, in the main, Gauss law?

text: There is an infinite cylinder surface which uniformly charged along and has a surface charge density, which can be represented as $$ \sigma = \sigma_{0}cos(\varphi ), $$ where $\varphi$ - polar angle in a cylindrical coordinate system (it means that z-axis is matches with cylinder axis). There need to find z-component $|\mathbf E|$. I tried to learn use Gauss law, but tasks, where density isn't constant on the surface, are still hard for me. Can you help?


title: In the classic viscosity definition, why does doubling the plate gap cause the force to halve (intuitive)?

text: I am puzzled by an artifact of the definition of viscosity and need an intuitive picture to help explain it. I know $\tau_{yx}=-\mu{dv_x \over dy}$ but I am looking for an intuitive picture of the specific question below. Take the prototypical definition/model of viscosity: Imagine two very large plates separated by a constant small distance with a fluid sandwiched inside. Apply a force $F$ parallel to the bottom plate so as to start moving the bottom plate at a speed of $V$. Once steady-state is reached, $F$ and $V$ are constant. The gap between plates is $Y$ and the flow is presumed/constrained to be laminar. One finds that the velocity gradient with respect to distance along $Y$ is linear with velocity at the top plate zero and the velocity at the bottom plate $V$ (no-slip condition). In equation form: $${F \over A}=\mu{V \over Y}$$ where $\mu$ is the viscosity. So far, so good. Now, consider the force $F$. It is proportional to $A$ which makes sense as a bigger plate will require more force to move due to the viscous contact with the liquid. It is proportional to $\mu$ which really is the definition of viscosity (higher viscosity fluid requires more force). It is proportional to $V$ which make sense since moving the plate at a higher velocity will require more force. Finally it is inversely proportional to $Y$, the plate gap. I can't quite understand this one intuitively. Why should doubling the plate gap cause the force to be halved everything else being equal? Conceptually, I get that the velocity gradient is half what it was and that this is what causes the force to be half but my intuitive picture is lacking. I just can't imagine myself exerting a smaller force on a doubled gap system. Taken to the extreme, at very large $Y$, $F$ is zero. This must be why my text _Transport Phenomena_ by Bird, Stewart, and Lightfoot says the gap must be "small". But what really determines what "small" is?


title: How to make the projected image smaller by adding one ore more lenses in front of the built in projector lense?

text: I have a projector that creates a large image, even if the distance to the screen is short. The device is very small, approximateley 10x10 cm if you look from above. The height is only 3 cm. ![enter image description here](http://i.stack.imgur.com/stIbK.png) I could remove and replace the builtin "lense pack" to make the size of the projected image smaller, but I'd prefer to keep the device unchanged. What type(s) of lense(s) _in front of the unchanged device_ would be reqired to make the image dramatically smaller? I have measured the following with the original device. ![enter image description here](http://i.stack.imgur.com/jo59s.png) If the distance d from the focal point to the screen is $300\,cm$, the width $w_1$ of the image on the screen is $\approx200\,cm$. And if d is as far as $900\,cm$ then $w_1= 600\,cm$. What can I do to make the image smaller, so it's new width $w_2$ would be approx. $100\,cm$ at a distance $d= 900\,cm$? ![enter image description here](http://img803.imageshack.us/img803/4397/project02a.png) Lenses, mirrors or prisms would be OK, but I somehow guess 1 or 2 lenses would do the trick. What type of lenses would this be, how do I choose and calculate the type and the parameters of the lense? Can you teach me how to choose and understand the formula? The quality of the image and the resolution should still be good afterwards. Of course I can have a $1\,m$ image if I move the projector close enough, but that is not the solution I am looking for. I tried also to find this out by myself already. I visited two camera stores and talked with the owners. I also visited optician retail stores. But no one there could answer this question. I have some basic knowledge of physics and also tried to solve this by reading books about optics and geometrical optics, but without any success. I also tried with some lenses I could get my hands on but I could not solve it. The problem I also had to deal with was that as soon as I moved the lenses a little further away from the device, they were too small to catch the whole beam.


title: Quantization and natural boundary conditions

text: The Euler-Lagrange equations follow from minimizing the action. Usually this is done with fixed (e.g. vanishing) boundary conditions such that we do not have to worry about any boundary terms. However, it's also possible to obtain boundary conditions which follow from the action itself, the so-called natural boundary conditions. In terms of some Lagrangian $L$ they take on the form $$ \frac{\delta L}{\delta x'(t)}\Bigg|_{x=a} = 0 $$ or for fields $$ \frac{\delta \mathcal{L}}{\delta \partial_x\phi}\Bigg|_{x=a} = 0~. $$ These boundary conditions minimize the action for a system with variable end- points. But what happen with these boundary conditions when we quantize the theory? Specifically, do states in the Hilbert space satisfy the natural boundary conditions automatically or is this something that needs to be imposed "by hand"?


title: Why did they add a Centripetal Force in the Free-Body Diagram?

text: problem ![](http://img196.imageshack.us/img196/6610/problmus.jpg)! solution ![](http://img838.imageshack.us/img838/391/solnd.jpg)! Why do they have the centripetal force in there? I understand that the normal force is the centripetal force here, but why would they say "or $\frac{mv^2}{r}$"? I thought it was wrong to include this. Also, on centripetal force, how come there is a net force inwards to the center, but there is no REAL force counterbalancing? Why do we have a fictitious force?


title: Why Do Hurricane Balls Spin So Fast?

text: I was wondering if anyone could offer an explanation as to why the balls described in this video spin so fast. Here's the setup: Two metal balls are wielded together. When spun with air, they acquire a massive amount of rpm.


title: How many electrons are displaced when combing hair?

text: Feynman talking about electricity: http://www.youtube.com/watch?v=qhh32JYkQPk When brushing your hair, Feynman mentioned that a "few" electrons were transferred to the brush (or vice versa, can't remember off the top of my head how the convention goes). Clearly this will vary depending on how much you comb your hair, so my question is what is the approximate charge of the brush which would be required so that it would be noticeable (i.e. I could make pepper dance with the brush, or jiggle a piece of paper at close range). Sorry if the question is silly :P Cheers, Rofler


title: Plausible explanations for 3 local space dimensions

text: Every now and then I see accounts of models that claim to explain why we experience only 3 space dimensions (locally, i.e. within cosmic horizons and outside black hole horizons). One such of course is string theory; but I wondered if there might be others, and perhaps even a survey of this topic along the lines of those expository articles one often sees in the ArXiV.


title: Motion of a plank hinged about a point

text: There is a plank hinged about a point O. At the free end of the plank a small impulse is given. Will the plank perform pure rotation or both rotation and translation?


title: How come a photon acts like it has mass in a superconducting field?

text: I've heard the Higgs mechanism explained as analogous to the reason that a photon acts like it has mass in a superconducting field. However, that's not too helpful if I don't understand the latter. Why does this occur, and how?


title: Gauge redundancies and global symmetries

text: It is often said that local (gauge) transformation is only redundancy of description of spin one massless particles, to make the number degrees of freedom from three to two. It is often said that these are not really symmetries because it means that there are only apparently different points in configuration space that are physically identical, but these are the same. Noether charge under the gauge symmetry is gauge dependent, so does this mean that only global symmetries make sense as symmetries? But why are local ones more fundamental? Under which conditions local symmetry implies the global one? Is conservation of electric charge due to local or global U(1) symmetry? Which symmetries are preferred by gravity, local or global? Is it general rule that some transformation is called symmetry if action is left invariant (classically)? Are there symmetries that strictly ask for Lagrangian to be left invariant? Are non-abelian symmetries "more" symmetries or redundancies in comparing to U(1)? Which sentences I wrote here are wrong, or which questions are not valid?


title: What is the capacitance of a superconductor?

text: I heard somewhere that the capacitance of a superconductor is much higher than regular conductors, but I haven't heard or seen anything yet proving or disproving such. So whats the truth? And why?


title: Curve of a rod bent by force on both sides

text: Suppose we have a flexible rod (i.e. it can be bent without breaking apart) and we excert a force on both sides, like this: ![Rod](http://i.stack.imgur.com/b4i76.png) If the force $F$ is not exactly horizontal, the rod will be bent and form a curve (like for example the dashed line). Is there an equation that describes the curve? How would I go about to determine it? Let's assume that we had some initial non-parallel force that caused the deflection in the beginning, but afterward the force is horizontal. Let's also neglect gravity.


title: Measurement of event time by different observers

text: Follow up question to Doppler redshift in special relativity A source of light pulses moves with speed v directly away from an observer at rest in an inertial frame. How will the time of the emission of a light pulse as measured by the emitter be different to the time as measured by the observer (how will it be different than the time of the pulse reaching the observer)? edit: to elaborate slightly, let us say $ t_{E_1} $ is the time of emission as measured by the emitter, $ t_{E_2} $ is the time of emission as measured by the observer, and $ t_O $ is the time of observation of the pulse by the observer. How are these three times different, how would they be measured in practice?


title: Dimension agreement in canonical transformation

text: In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical transformation. But here I'm want to ask if the transformation is valid, because the dimension of $\sqrt{p}$ is not always same with that of $\sqrt{q}$. If these two dimensions are different, how can they be connected through a minus sign, and what is the dimension of the result?


title: Puzzled by $\mathrm p = -i\hbar \nabla = m \dot{\mathrm r}$?

text: I'm working on _Physics of Atoms and Molecules_ by Bransden and Joachain. And I've come across the following statement, which I don't understand (p.195, ch. 4.3, "The dipole approximation") > ... In terms of the momentum operator $\mathrm p = -i\hbar \nabla = m > \dot{\mathrm r}$, we can also write $$\langle \psi_b |\mathrm p| > \psi_a\rangle = m\langle \psi_b |\dot{\mathrm r}| \psi_a\rangle$$ ... Here I'm puzzled by the operator $\dot{\mathrm r}$. First of all, I don't see how it makes any sense? $\mathrm r$ is just given by $$\mathrm r = \begin{pmatrix} x \\\ y\\\ z \end{pmatrix}$$ in cartesian coordinates. So I don't see any time dependence here that would allow me to differentiate. Maybe their equality is to be taken in an average sense? For example is what they mean: $$\langle \psi_b |\mathrm p| \psi_a\rangle = m\frac{\mathrm d}{\mathrm dt}\langle \psi_b |{\mathrm r}| \psi_a\rangle$$ ? This would seem to make sense, since it would in particular give us $\langle p \rangle = m \frac{\mathrm d}{\mathrm dt} \langle r\rangle$, which is the classical expression for momentum. If my guess is correct: how can I derive this equality? Thanks for your help!


title: Supermassive Black Holes = expansion origin?

text: (Sorry, couldn't resist the title) An article on Physics Today brings news that there is a new record for largest black hole (not) seen in space, some 300 million light-years away. It hints at something in the last sentence: > Also, because of their unusually large mass, they may have evolved > differently from smaller black holes. Could a significant proportion of the mass of the universe be bound up in primordial black holes, which are still moving away from each other with their incredible momenta, warping spacetime as they go? I can't quite square this with the near-universal red-shift, however; if it was the action of black holes dragging us away from other galaxies, we would see some galaxies drawn towards these black holes with blue-shift instead. And yet... the peculiarities of spacetime around black holes could account for hyperinflation, couldn't they? If the universe began as one or many black holes, then there would need to be some sort of abnormal physics going on to separate them in the first place. The joys of ignorance. Thinking about CMB, there would be some sign in that that there were these black holes, unless they were so numerous that the distribution was nearly uniform. What do you think?


title: (air pressure and displacement) Isn't this image wrong?

text: ![enter image description here](http://i.stack.imgur.com/lefCS.png) Isn't this figure wrong? P(x,t) = -B(dy/dx) . If the derivative of air displacement has a maximum, then this is where the pressure is minimum, not maximum as this figure suggests. Could someone clarify this for me? Thanks. If anything this is the source for the image: http://hyperphysics.phy- astr.gsu.edu/hbase/waves/standw.html y(x,t) is the function for displacement. B is the bulk modulus P(x,t) is the function for pressure.


title: Why model space with real numbers?

text: Are there any good papers discussing why we use $\mathbb{R}^{3}$as a model for space? More specifically are there any that explain why we don't use other number systems such as extensions of the real line (hyperreals, surreals, etc.)?


title: Measurement of heat in electrons

text: Is there any method to measure the heat of an electron and if there is; then is it a constant or a variable.


title: A thought experiment to understand (literal) bootstrapping

text: The common explanation of why you can't lift yourself off the ground by pulling your feet up with your hands, or in more cliched terms "pull yourself up by your own bootstraps", is that if you consider your whole body as a system, the pull of your hands on your feet is just an internal force, so it can't pull your body off the ground. But what about this thought experiment: let's say you say cross-legged with only your feet on the ground. Then you place your hands under your feet, so that only your hands are on the ground. Then you rapidly slide your hands way from under your feet and bearing your hands up. What would happen? Well, if your hands were 1 cm thick, then, the moment you removed your hands from under your feet, your feet would be 1 cm off the ground. So wouldn't you have managed to successfully lift yourself off the ground, if only for a moment before you crash back down?


title: Calculate the current in mA which will be flowing through the two resistors

text: Three resistors of 3,4 and 5 ohms are joined in parallel in an electric circuit. If a current of 150mA flows through the resistor of 4 ohms,then calculate the values of the current in mA which will be flowing through the other two resistors. I could not comprehend the question completely.Thanks in advance.


title: Action for a point particle in a curved spacetime

text: Is this action for a point particle in a curved spacetime correct? $$\mathcal S =-Mc \int ds = -Mc \int_{\xi_0}^{\xi_1}\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi$$


title: How Scientist Came to Know the Shape of our own galaxy without going out of it?

text: As far as I know, scientists have been able to see a lot of differently shaped galaxies in our visible universe through modern age telescopes. But I was wondering how it was possible to know how our own galaxy looked like without going out of it to see it's shape?


title: Why is Near Field Communication (NFC) range limited to about 20cm?

text: Near Field Communication (NFC) operates at 13.56 MHz. Near Field is the region situated at a distance r << λ λ = c/f               c = 299 792 458 m/s in the vacuum.     f = 13.56 MHz = 13.56*10^6 Hz      λ = 299.8/13.56 = 22.11 m Of this distance, the Reactive Near field is commonly considered λ/2π. For NFC, this would be 3.51 m. So where do the 20 cm come from?


title: Pseudo force in rotational motion?

text: If a cylinder is in combined rotation and translation on a moving surface(say a plank with some acceleration), while solving for the acceleration of the centre of mass of the cylinder, do we consider a pseudo force due to the movement of the surface of the cylinder, I.e. The plank?


title: Double pendulum find first integral

text: Consider the following situation of a double pendulum in 2D. ![doppel pendulum](http://i.stack.imgur.com/xcU92.png) We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\\ \ddot{\theta}_2=-L_2\sin\theta_2. $$ Respectively as a system (*) $$ \dot{\theta}_1=w_1\\\ \dot{\theta}_2=w_2\\\ \dot{w}_1=-L_1\sin\theta_1+\frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1)\\\ \dot{w}_2=-L_2\sin\theta_2. $$ Now the task is to find a first integral of the system (*). As a hint it is said to find formula for the kinetic and potential energy of the system by working in cartesian coordinates. I am studying mathematics and not physics. So I do not know how to determine kinetic and potential energy. Can you help?


title: Getting a given wavelength radio signal given an antenna with real-world constraints

text: Supposing you are given a transmitting antenna of whatever type of metal is most commonly used these days, and supposing that you are applying an AC current with the intent of transmitting a 1 m wavelength radio signal, about how far is one electron likely to make it in one direction along the antenna conductor in one half cycle from a negative peak to a positive peak, factoring in the likelihood of the presence of antenna material atoms affecting the electron's path? Can individual charge carriers actually make it very far (on the order of meters) through metal? But if the answer is that individual charge carriers actually only make a microscopic or very small displacement in space, how could a 1 m electromagnetic wave be produced? I don't see how a large number of charge carriers each only making a very small displacement in space can add up to a 1 m wave. This site, especially Fig. 2 there, is where I'm getting my basic understanding of how alternating current gets an EM wave transmitted. From that is where I got the feeling that charge carriers would actually need to make a displacement in space on the same order as the intended EM wave.


title: Can we measure rates in real time?

text: I know what it means to say that my position is "X" at a particular moment in time. I can easily take a picture of my motion and observe my exact location at the instant the picture was taken. That is to say, my _instantaneous_ position can be measured. However, could I ever measure my velocity at an exact point in time? At best, it seems I can only estimate it based on the approximate derivative of my position or approximate integral of my acceleration over some measure of time. Each of these methods, however, requires some elapse of time to perform the calculation, and even then it's only approximate. Neither allows me to know what my precise velocity is at _an exact_ point in time, i.e. my _instantaneous_ velocity is not being reported. We obviously have devices like speedometers, tachometers, heat sensors, and dynos which ostensibly report quantities like velocity, heat flux, and power in real time. But really, how could they? Each of these quantities is itself the time derivative of another quantity and thus requires the passage of time to estimate (although velocity may be a bad example because we have things like pitot-static tubes). Am I correct in surmising that rate-based instruments are not reporting in real-time? And if this is true, could we ever construct a device which _did_ measure derivative quantities directly? What exactly would it measure?


title: Does the pressure increase under an upturned glass?

text: During my children's bath time we were playing with an upturned plastic jug filled with water and making some plastic divers float up inside the jug - see image. ![water in upturned glass](http://i.stack.imgur.com/eW4i5.png) If one of the divers swam from A to B to C, would the water pressure on them remain constant? Or would it increase at B because of the higher column of water above?