title: How do I compute the eigenfunctions of the Fourier Transform?

text: I read today (ref) that the Continuous Fourier Transform has four eigenvalues: +1, +i, -1, and -i. Associated with each eigenvalue is a space of eigenfunctions: functions which retain their form after undergoing the Fourier transform. Perhaps the best known example is the Gaussian: the Fourier transform of a Gaussian is again Gaussian. A more general example is the Hermite-Gauss functions (Gaussian multiplied by Hermite polynomial). These are also eigenfunctions of the Fourier transform, which is why TEMxx modes (with Hermite-Gauss transverse profiles) are stable modes of laser beam propagation. This leads me to wonder a number of things:   * How do I show that the Fourier transform has just these four eigenvalues?   * Given an arbitrary function, how do I find the projections onto the four eigenspaces?


title: Advantages of high-energy heavy-ion collisions over proton-proton collisions?

text: Some high-energy experiments (RHIC, LHC) use ion-ion collisions instead of proton-proton collisions. Although the total center-of-mass energy is indeed higher than p-p collisions, it might happen that the total energy _per nucleon_ is actually lower. What are the advantages of using ion-ion collisions (e.g. gold-gold or lead-lead) instead of proton-proton collisions, considering the same accelerator?


title: Positrons versus holes as positive charge carriers

text: From Wikipedia: [The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.] and: [Dirac's idea is completely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice — that cancels out the electric charge of the sea.] It always confused me to think of holes as positive charge carriers in semi- conductors as **not being real** : real electrons move from one lattice- position to another lattice-position, which effectively _looks like_ a positive hole in the lattice that is moving in the other direction, but in reality a real electron moves, the hole is kind of an "illusion". On the other hand the positrons are always introduced as **real** hard-core particles. The quotes from the Wikipedia article make me unsure: how should I look upon these phenomena? Edit: holes in a Dirac sea give rise to real pos. entities in one case and to unreal pos. entities in another - how can we distinguish, is it a matter of formalism?


title: Accidental, unplanned breakthroughs in physics

text: There is possibly some idioms or saying like this, ``If you try too hard for something, you will never get it. If you do not aim for something, it may fall on you accidentally, not as you originally planned.'' **In physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts.** Moreover, there are some cases that originally having attempts for one specific question or a goal, but solve another seemly unrelated problems, or reach seemly the opposite goals. There are these kinds of moments leading to breakthrough of physics or science. For example, (1) **Yang-Mills theory** original paper has attempt to explain a theory of nucleon, such as neutron and proton interactions with isospin symmetry, but it turns out that Yang-Mills theory as a non-Abelian gauge theory suitable to describe a more fundamental subject, the gauge fields coupled to quarks and leptons in the standard model (EM, and especially the weak interaction and the strong interaction.) (2) **$Z_2$ topological insulator in 2+1D** : the original paper of quantum spin hall effect from Kane and Mele is set for graphene. But it turns out the phenomena is not in graphene(C Kane jokes on this accident himself.), but the physics is profound and correct, and later realized in CdTe/HgTe/CdTe quantum wells with a 2D film HgTe sandwiched. **[Question]: Can any of the readers here list more?** Both in theories or in experiments. Making an inspiring list for eager-mind scientists at Phys.SE here. To give us some inspiration and high motivations to be subconsciously aware of those random accidental moments.


title: Displacement of a body in free fall

text: Suppose a ball is thrown up from the ground with some velocity. Let vertically up be the positive direction and vertically down be the negative direction and let the thrower be at the origin. What will be the sign of displacement after it has reached it's maximum height and is still descending? I think it will be negative since it's displacement will be in the negative direction, although it's position will always be positive. Am I correct?


title: Where does the idea gravity=curvature of spacetime really come from?

text: I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I know that the starting point for Einstein's considerations was the equivalence principle. Nevertheless I can't make the connection from there, to why gravity is the curvature of spacetime. Any help or reference to where this is explained would be much appreciated.


title: Relative strength of a dome

text: Is there a rough way to determine how the height of a dome affects the load that dome could support? For instance, assuming the bases of two domes are 24" in diameter, and one dome is 2" high while the other is 4" high. Is there a way to understand how much stronger the 2nd one is compared to the first one?


title: Definition of force, kinetic energy and momentum

text: **I've edited the post. Q1 and Q4 are the important ones but I didn't delete Q2 and Q3 since some older answers would not make sense anymore.** To begin with, the formula of the kinetic energy $T$ is $\frac{mv^2}{2}$. Furthermore momentum is conserved $\Sigma m_{i}\vec{v_{i}}=const.$ Then you have the definition that force is the change of momentum with respect to time $\vec{F}=\dfrac{d(m\vec{v})}{dt}$. I've read the chapters concerning mechanics of Physics for scientists and engineers by Giancoli and the Feynman Lectures. Giancoli introduces the arbitrarily work as $W=\int\vec{F} \cdot d \vec{s}$. From this definition of work he derives the kinetic energy to be $\frac{mv^2}{2}$. In contrast to that, in the Feynman lectures you never get a derivation of $\frac{mv^2}{2}$, but it is shown that $\dfrac{dT}{dt}=\vec{F}\cdot\vec{v} = \vec{F}\cdot\dfrac{d\vec{s}}{dt}$. Then it is shown that $dT=\vec{F}\cdot d \vec{s}$ and as a consequence $\Delta T = \vec{F} \cdot \vec{s}$ which is called work. http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S1 **Now I've got some questions:** Q1 _Is $F=\dfrac{d(mv)}{dt}$ **just an arbitrary definition** or is there something "more" behind the formula for force?_ Q2 _Is $W=\int\vec{F} \cdot d \vec{s}$ **just a definition** or is there something more behind? I mean, can you derive the formula for work not by taking the formula for the kinetic energy as given._ Q3 **How to derive the formula for the kinetic energy and work only form the conservation of momentum $\Sigma m_{i}\vec{v_{i}}=const.$?** Q4 **How are work and kinetic energy defined?** I have found both: A) Kinetic energy is doable work and work is $F=\dfrac{d(mv)}{dt}$. B) The formula for kinetic energy is: $\frac{mv^2}{2}$. Then, after some maths it follows that $\Delta T = \int \vec{F} \cdot d\vec{s}$. **It doesn't make sense to define 2 things in that way. This would be circular logic.** P.S.: I am not a native English Speaker so feel free to edit.


title: How much more energy would I spend if my bicycle was 2 kilos heavier?

text: I have a bicycle weighing 8 kilos. An 10 kilometer ride averaging 20 km/h requires `z` kWh's of energy. How much more energy would I spend if I added 2 kilos to my backpack? Or turning it around: If I used the same amount of energy, how much slower would I get? UPDATE: I'm asking out of curiosity, not out of some homework assignment. It's a long time since I've had those :) I have friends arguing that a 2 kg lighter bike (e.g. a racer instead of a cyclo cross) will give me serious advantages because of the decreased weight, even though I'm an amateur when it comes to bicycling. Me, on the other hand, can't really see why this would matter all that much until I get seriously more fit. Anyway, I'm not familiar with the physics needed to calculate this. If it helps, let's assume a 5 degree rise from 0-5 km in 10 km/h and a 5 degree descent from 5-10 km in 40 km/h.


title: Can an object move without relying upon Newtons third law?

text: Pretty much everything we do appears to boil down to practice of Newton's Third law. Even this rug I'm seated upon relies upon reaction from the floor to keep me seated. Amongst others the same reaction helps me move, sleep/recline .. and so on. What I wish to know - is it possible for a body to move without relying upon a reaction from the objects/whatever in the direction it wishes to move?


title: Are Poisson brackets of second-class constraints independent of the canonical coordinates?

text: Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\\{\chi_N(q,p),\chi_M(q,p)\\}_P$ . Is the matrix $C_{NM}$ independent of $q,p$? The motivation of the question is that when I tried to verify Dirac brackets satisfy Jacobi identity, it seems I need to assume $C_{NM}$ doesn't depend on canonical coordinates(To be honest, I'm not 100% sure if this assumption is necessary, because I did not try very hard on the verification without this assumption because one side of the Jacobi identity will become horribly complicated). Besides, for some simple examples(e.g. a massive vector field), it does turn out to be the case. EDIT: I think I know where I made mistakes, somewhere in the calculation I needed some terms to be 0, I was searching in my brain only all the strong equalities, while in fact some weak equalities could help.


title: Thrust center in space

text: I have this dilemma: Suppose you have a space ship somewhere in deep space, where there is no drag force or substantial gravity. If the ship has a single engine situated in such a way that the center of thrust is off-axis with respect to the center of mass, will the ship fly straight when the engine is turned on or will it begin to rotate? I firstly thought that it would fly straight, since there is no drag or weight to pull the ship on other axis, but I found this question Force applied off center on an object which suggest that I'm wrong, although I don't fully understand the answers there. Can anyone explain or provide links to material that could shed some light for me in the physics behind this?


title: Magnet and energy conservation

text: If we consider a steel ball falling under gravity in a cup (potential well) and being stopped at the bottom by an obstacle then energy conservation implies that the gravitational potential energy has first been transformed into kinetic energy and then in heat. After the experiment the potentiel well's height is decreased because there is a steel ball standing at the bottom. So I can repeat the experiment and each time I will extract less and less energy until I extract nothing when the well is filled up with balls. Now consider a magnet and a steel ball. The magnet accelerates the steel ball until they stick together and the kinetic energy is dissipated by heat emission. Energy conservation implies that the energy of the magnet+field+ball has changed (decreased by the amount of heat produced). My question is how is it decreased ? Is it the magnetic moment of the magnet which has been decreased, or should we take into account the induced magnetic moment in the steel ball to recompute the energy of the field ? The problem can be simplified (for computation) but remains open if instead of a magnet and a steel ball one uses two coils, each one connected to a generator. If someone can write down the details of the calculus, it would be much appreciated. (Problem solved for the coils in Griffiths p211. The generators produce the energy dissipated by heat.) So it remains to solve the problem for a magnet and the steel ball. Is there a domain reconfiguration ?


title: The possibility of free electrical energy?

text: Please excuse my lack of knowledge/understanding. **Question:** Why Nikola Tesla's Free Energy concept was never worked upon? Even today. **Context:** Now that we know Nikola Tesla was a genius and did exceptionally well with his works and the **Free Energy** concept. Why his idea was never adopted or was worked upon, even now? I know, back then, Edison didn't let him do much. But now, in our times, why GE or any other company or well the physics `PhD`s aren't following up? Why CERN is not working over it? By the articles that I have read over the internet I am under an impression that everyone (now) feel that we have "missed" the free-energy _chance_ that we had as a planet, that Nikola Tesla was trying to bring in. I see a lot of _brag_ (to an extent, as well) which leaves no doubt that his Free-Energy concept would have been **must-have**. But if I look at the actual followups, no one _it seems_ got **actually** interested to pursue his theories; or is it that no one is capable enough to do it? Which would be strange, given that we are at the peaks of technological advancements. **_How hard is it to build a prototype? Really?_** Or people might have worked over it, but Tesla's blue-prints were not correct, and that is why no one came up with the actual implementation of it? Or is it that he was _really_ a crackpot as politically-publicized by Edison? I would really appreciate an answer, because this is beyond me, to understand why no one followed up on this concept? Thank you, in advance.


title: Can alimentary packaging film be used to make a Fabry-Perot interferometer?

text: An alimentary packaging film consists on a thin plastic layer. If we put two of this films one on the other, could this acts as a Fabry-Perot interferometer? (I don't have the appropriate material at home to try and it doesn't work with incoherent light). As we don't usually see a "rainbow" on a such film of plastic, I was wondering if this was related to the non-conducting properties of it (but glass doesn't conduct too). What else could it be?


title: Divergent Series

text: Why is it that divergent series make sense? Specifically, by basic calculus a sum such as $1 - 1 + 1 ...$ describes a divergent series (where divergent := non-convergent sequence of partial sums) but, as described in these videos, one can use Euler, Borel or generic summation to arrive at a value of $\tfrac{1}{2}$ for this sum. The first apparent indication that this makes any sense is the claim that the summation machine 'really' works in the complex plane, so that, for a sum like $1 + 2 + 4 + 8 + ... = -1$ there is some process like this: ![enter image description here](http://i.stack.imgur.com/slTT9.png) going on on a unit circle in the complex plane, where those lines will go all the way back to $-1$ explaining why it gives that sum. The claim seems to be that when we have a divergent series it is a non- convergent way of representing a function, thus we just need to change the way we express it, e.g. in the way one analytically continues the Gamma function to a larger domain. This claim doesn't make any sense out of the picture above, & I don't see (or can't find or think of) any justification for believing it. Futhermore there is this notion of Cesaro summation one uses in Fourier theory. For some reason one can construct these Cesaro sums to get convergence when you have a divergent Fourier series & prove some form of convergence, where in the world does such a notion come from? It just seems as though you are defining something to work when it doesn't, obviously I'm missing something. I've really tried to find some answers to these questions but I can't. Typical of the explanations is this summary of Hardy's divergent series book, just plowing right ahead without explaining or justifying the concepts. I really need some general intuition for these things for beginning to work with perturbation series expansions in quantum mechanics & quantum field theory, finding 'the real' expanation for WKB theory etc. It would be so great if somebody could just say something that links all these threads together.


title: Condition for closed orbit

text: I'd like to know when an orbit is closed. I know that, to have a closed orbit, there is a ratio that must be a rational number, but I don't know other things..


title: Fate of largest scale structures?

text: In $\Lambda\mathrm{CDM}$, structures form "bottom up" with larger structures forming later. Structures are generally speaking supported by the velocity dispersion of their constituent objects (e.g. elliptical galaxy supported by velocity dispersion of stars, galaxy cluster by velocity dispersion of galaxies)$^1$. More massive virialized structures require higher velocity dispersions to support them. What happens when the velocity dispersion required to support a structure becomes relativistic and eventually exceeds $c$? Does the structure simply fail to collapse? Collapse to a black hole? Something else? It occurs to me that the $\Lambda$-driven exponential expansion that is currently thought to be getting under way in our universe might be rapid enough to cut off structure collapse at some scale, avoiding the scenario I described above. For the purposes of this question, let's assume for convenience a model where the Universe continues to expand with $\lim_{t\rightarrow\infty}\dot{a}(t)=0$. $^1$ With the notable exception of systems where dissipation is important, allowing the formation of a rotationally supported disk.


title: A small object on the opposite side of the earth's orbit with 0 velocity,"in line" with the core - would it collide with the earth?

text: If a small object (say mass 1 kg), at the opposite side of the earth's orbit (six months away), was placed in the same orbital position as the earth's core, at zero velocity in all directions, would the object collide with the earth or would it have been drawn by the sun's gravity away from a collision with earth? A naive answer would simply calculate how far "inward" the object would travel in six months, when accelerated by the sun's gravity, and if the distance covered is greater than the earth's radius, a collision would be averted. But I started thinking about the earth's own gravitational influence as it orbited closer to the object. How would we get an answer to this question?


title: How did Jackson get the exact solution to potential in polar coordinates

text: On page 45 of Jackson's Classical Electrodynamics (3ed), He solved a polar potential problem with known source distribution. $$ \frac{1}{\rho}\frac{\partial}{\partial \rho}\left( \rho\frac{\partial \psi}{\partial \rho} \right)=-g(\rho)\\\ \psi(\rho)\bigg|_{\rho=1}=0\\\ $$ with the given source $$ g(\rho)=-5(1-\rho)+10^4\rho^5(1-\rho)^5 $$ To me, this second order Sturm-Liouville ODE can only be solved if we have two boundary conditions, but in this case, we don't know the other BC.


title: How metallic surfaces states can emerge in topological insulators?

text: Topological insulators are materials known to have bulk insulator and metallic surface states. But, what is the origin of these metallic surface states? And how the topology of band could help the understanding of these states?


title: Quickest "flights" across Atlantic and round the world

text: With the dawning of the private space industry, if someone was to build a rocket-based craft that powers you up and then glides in for a landing, what would be the quickest you could get across the Atlantic? Or half way around the world? (Assume a typical flight trajectory - e.g. don't go through the earth or travel at ground level + 1 meter!)


title: Must every isometry have an associated Killing vector?

text: I understand that the flows of Killing vector fields are isometries, and that one-parameter groups of isometries have an associated Killing vector which generates them, but are your Killing vectors guaranteed to give you all possible isometries of the manifold? I guess what I'm trying to ask is, do isometries necessarily come in families, or can you have "isolated" isometries which can't have an associated Killing vector field?


title: The Matter-Vacuum Boundary in General Relativity

text: A previous Stack question (before I joined) asking about continuity in GR received replies which suggested that Curvature would be discontinuous at say a planetary boundary (assume no atmosphere for simplicity). I will analyse some basics of this and then return to that question. It is true that the Stress-Energy Tensor $T{_a}{_b}=0$ outside the body and is nonzero in the interior resulting in a discontinuity at the surface. This would imply that the Ricci Tensor $R{_a}{_b}$ is also discontinous at the boundary, and zero in the vacuum part as expected from the Einstein equations. However the Riemann Curvature Tensor $R{_a}{_b}{_c}{_d}$ (which generates the physically measurable accelerations) has contributions from the Weyl Curvature Tensor $C{_a}{_b}{_c}{_d}$ as well. In fact the Ricci Tensor "hands over" to the Weyl Tensor at the boundary: thus the Riemann Tensor stays non-zero there. However this "hand over" does not imply continuity, unless there is some GR Theorem which says that the Riemann Tensor stays continuous in this region. Also in the Newtonian approximation the analogous role is played by the gravitational potential $\phi$ in the Poisson equation $\nabla^2 \phi = 4 \pi G\rho$. Clearly this shows a discontinuity too as the density $\rho$ suddenly drops off at the boundary. However the discontinuity is in the second derivative of the potential: the potential itself is continuous. This means that in exiting a planetary cave or mine one does not suddenly meet a change in Gravitational potential. However I do not know any theorem in GR which guarantees such continuity. The applicable in-the-large scenario might be the surface of a neutron star; there may be in-the-small particle models too.


title: Why is a plastic bag transparent in infrared light?

text: This is a classic trick to do with a IR camera: ![enter image description here](http://i.stack.imgur.com/Rh1fb.jpg) Bu why is the plastic bag transparent, while the glasses aren't? I've also heard that water is not transparent in IR light. What causes this phenomena?


title: Can modern times physics explain Hutchison phenomenon?

text: I saw this video and trying to understand the phenomenon. But as you can see the electro-magnetical device lifts the non-polar material.(1:35~2:00) How can it be possible?


title: Baryon masses in Wetterich's new cosmology

text: Christoph Wetterich has put out a paper in which the universe isn't always expanding; it can be static or expanding just some of the time or even shrinking. And then there is an interaction which makes the masses of fundamental particles change in a complementary way, so as to preserve the properties of atoms, etc. Now here is what I don't understand. An electron gets its mass through the Higgs mechanism. A nucleon gets its mass through QCD effects. The quarks also get their masses from the Higgs mechanism and that makes a very small contribution to the nucleon mass, but mostly the nucleon mass arises in a different way. So I just don't see how any simple mechanism of varying mass can preserve e.g. the electron/proton mass ratio. Is this a tremendous problem for his theory, or is there something I have overlooked?


title: What is the cut-off for quantum superposition?

text: Is there an explanation as to how macro objects aren't in superposition? At what size do objects stop being in a state of superposition?


title: Twin Paradox, calculating spacetime intervals from both perspectives

text: I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me. The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path. * * * **According to Alice:** $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$ Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$. * * * The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations. * * * **According to Bob:** $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$ Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated!


title: Cooling - in vs out?

text: Suppose I have a 20x20x20 c"m plastic box where the top cover is open. On the top cover there is a fan ( 20x20 cm) which suppose to cool a cpu which is in the box . At the bottom there is a CPU which is getting warm. Now - in order to cool the CPU - I have 2 options :   * put the fan to suck air from the box towards outside   * put the fan to push air from outside towards the box. **Final goal :** Max - Cooling the cpu. **Question:** Which is the preferred approach ?


title: Anti-gravity in an infinite lattice of point masses

text: Another interesting infinite lattice problem I found while watching a physics documentary. Imagine an infinite square lattice of point masses, subject to gravity. The masses involved are all $m$ and the length of each square of the lattice is $l$. Due to the symmetries of the problem the system should be in (unstable) balance. What happens if a mass is removed to the system? Intuition says that the other masses would be repelled by the hole in a sort of "anti-gravity".   * Is my intuition correct?    * Is it possible to derive analytically a formula for this apparent repulsion force?   * If so, is the "anti-gravity" force expressed by $F=-\frac{Gm^2}{r^2}$, where $r$ is the radial distance of a point mass from the hole? * * * Edit: Video here (start at 7min): http://www.disclose.tv/action/viewvideo/45729/Stephen_Hawking__The_Story_of_Everything_pt_2_9/


title: How fast do you have to spin an egg to have it standing?

text: If you take a hardboiled egg and put it on a table and start to spin it, if you spin it fast enough it will start to spin in an upright position. What is the angular velocity needed for this transition to occur? Is energy conserved in the transition? Must there be some imperfection in the egg for the transition to occur? Assume a prolate spheroid egg of constant density. If that doesnt work, assume its center of mass is shifted abit, if that dont work, assume an egg. Edit: One is allowed to use simpler shapes of the same symmetry type, to avoid messy integrals. And most importantly, can you explain without equations, why does the egg prefer to spin in an upright position ?


title: Cantilever Beam - Maximum Shear of the Beam

text: > A cantilever beam $3\ \text{m}$ long is subjected to a moment of $10\ > \text{kNm}$ at the free end. Find the maximum shear of the beam. The answer is "There is no vertical load, shear is zero" How come? The shear should be sliding upward at the end of the wall and downward to the free end.


title: Why is carbon dating limit only 40,000 years?

text: For an example, when they tried to get the carbon dating for presence of Aboriginal people in Australia they get to the number 40,000. But it could be much earlier. Why is that 40,000 years limit for carbon dating methods?


title: Boltzmann distribution: derivation from canonical distribution

text: I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not sure what's wrong with it :). Any guidance would be much appreciated! I believe that the probability density function for the energy, $E$, of a random particle in a system at equilibrium is meant to look like this: $$f_E(x) = Ae^{-\frac{x}{kT}}$$ To simplify things I'll just consider the particular case: $$f_E(x) = e^{-x}$$ The cumulative distribution function is then: $$Pr(E\leq x) = F_E(x) = 1-e^{-x}$$ Now suppose all the particles in our system are of mass $1$, so, ignoring potential energy, the random variable governing the speed of a given particle is $V = \sqrt{2E}$. Now we want to find the distribution of the speeds. We can do this as so: $$F_V(v)=Pr(V\leq v) = Pr(\sqrt{2E}\leq v) = Pr\left(E\leq \frac{v^2}{2}\right) = F_E\left(\frac{v^2}{2}\right)$$ and so $$F_V(v)=1-e^{-\frac{v^2}{2}}$$ and finally $$f_V(v)= ve^{-\frac{v^2}{2}}$$. This is _almost_ right, however, the Maxwell-Boltzman distribution would predict a pdf with $v^2$ rather than $v$, something like: $$f_V(v)= \sqrt{\frac{2}{\pi}}v^2e^{-\frac{v^2}{2}}.$$ Can anyone point out the mistake in my logic? Thanks!


title: Where does the mass term come from in the Proca Lagrangian?

text: There are many good books describing how to construct the Lagrangian for an electromagnetic field in a medium. $$ \mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu} $$ When moving to the Proca Lagrangian (and a massive photon), I know what the mass term looks like but I have know idea where it came from. $$ \mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu}+\frac{\mu^{2}}{8\pi}A_{\mu}A^{\mu} $$ Why is $A_{\mu}A^{\mu}$ the correct term to include? I guess that it must be Gauge and Lorentz invariant so why wasn't it included in the original Lagrangian? Why is the factor of $\frac{\mu^{2}}{8\pi}$ needed?


title: Commutator problem

text: I am trying to calculate the following commutator $$[\mathcal{H}_0(r',t'),\psi(r,t)]_-$$ where $\mathcal{H}_0 = (\frac{1}{2m}\nabla^2 + e\mathbf{A}(r',t'))^2 + e\phi(r',t') - \mu$, and $\mu$ is the chemical potential. It is supposed to be $0$, but I don't understand why it is not a a delta- function $\delta(r-r')[\mathcal{H}_0(r',t'),\psi(r,t)]_-$ which is not necessarily $0$. Could someone please help me understand why the anticommutator is 0?


title: How is extremely low frequency (ELF) radiation collected by a submarine antenna?

text: The U.S. Navy Project ELF managed to generate extremely low frequency (ELF) radiation at down to $\approx 76$ Hz (implying a wavelength of $\approx 3,945$ km!). I was curious, what kind of receiving antennas do submarines use to actually pick up these ELF signals? What are the size requirements? Do submarines just trail a very long cable that approximates a monopole or dipole antenna, use electrical lengthening to shorten the antenna to some length $\approx 3,945/k$ km, where $k \in \mathbb{Z}$, and then travel along an optimal vector to pick up a transmitted signal?


I know that I can detect a vehicle by using magnetic field. I can see obviously difference in signal when a vehicle pass. But I don't see that when measuring in a car. Is it possible to measure outside magnetic field from inside a car? If possible, how could I extract that signal?


title: Ideal Gas Constant

text: Wikipedia states that the ideal gas constant relates the energy scale to the temperature scale. It serves as the constant of proportionality. This is obvious from the units. If temperature is a measure of average kinetic energy, then why is the internal energy of an ideal gas, (which is just KE), $ \cfrac{3}{2} nRT $ and not just $ nRT $? My first thought is that this constant was defined in terms of the ideal gas law, and that this relationship was determined before the KE of an ideal gas was known. Regardless, how can it be said that R is the constant of proportionality between the two units (K and J), when in fact it is off by a factor of 3/2?


title: leaving 2-norm propelled probability implications

text: I am curious about why there are no further generalized probability structures used in Physics. The great revolution was moving away from one-norm system to a two-norm system. What happens if we attempt to generalize to some n-norm system and formulate discover some rule for appropriate representations. Is it possible to have similar physics in some limit? ps please try to reason with me here. otherwise; $P_{R} \rightarrow$ classical $P_{C} \rightarrow$ Quantum $P_{n} \rightarrow$ ? I really don't know . . . .


title: Why is there a Majorana zero mode in the $\pi$ flux core of the p+ip superconductor?

text: In this review paper (http://arxiv.org/pdf/1202.1293.pdf), the author shows that threading a $\pi$ flux through a 2D $p_x+ip_y$ superconductor will trap a Majorana zero mode at the flux center. The deduction involves solving differential equations, and is hard to generalize to arbitrary dimensions or arbitrary gauge defects. Since Majorana zero mode is topologically robust, I would expect a simpler and less detail-dependent way to show that there is indeed a Majorana zero mode in the $\pi$ flux of a $p_x+ip_y$ superconductor. Does anyone know such a simpler approach? PS: The answer I expect should be general enough that can be applied to $\pi$-flux tubes in 3D topological insulator or topological superconductors as well.


title: Why does the index of refraction change the direction of light

text: I've been studying in optics the macroscopic maxwell's equations, and how electromagnetic fields propagate through different mediums. Over there, the index of refraction appears, as a complex function that generally depends on $\omega$, with both refraction and absorbtion terms: $n_c=n+i\kappa$. I understand how this affects the speed at which light propagates, both macro and microscopically, and how this affects to how light is absorbed when it transmits through the medium, but I don't get yet how this index changes the direction of light, producing dispersion. I mean, all time I'm seeing those effects as something that separates dispersive from non dispersive mediums and all that stuff, and I don't know how light is actually dispersed. As it's something that happens when changing medium, I guess it's an interface effect, and we haven't seen those effects yet, but I would like to know an explanation.


title: Why is the energy spectrum continuous for a plane wave when it has energy less than the potential barrier?

text: Please explain it in the context of this task: we have a potential barrier that looks like $\prod$, with $E<U$. There are 3 regions: 1) no field   2) barrier   3) no field Solution could be found as * * * 1) $\psi_{1} = A\cdot e^{-ikx}+B\cdot e^{ikx} $   2) $\psi_{2} = C\cdot e^{-ik_{0}x}+D\cdot e^{ik_{0}x}$   3) $\psi_{3} = 0 +F\cdot e^{ikx}$ (cause we haven't got a reflected wave here) where $k_{0}^{2}=2m_{0}(U_{0}-E)/h^{2}$ and $k^{2}=2m_{0}E/h^{2}$ * * * So why in this case spectrum is continuous?


title: Change of variables, Fermi Integral

text: This is a really basic question, but I'm kind of confused. I have this integral $$\int_{0}^{\infty}\frac{p^{2}dp}{e^{\alpha+\beta p^{2}/2m}+1}$$ where $p:=|\mathbf{p}|=\left(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right)^{1/2}$ is the magnitude of the momentum of a particle of mass $m$. This integral represents the total number of particles in a Fermi-Dirac gas. Now, I want to convert this integral to something of the form $$\int n(p_{x})dp_{x}$$ where $n(p_x)$ is the number of particles with momentum between the interval $(p_x,p_x+dp_x)$. To this end I must use cylindrical coordinates but I don't know how to convert the term $dp$ in something like $dp_xdp_r$. Can you help me please?


title: Why are von Neumann Algebras important in quantum physics?

text: At the moment I am studying operator algebras from a mathematical point of view. Up to now I have read and heard of many remarks and side notes that von Neumann algebras ($W^*$ algebras) are important in quantum physics. However I didn't see where they actually occur and why they are important. So my question is, where they occur and what's exactly the point why they are important.


title: Question about the parallel axis theorem

text: I've a question about the parallel axis theorem. So I'm perfectly OK with the derivation of the proof in 2D. However in the typical derivation they just say this in my textbook "Because the Zi coordinates are not taken into account into this calculation, we can generalise the summations to include all particles for all slices" How is this possible? I don't get it how you can generalise for all slices! Please help me, and thank you very much.


title: Is there a mistake in this example in my mechanics textbook?

text: This is a worked example in my mechanics book: ![enter image description here](http://i.stack.imgur.com/1FKOF.png) ![enter image description here](http://i.stack.imgur.com/rxymK.png) But I think this worked example is wrong. In the 6th line. shouldn't $X=4R$ as the expression for distance above the surface is $(X-R)$ and when the particle P is a distance of 3R ABOVE the surface, $v=0.5(gR)^{0.5}$ could someone please tell me if I am wrong or the book is wrong... I mean books a re rarely wrong and so I am not sure what to follow


title: Particles scattering on fluids: breakdown of the effective continuum description

text: When does the macroscopic continuum description of a medium like a fluid break down? Say I'm interested in a scattering process of some particles with momentum $p$ and energy $E$ off a fluid of temperature $T$, volume $V$, and pressure $p$: when should I consider the single fluid particles rather than the collective modes? For a solid with a lattice, there is a natural cutoff, but is there for a fluid?


title: Could energy be stored into (not extracted from) the quantum zero point field (like a battery)?

text: In order to explain the question clearly, I will make a short introduction. In 1962, Josephson predicted that for a sufficiently thin insulating layer, it should be possible for Cooper pairs to tunnel between two pieces of superconductor. With a potential difference $V$ across the junction, an alternating current should flow at a frequency $f$ given by: $$h\,f = 2\,e\,V.$$ For a potential difference of up to $10^3\text{ volts}$ (just an example), the frequency of these alternating currents would be up to around $10^{18}\text{ Hz}$ (for a potential difference of only $1\text{ volt}$, these alternating currents would still be in the petahertz region). There are diodes that operate at petahertz frequencies. One example is the metal/double insulator/metal tunneling diodes. These designs are based on the idea of harvesting vacuum fluctuations using down converter and antenna - coupled rectifier. Note that I am proposing a completely different idea (since most of the designs just mentioned have been proved not feasible). According to stochastic electrodynamics (SED, references [2], [3], [4]), the energy of classical electron orbits in atoms is determined by a balance of emission and absorption of vacuum energy. Electrons emit a continuous stream of Larmor radiation as a result of the acceleration they experience in their orbits. As the electrons release energy, their orbits would spin down were it not for the absorption of vacuum energy from ZPF (zero - point fields). This balancing of emission and absorption has been modeled and shown to yield the correct Bohr radius in hydrogen. There is some coupling here with ZPF. * * * After this introduction, I can probably state my question more clearly. As I said above, I am thinking about a network of Josephson tunnel junctions under high voltage, that can also be seen as vacuum gap (instead of a dielectric) superconducting capacitors. Is it possible that the phenomenon of resonance could play a role in this setup? More appropriately, we should probably consider some sort of coupling with the zero point field. I am not necessarily saying that we would get any "free lunch" (by this I mean energy from nothing), but this kind of coupling with ZPF should at least open some designs of systems that could at least store the input energy, for later use (battery design). I say "at least", because other interesting phenomena might appear in this context. By the way, I am drawing a parallel here between stable atoms and various vibrational modes involving ZPF (think about some sort of 3D Chladni figures, as an analogy). Following this analogy, only radiation of certain frequencies will couple with the ZPF (I would call them Larmor frequencies). These interesting frequencies should be in the range of Josephson tunnel junctions under high voltage, mentioned above (range as far as frequency is concerned, we are dealing here with alternating currents, not direct radiation). * * * References: [1] G. Moddel, B. Eliasson, "High speed electron tunneling device and applications", US Patent No. 6563185 (13 May 2003). This design is related to energy extraction, and its feasibility is questionable. [2] T. H. Boyer, "Random electrodynamics: the theory of classical electrodynamics with classical electromagnetic zero point radiation", Phys. Rev. D 11, 790-808 (1975). [3] L. de la Pena, M. Cetto, "Quantum phenomena and the zero point radiation field II", Found. Phys. 25, 573-604 (1995). [4] D. C. Cole, Y. Zou, "Quantum mechanical ground state of hydrogen obtained from classical electrodynamics", Phys. Lett. A 317, 14-20 (2003).


title: Audio anomaly identification

text: I left an audio recorder in the forest in many different locations for fauna identification. I am collecting subjects on my audio recorder that are clearly not in the frequency range of any animal collected so far. An average 35 kg wallaby hopping has a high frequency range of between 2000 and 13000 Hz. The unknown subjects have a frequency range of between 46 and 100 Hz; very heavy. I've done a drop test of a 10 kg dumbbell onto the forest floor at 5 different heights and at 1-50 metres. 250 drops collected on audio for comparison. At a height of 1 metre, the 10 kg dumbbell bounces off the forest floor 14 mm. Apparently this is an impact of 700 newtons but I've been told that this is inaccurate without taking into account the elasticity of both forest floor and dumbbell. How can I accurately determine distance and weight of the unknown subject without spending money on impact plates or is there another way to do it?


title: Do anti-photons exist?

text: I know what anti-matter is and how when it collides with matter both are annihilated. However, what about anti-photons? Are there such things as anti- photons? I initially thought the idea preposterous. However I am curious because, if anti-photons don't exist, then anti-matter could theoretically transfer its energy to normal matter - through the mechanism of light...right?


title: How to count quarks using Deep Inelastic Scattering?

text: The Wikipedia article on deep inelastic scattering suggests that the experiment shows baryons have three point of deflections (corresponding to three quarks) and mesons have two points of deflection. How are the electrons fired in this experiment being detected, and how exactly do the two or three points of deflection appear in the data? Are they fired at a target consisting entirely of baryons, or are collisions with non-baryons somehow filtered from the data?


title: What is the difference in application for near infrared camera data capture as opposed to infrared data capture?

text: I am trying to understand why I would use a near infrared as opposed to an infrared camera for data capture of vegetation health monitoring.


title: Question about De Broglie Wavelength

text: I read that: > The smallest wave packet we can build has a size on the order of the de > Broglie wavelength $\lambda$ of a free particle moving with the same speed > $v$. I haven't been able to find a clear explanation of why this is. The uncertainty or spread (standard deviation) in position should be something like $$\Delta x \geq \frac{\hbar}{2\Delta p}$$ but I don't see why this necessarily directly relates to the de Broglie wavelength of the momentum (or average of the momentum as it only has a definite value for an infinite plane wave I suppose), and why it means that particles moving more quickly can have more localized wavepackets. Any clear explanation would be much appreciated. (I don't mind a bit of Fourier analysis but I am a bit rusty!)


title: How is electromagnetic wave variation distributed in space?

text: Imagine an electromagnetic wave (a monochromatic one for example) The electric field amplitude, and its variations travel in the propagation direction. So, if there really exists a propagation direction, what happens in other directions? I mean, is there an oscilation in those other directions? Does something (electric or magnetic field) move in those directions? Does it's speed in those direction depends on frequency,or is this wave infinitely thin? Is it not covering any space in other directions? I have seen many graphs showing that the field variation is not in the direction of propagation, that "means", based on polarization experiments that it's not a Longitudinal wave, It's a transverse wave. If I accept field is not variating in the propagation direction (it just go forward in that direction), then I could think it's variating in Other direction, so ok what direction? Perhaps we can't say it's variating in any direction, it's not a spatial variable!!! I can't imagine how the field variation is distributed in space. (If it were distributed..) Any information welcome, thanks! (I've already asked this at old site physic overflow, still searching an answer) **EDIT** some options 1-Electromagnetic waves just move in one single spatial propagation direction.. then all spatial concepts that take account other directions are wrong, like "polarization" and "transverse wave" definition 2-Electromagnetic waves moves in others directions too, then speed of propagation in those other directions could depend on frequency, and thats weird, but who knows! 3,4-- others.. .. **EDIT 2** (Add my comment as part of the question) **If there are infinite wave planes, then imagine a light that is turned OFF and then is turned ON, comparing the EM, first it is covering nothing, and then suddenly reach "infinity" (or at least very far..in fact any distance is enough), what is the speed of THAT propagation? because it is NOT a wave propagation, because is not in the direction of propagation, that infinite or very large field extends itself at infinite speed?**


title: Can Outer Space be used for Waste Disposal?

text: If harmful toxic, nuclear and other wastes are dangerous if improperly disposed on Earth, can't they be launched into space? If a large, tightly bound mass of waste were to be propelled into outer space, according to the law of Inertia, it would never stop going forward, unless obstructed by something. Also, the chances of it running into a space shuttle are almost zero. So can wastes be launched into space? If not, what are the demerits of such a practice? Or has such a project already been undertaken?


title: Temperature in CGS (Gaussian) units

text: I've been struggling with conversion from Gaussian to SI units for sometime, trying to figure out how derived units in CGS (current, charge etc) relate to the SI units. But I couldn't find any reference that spoke about temperature. I've read that the only base units in CGS are the centimetre, gram and second. How does one define temperature (Kelvin) in this unit system? Or do we have to define another base unit?


title: Electric arc due to static discharge in a T-Shirt possible?

text: Yesterday, when I came home, I went to the bathroom (lights off) and i took off my T-Shirt (100% cotton) which I wore under a Shirt (50% cotton, 50% polyester). I _believe_ to have seen a small but well visible light arc. **Question:** Is the amount of static discharge in T-Shirt big enough to produce a visible light arc?


title: Are there any useful site/papers that provides data about variability in a composites material properties such as G1C,G12,E11,E22,V12 or fibre angles?

text: I realise this is a question that borders on Mechanical Engineering, but I was hoping someone could help. I am looking at finding experimental data that will point to variability in a composites material properties such as G1C,G12,E11,E22,V12 or fibre angles.For example, if G1C has a design value of 550J/m^2, I am looking at information that would point to the uncertainty in this value in terms of standard deviation in experimental data. Thanks


title: Can it be considered as time travel?

text: India is about 8.5 hours ahead of US. If some person starts from India at 9 pm(according to his wrist watch) and reaches US in 2 hours(just assume),at that time his wrist watch will b showing 11 pm but as he his in US so he will have to match his watch according to US time so he will adjust his watch to 4:30 pm.This means that he is goin to live those 6.5 hours again in his life,but this time in US.So can this be considered as time travel or i am wrong.Please explain.


title: Relative Velocities and Conservation of Kinetic Energy

text: An object of mass m moves with velocity $v$ towards a stationary object of same mass. Impact is an elastic collision. $v_1$ is the velocity after impact of the mass originally moving $v_2$ is the velocity after impact of the mass originally stationary Elastic collision means K.E. is conserved, so: $$\frac{1}{2}mv^2=\frac{1}{2}m(v_1)^2+\frac{1}{2}m(v_2)^2$$ Thus, $$v^2=(v_1)^2+(v_2)^2$$ However, using relative velocities: $$-v=(v_1)-(v_2)$$ Squaring both sides gives a different value for $v^2$. Instead of $v^2=v_1^2+v_2^2$, $v^2=v_1^2-2v_1v_2+v_2^2$. How come?


title: $\mathcal{N}=2$ spin $1/2$ supermultiplet

text: In Freedman and Van Proeyen's _Supergravity_ , in the footnote on pg. 128, they say > There is a subtle hermiticity requirement for $\mathcal{N}=2$, which > requires the multiplet $(-1/2,0,0,1/2)$ must be doubled although it is self- > conjugate. What exactly are they referring to here?


title: Topological Quantum Field Theories

text: I've asked this on Math.SE, but with no avail. So, I decided to ask it here. I was wondering about the following after reading the Wikipedia article on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: > Why is this so? Secondly, consider the Schroedinger equation: $$i\hbar\dfrac{\partial}{\partial t}\Psi=\hat{\mathcal{H}}\Psi=0$$ So, there are no dynamics or propagation. This leads to my second question: > What does one mean by "nontrivial dynamics/propagation" (as mentioned in the > Wikipedia article)? I've understood the TQFT must be invariant under time reparameterizations, the reason being pretty obvious. Thank you!


title: Why is air invisible?

text: I think that something is invisible if it's isolated particles are smaller than the wavelength of visible light. Is this correct? Why is air invisible? What about other gases and fumes which are visible?


title: Why glass is transparent?

text: Once I asked this question from my teacher and he replied "because it passes light", "and why it passes light" I asked and he said "because it is transparent". Same question again, Why glass is transparent. Why it passes light while opaque objects do not?


title: Friction as a Centripetal Force

text: When an object is moving in a circle, won't the frictional force oppose it's motion? And if the velocity is always tangential to the path, there is no component of it towards the center, so there is no component of friction towards the center (since the frictional force is antiparallel). So how can friction be a centripetal force, like in the case of a car rounding a corner?


title: Is it possible to see light intensity fluctuate?

text: Solutions to Maxwell's equations shows that the $\vec{E}$ and $\vec{H}$ component are of the same phase, which means they go to maximum and zero together, therefore the intensity of the electromagnetic wave should be 'lumpy'. My first question is, if I have a fast enough light intensity detector, or say detecting low frequency like radio wave, can I see the fluctuation in the signal of intensity? I always think it is reasonable to observe that, until I learned in quantum field theory, that telling me electromagnetic interactions obey the phase gauge transform invariance, phase is always relative, absolute phase is not observable. Why absolute phase is not observable? Of course we can see the absolute phase in alternating current, so I'm confused, what is the quantum field theory trying to imply? * * * UPDATE: For the second question, it looks like I'm mixing up the complex amplitude in optics and the wave function in quantum mechanics, complex amplitude $u=e^{i\varphi}$ is actually a representation for $\cos \varphi$, while wave function $\psi=e^{i\varphi}$ doesn't have a meaning because only the $|\psi|^2$ is observable.


title: Finding sphere of influence in multibody system

text: I would like to draw a boundary illustrating the sphere of influence of each body in a 3 body system. What is the best way to find such a boundary? My understanding is at the edge an SoI is the minimum of the sum of gravitational forces. My current method is, for each body, project a line from the body to some fixed point well beyond the size of the system, and find the minimum net gravitational force by binary intersection. Is there a faster way of doing this?


title: Why does bringing N 1-orbital atoms together yield N levels?

text: A common example of this is that when bringing N hydrogen atoms together into a ring. Far apart, assume each electron exists in the 1s state. As we bring them together, instead of each electron staying at the original 1s level, or all of them changing by the same amount, the 1s level fans out into N. For the case of 2 atoms, I can understand this as bonding or anti-bonding of the atoms. i.e., do the wavefunctions add between the protons, meaning each electron can share in the potential of both protons (bonding) or do the wavefunctions destructively interfere between the protons (anti-bonding). With 3 atoms, I can't find 3 levels. Assuming Gaussian shaped wavefunctions, note that the sign of each wavefunction between any two atoms defines the wavefunction on the rest of the ring. Since the signs of the wavefunction are independent, there should be 2^3=8 possibilities since each wavefunction can be + or -. Yet, there are really only 2 energetically distinct arrangements that I see: all have the same sign (two cases) or 2 of 3 have the same sign (2*(3 choose 2)), to account for both sign cases). So I get 3 atoms yield 2 levels. Can somebody shed light on what I've done incorrectly? Or is 3 too small to work correctly? Is there an argument about the shape of the orbitals I've neglected? Thank you.


title: Calculating current used by individual resistor?

text: I just recently finished a test on Physics, and when receiving my scores back I missed a question. Here's what it was if I can remember it correctly: > Three resistors (10.0, 15.0, and 6.00 ohms) are parallel and the circuit is > powered by 12.0V. How much current (in amps) is the 6.00 ohm resistor using? So:               |-----------------|----------------|---------------|     |                 |                |               |     --(12.0 V)        <(10.0 ohms)     <(15.0 ohms)    <(6.00 ohms)     -                 >                >               >     |                 |                |               |     |                 |                |               |     ----------------------------------------------------      So, I assumed that since you can combine parallel resistors into one equivilent resistor, they use the same amount of current. So I just found the current used ( **4.00 A** ), and assumed that the 6.00 ohms resistor used the same current. The teacher, however, stated that the correct answer was **2.00 A**. Am I right or wrong? Why?


title: The Large Hadron Collider produce material residues?

text: In the LHC particles are accelerated until they collide, producing energy and this make new particles. My question is what happened after this. What happened to new particles and to old particles that aren't destroyed in collision, they dissolve in the nether are absorbed by the container or simply are so insignificant that are leave inside?


title: Recommended progression with which to learn physics for fun

text: I enjoy reading up on new topics in physics and astronomy, and learning the concepts and ideas in the major theories, like relativity and quantum mechanics. Unfortunately, one thing that I lack completely (and I envy others for) is the mathematical understanding. I would like to start studying the mathematics as well, from Newtonian gravity to more advanced problems, but I was wondering what the best progression of topics would be. How should I learn the mathematics of physics? I know that there are other questions on how to learn physics, and I apologize if this seems like a duplicate. However, I don't think that the mathematics have specifically been addressed in previous questions.


title: Calculating eccentricity from semi-major axis, position and velocity of particles

text: I'm trying to calculate the eccentricity of a binary system's body knowing the bodies' velocities, positions, masses and the semi-major axis. I am only trying to calculate the eccentricity for one of the bodies but can't seem to get the formulas combined right to get the eccentricity only using these values. Any advice would be brilliant, thanks!


title: Find the dispersion of the slab?

text: I've been having trouble with this for a few days and already used up all my tries on the homework for it, but have the final is coming next week. Thus I would like to know how do this, however I cannot figure out what the right equation is for the slabs dispersion. A light beam containing red and violet wavelengths is incident on a slab of quartz at an angle of incidence of 61.5°. The index of refraction of quartz is 1.455 at 600 nm (red light), and its index of refraction is 1.468 at 410 nm (violet light). Find the dispersion of the slab, which is defined as the difference in the angles of refraction for the two wavelengths. Would appreciate any help, thank you.


title: No attraction radially in an cylinder of spherical magnets

text: I have a set of small magnetic spheres the size of ball bearings. When many of them are built into a cylinder such that they are hexagonally packed, there is no magnetic attraction radially (between the walls inside). ![Cylinder of magnetic balls](http://i.stack.imgur.com/QRJ9U.png) That is, if I try to squish the cylinder, ![getting squished by pencil](http://i.stack.imgur.com/cWYNx.png) the walls do not stick together; ![enter image description here](http://i.stack.imgur.com/ltC6P.png) rather it easily resumes its cylindrical shape. However, if I force a little ball inside, it is attracted to the walls, ![ball inside cylinder](http://i.stack.imgur.com/yW5LI.png) such that when I flatten it, it stays flattened. ![staying squished on its own](http://i.stack.imgur.com/tFELi.png) What is the magnetic field doing in this situation? Why is there no apparent attraction (and even some resistance—though that may be frictional) radially?


title: Find vector position in time give this graph?

text: Graph is position x position. There are 3 points, $A$, $B$ and $C$.   * $A(0,2)$    * $B(4,2)$    * $C(6,0)$  Particle travels from $A$ to $B$ and from $B$ to $C$ at constant $v = 2 ~m/s$. Find vector position in time for the $BC$ part of the motion? For the AB part I did this: $v$ is constant, so position in time is given by $2\hat{i}\cdot \mathrm{t} + 2\hat{j}$. Velocity in $i$ direction x time, plus the constant $2\hat{j}$ vector component. For BC the "angled" displacement vector is making things a bit harder. Displacement vector has two components, $2\hat{i} + (-2\hat{j})$. Absolute value of it is $2\sqrt{2}$. Tried to decompose the velocity vector in two: $\hat{i}\sqrt{2}$ $-\hat{j}\sqrt{2}$. Then integrate each component and add up the two. But the result is not quite right... Integrated each component independently: $\hat{i}\mathrm{t}\cdot\sqrt{2} + c$ and $-\hat{j}\mathrm{t}\cdot\sqrt{2} + c$


title: Operators Uncertainty

text: $\hat A$ is an operator. The uncertainty on $\hat{A}$, $\Delta A$ is defined by: $$\Delta A=\sqrt{\langle\hat A^2\rangle - \langle\hat A\rangle^2}$$ what is difference between $\langle\hat A^2\rangle$ and $\langle\hat A\rangle^2$ that leads to Uncertainty Relation between two Operators? more details: $$ \langle\hat A^2 \rangle=\langle\psi|\hat A^2|\psi \rangle$$ What is the name of difference between absolute value of these two complex conjugates


title: Evaluating angle of refraction

text: Can someone show me how to evaluate the angle of refraction for white light where the angle of incidence is 30 degrees if medium A is air and medium B is water. The answer is 22 degrees. I don't know how this works. What formula do I use?


title: Why do Engineers manipulate coordinate systems?

text: When tackling a physics problem, An Engineer will manipulate the axes/coordinate system where a Mathematicians and/or Physicists will use the original coordinate system and math. Why do Engineers think differently? I know its likely because that is how they are taught, but why are they taught that way?


title: Geostationary orbit and rotation speed of a planet

text: Assume for planets similar to Earth but with different rotation speeds (including zero), and we put a satellites at altitude of 35 786 km/hr (similar to the Geostationary orbit or earth), with speed 11 068 km/hr, similar to the speed of communication satellites. these satellites, can rotate around the center point of the planets at any direction regardless of it's rotation axis, as long as it's perpendicular to the line from it's center and the planet center. For any planet, if we consider it as the reference point (we will exclude the rotation speed), it has multiple satellites at same altitude but with different speed but they don't fall. How is that possible? or we should consider another reference point, with absolute zero of speed and rotation, which does not exist? or there is another factor missing? **rephrasing:** I put an assumption that the planets are similar to earth to ease calculation The satellite at orbital speed of 11 068 km/hr, can rotate around the planet in any direction as in the image ![enter image description here](http://i.stack.imgur.com/zbbsd.png). it doesn't have to be around equator. only the one around the equator, will look as fixed in the space.


title: Why is a material such as plastic air-tight?

text: I'm not too familiar with fluid mechanics, but i always wondered how let's say plastic containers would be able to contain a fluid, such as gas or water. How does plastic prevent the molecules of the fluid from crossing it? Is it because of a high density relation at the boundary? Does it have anything to do with energy levels?


title: Optimal Airplane trajectory

text: The last time I took a plane the following problem crossed my mind. Setting: take the Earth and neglect its rotation around the Sun. It then only rotates on itself with angular velocity $\Omega$. Assume an airplane is flying with constant velocity $v$ in this absolute reference frame, at a constant altitude, i.e. we consider a constant radial coordinate $R$. The angle variables are $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$ . The non rotating relative frame would be $\theta'=\theta$, $\phi'=\phi+\Omega T$. Given a starting point $(\theta_0,0)$ and an endpoint $(\theta_f,\phi_f)$, what is the path $(\theta,\phi)$ which minimize the flight time $T$ ? I think we have: $$ T = \dfrac{1}{v}\int ds = \dfrac{R}{v}\int\limits_0^{\phi_f + \Omega T}\underbrace{\sqrt{(\dfrac{d\theta}{d\phi})^2+\sin^2(\theta)}}_{\mathcal L}d\phi $$ which at first variation is $$ \delta T = \dfrac{R}{v} \left\\{ \int\limits_0^{\phi_f + \Omega T} \left(\dfrac{\partial \mathcal L}{ \partial \theta} - \dfrac{d}{d\phi} \dfrac{\partial \mathcal L}{ \partial \dfrac{d\theta}{d\phi}} \right)\delta\theta d\phi \+ \left[ \delta\theta \dfrac{\partial \mathcal L}{ \partial \dfrac{d\theta}{d\phi}} \right]^{\phi_f + \Omega T}_0 \+ \delta T \Omega \left[ \mathcal L \right]_{\phi_f + \Omega T} \right\\}. $$ Given the implicit character of the last equation, I think we should use some perturbative approach. Intuitively, the small expansion parameter is $R\Omega/v$. However I am unsure as to how to proceed. Indeed setting $\delta T = 0$ leave only two terms, but unlike classical equation of motions, I think we can't choose $\delta \theta$ such that the second term (boundary term) vanishes. This is because changing $\theta\rightarrow\theta+\delta\theta$ changes the boundary condition as well, because $T=T(\theta)$. This is further supported that by doing so we are back to standard non rotating case, i.e. shortest path on a sphere. Does anyone have any idea to get some (even approximate) solution to this problem ? Is there any more elegant way to proceed ?


title: How does someone move space?

text: I've heard that we've known components of the universe appear to be flying apart faster than $c$, and that this can happen because the space containing these materials are also moving apart. How does someone move space? How efficiently can it be done? I would like to get on a rocket and then move the space very close to Sedna and see what it looks like. What are the physical barriers to doing that?


title: Convention in physics for [],{} and operators (QM)

text: I got a little mixed up with the convention in physics. Usually a hat means an operator. For a given electron-ion Hamiltonian $\hat{H}_{e-n}$, what are the difference between these: 1) $\hat{H}_{e-n}[{\bf{R}}]$ 2) $\hat{H}_{e-n}[{\bf{R}}](\bf{r})$ 3) $\hat{H}_{e-n}[{\\{\bf{R}}\\}]$ 4) $\hat{H}_{e-n}[{\\{\bf{R}}\\}](\bf{r})$, 5) $H_{e-n}[{\\{\bf{\hat{R}}}\\}](\bf{\hat{r}})$, with $\bf{R}$ the position of the nucleii and $\bf{r}$ the position of the electrons. From what I know, I think $[\bf{R}]$ means a parametric dependence on the ionic position (i.e. given some atomic position you can apply the operator and it does not explicitely depends on atomic position). For the curl bracket $\\{\bf{R}\\}$ I think it means that you cycle over all possible type of atomic position? Also, what is the difference between putting the hat on the operator H and on the positions knowing that the position operator resolve itself by: $\bf{r} = <0|\hat{r}|0>?$ EDIT: Something more traditional is the notation for the exchange-correlation functional $V_{xc}[\rho](\mathbf{r})$


title: Thought Experiments on black holes, (im)possible perpetual motion and minimum mass for photographic plates

text: Consider a black hole in vacuum at Temperature T. The setup that I am interested is a one that collects thermal photons from a black hole by enclosing in a spherical photographic plate and then lowering it back into the black hole. If one assumes that photographic plates are massless.(which they are not). Then the very fact that I can lower the photographic plates(after the photons are collected) back into the black hole allows for a possibility of for the creation of a perpetual motion machine, from the potential energy generated by lowering the collected photons back into the black hole. The mass of the photons emitted is equal to the mass of the black hole that has evaporated away. When I drop the photons back in the black hole, The mass of the black hole is equal to the original mass that we started off with. Which is why i thought it could be perpetual. This can continue for ever. Of course I know something is wrong with such a setup. I think It can be traced to my assumption about photographic being massless. My understanding of black hole thermodynamics is pretty elementary. Perhaps some one could comment on what is going on?


title: A water balloon in vacuum: does it boil?

text: If I put water in a vacuum it will boil. But what if I put this water inside a balloon ? I searched for answers and fount this video: http://www.youtube.com/watch?v=9q8F3ClUuV0 It seams that the water is not boiling. But I am still not satisfied. The vacuum on the video could be just not enough to make the water boil at its temperature. Or it might be because of the pressure of the balloon. If I heat up a water balloon over 100°C at atmospheric pressure, it will boil and grow. So why not at 20°C in a vacuum ?


title: Electric field at a point $P$

text: > **Problem**   >  Charge is distributed uniformly over a large square of side $l$, as shown > in the figure. The charge per unit area ($C/m^2$) is $\sigma$. Determine the > electric field at a point $P$ a distance $z$ above the center of the plane, > in the limit that $z<<l$.   >   >  ![Figure 1](http://i.stack.imgur.com/GP4JG.jpg) Assuming that $dQ=\sigma dxdy$ I said that \begin{align} E_z=\dfrac{1}{4\pi \epsilon_0}\int\limits_{-l/2}^{l/2}\int\limits_{-l/2}^{l/2}\dfrac{dQ}{r^2}\cos\theta&=\dfrac{\sigma}{4\pi \epsilon_0}\int\limits_{-l/2}^{l/2}\int\limits_{-l/2}^{l/2}\dfrac{1}{x^2+y^2+z^2}\dfrac{z}{\sqrt{x^2+y^2+z^2}}dxdy\\\ &=\dfrac{\sigma z}{4\pi \epsilon_0}\int\limits_{-l/2}^{l/2}\int\limits_{-l/2}^{l/2}\dfrac{1}{(x^2+y^2+z^2)^{3/2}}dxdy. \tag{1}\end{align} To begin with I think that $E_x$ and $E_y$ are 0, but I dont know if $(1)$ is correct or not. If it is correct, how can I calculate this integral. And is there an alternative method for calculating the electric field at point $P$?( maybe by integrating only in one dimension $dy$?)


title: Are there "gaps" in light, or will it hit everywhere?

text: Not sure how to word my question. Picture a light source in vacuum, so nothing disturbs the light (or similar conditions), 2d.   If I move very, very far away, will it happen that some of the light hits right beside me (both left and right), but not exactly at where I am?   Seeing as there will be a fixed number of photons sent away from the light, and a huge circumference for them to spread, they cannot possibly hit every point of the circle? If it was a person kicking footballs in all directions instead of the light, the footballs would hit everywhere on a circle with small radius (red circle). If the radius is bigger, the distance between the balls traveling will be greater, and there will be gaps between the balls where I can stand without being hit (red arc). ![footballs in space](http://i.stack.imgur.com/OQySk.png) I'm unsure how this would work with light, being both like footballs but also a wave.   Thanks


title: Collimating and homogenizing a divergent tunable-spectrum light source

text: I have a light source (Xenon lamp) which has the possibility to be tuned over spectral range of 390 - 740 nm with 1 nm spectral accuracy . Bandwidth at each wavelength is about 20 nm . Output intensity is 350 mW full spectrum at output of 5 mm liquid light guide (1.5mW/nm typical , 1.8 mW/nm peak)   Output light is divergent and to some extent not coherent. What I'm looking for actually is to illuminate a few square cm (1-3) of surface with a well **collimated** and **homogenous** light coming from mentioned light source.(power is not of importance since several micro Watts would be sufficient) Which setup can I utilize with lab instruments like pinholes , lenses and ...?   Which task should be done 1st (collimation or homogenization)(Or may be it can be done simoltanously)?   Do you have any suggestion what is the best way to remain in good collimation and homogeneity with proposed setup when we sweep over the spectrum ?   (Both experimental as well as theoretical approaches are of interest)


title: Symmetry in resistor circuits

text: > Given 6 points that are connected with each other with a resistor of > resistance $R$, find the resistance between **any two points**. _(Answer: > $R/3$)_ ![Illustration of the problem](http://i.stack.imgur.com/nltIP.png) (All the conducting wires have the same resistance $R$.) I know that such a wording immediately implies that these 6 points are absolutely identical, which makes it possible to apply symmetry arguments that will help in reducing the network to a simpler one. That is, after choosing any two points in the network, the remaining four points will still be identical, so we can swap any of them and the network will remain the same. Thus, we can remove the resistors which are connected between these other four points, since the points are identical. However, we can _also_ swap the two chosen points, and the system will still remain the same. **So why can't we also remove the resistor between the two chosen points?** I'm told of the following analogy: The system of these 6 points is like a system of 6 absolutely similar balls painted, say, in white. By choosing two points, we paint them in black, thus the system loses some symmetry level, but its certain elements are still symmetrical under certain rearrangements. Specifically, any two of the white balls can be swapped without changing the system in any way, so all the white balls are identical and we can ignore any resistors between them. But swapping the two _black_ balls still won't change the system, so why can't we follow the same logic and ignore the resistor between them as well? I'll generalize the question a little bit: **why don't we care about other symmetries in the system?** (I'm looking forward a simple explanation, without involving advanced math, as I'm just a self-taught beginner and I'm only familiar with calculus. So I try to avoid matrices and anything advanced which students learn in advanced courses of electronics. I just want to get the idea and the concept itself.)


title: How can we calculate (relativisticly correct) ratio $\lambda_e/\lambda_p$ if proton and electron have same speeds

text: This question is about group/phase velocities and also De Brogilie wavelength. What I would like to know is how to derive ratio $\lambda_e/\lambda_p$ ($\lambda_e$ and $\lambda_p$ are De Broglie wavelength for electron and proton) if we know that electron and proton have same velocities? I know that when we say "velocity" $v$, this velocity is the same as "group velocity" $v_g$. So for start I can calculate ratio for group velocities: \begin{align} \boxed{v_e = v_p} \longrightarrow \frac{v_{ge}}{v_{ge}} = \frac{v_e}{v_p} = 1 \end{align} and similarly I can do for phase velocities $v_p$: \begin{align} \frac{v_{pe}}{v_{pp}} = \dfrac{\tfrac{c^2}{v_{ge}}}{\tfrac{c^2}{v_{gp}}} = \frac{v_{gp}}{v_{ge}} = 1 \end{align} But when i try to calculate the ratio for wavelengths I got stuck: \begin{align} \frac{\lambda_{e}}{\lambda_{p}} = \frac{\tfrac{h}{p_e}}{\tfrac{h}{p_p}} = \frac{p_p}{p_e} = ~\longleftarrow \substack{\text{I got stuck here where i don't know}\\\\\text{how to use relation $\boxed{v_p = v_e}$}} \end{align} Can anyone give me a hint? I was trying to use relation $p = \frac{\sqrt{{E_k}^2 + 2E_0E_k}}{c}$ but I got lost...


title: How to heat swimming pool with physics principles

text: I'd like to apply physics principles to a home-experiment relating to my swimming pool. The water should flow inner to 100 meters of black pipe and so it should become warm. Then it should return in the swimming pool. But how can I carry the water from the swimming pool to the pipe? if I use a tight pipe, for the capillarity and putting the pipe vertically from the bottom of the swimming pool, the water should come up.. but it has to pass about 30 cm... is it possible to pass a similar hight using capillarity? (the pipe should be 30 cm over the water level) Which should the pipe diameter be? If the idea is impracticable, which principle can I use?


title: Why is infrared imagery of Mars' surface sharper / richer in contrast than that from the visible spectrum?

text: Looking at the terrain of Mars through Google Mars, it seems to me that the infrared imagery is much crisper, the images from the visible spectrum seem to be somewhat hazy. I'm wondering if that is just an artifact of how the images are made, or if, maybe, the Martian surface really _is_ more blurry in the visible spectrum. Just to give an example of what kind of effect I could imagine, let's say there's fine dust blowing about that is invisible in infrared, so that in that spectrum, you see more of the jagged, more pronounced features in the rocks below.


In electrostatics total flux linked from the closed surface enclosing the charge is equal to $Q/\varepsilon_0$. This is according to Gauss Law. Is this the experimental value or defined value. If experimental what type of experiment is done by Gauss? Why this value is chosen?


title: Resistance of a cylinder contacted by two smaller circular faces

text: Suppose we have a solid homogeneous cylinder with radius $a$, heigth $h$ and conductivity $\sigma$. The top and the bottom face is contacted with a smaller circular face with radius $b$. How can I calculate the resistance? I know the two boundary cases:   * if $a = b$, then $R = \rho \cdot \frac{h}{A}$ simply $\propto \frac{heigth}{Area}$   * if $b = 0$, then $R = \infty$ because of the singularity of both points. I also know how to calculate the resistance of a truncated cone: Divide cone into slice and sum up the resistances $d R$ of all the slices along the heigth. Also works with any other $r(h)$ dependency. But I do not know what happens, when the contact face is smaller than the face of the body. I guess if $b \gg h$, we can approximate, that the electric field for $r > b$ is $0$. Like the approximation for a parallel plate capacitor. Then the material outside the cylinder with radius $b$ would be irrelevant and the resistance $R = \rho \cdot \frac{h}{b^2 \pi}$


title: Orthochronous Lorentz transformations are time-preserving and $SL(2,\mathbb{R})$

text: Let's consider the psuedosphere/hyperboloid in $\mathbb{R}^{1,2}$ given by $$x^2+y^2-z^2=-R^2.$$ We know that the Lorentz group $$O(1,2)=\\{ A \in Mat(3,\mathbb{R}): A^tGA=G \\},$$ where $G=diag(-1,-1,1)$ leaves the pseudosphere invariant. Now we are interested in the following facts:   1. How can we show that the orthochronous Lorentz group $O_+(1,2)=\\{ A: a_{33}>0 \\}$ is subgroup and, more important, maps upper cone to upper cone?   2. What is the relation between groups $O_+(1,2)$ and $SL(2,\mathbb{R})$?


title: When we take time derivative of a function of time, then is the result another function of time, again?

text: (I'll try to explain my question by one known example), for example where the velocity is a function of time v(t) then its time derivative (which is acceleration: $a=\frac {dv}{dt}$) is another function of time a(t)? (because according to defination of integral it must be another function of time: $v=\int a(t)dt$) and then what will be time derivative of acceleration?


title: Deriving the Schwarzschild solution

text: I am CS student and i want to create graphical simulation of black hole, so I need to use The Schwarzschild solution to calculate possible coordinates of given body every second. First try in 2D space+ time. So, my question is how using Schwarzschild solution I can calculate Cones like in this picture: ![enter image description here](http://i.stack.imgur.com/3ej3Y.gif)


title: Uniform load applied to a parabolic curve

text: As I have to design the vertebra bone and its natural boundary conditions I came across a problem. How can I applied a uniform load if the place where the force is applied is a parabolic curve. Ok so I get the curve will have this value $$y(x) = -y_1/x_1 \cdot x^2 + y_1$$ So I did the derivative $\frac{dy}{dx}$ to discover the tangent of the curve so it can be possible to calculate the normal force so I can apply the normal force to the surface. My question is how is it applied to the surface?


title: Is heat flux density and heat flux the same thing?

text: Heat flux and heat flux density is the same thing, while electric flux density and electric flux is not the same thing? It makes me confused since we compare Fourier's law with Ohm's law. Here is a statement from Wikipedia. > To define the heat flux at a certain point in space, one takes the limiting > case where the size of the surface becomes infinitesimally small. Is heat flux defined at a point or on a surface? I have never been found any defintion of heat flux or heat flux density. As a mathematical concept, flux is represented by the surface integral of a vector field, $$\Phi_F = \iint_A \mathbf{F}\cdot \mathrm{d}\mathbf{A}$$ where $\mathbf{F}$ is a vector field, and $\mathrm{d}\mathbf{A}$ is the vector area of the surface $A$, directed as the surface normal. Heat is often denoted $\vec{\phi_q}$ and we integrated the heat flux density $\vec{\phi_q}$ over the surface of the system to have the heat rate but we integrated the $\mathbf{E}$-filed to get the electric flux? Thanks.


title: Confusion in connected pulleys problem

text: I'm having some troubles in understanding why my reasoning doesn't work in the following problem: **Problem** Two pulleys of mass $m_1$,$m_2$ and radius $r_1,r_2$ are connected by a belt (like chain rings in a bycicle) of negligible mass. Torques $\Gamma _1$ and $\Gamma_2$ are applied to $m_1$ and $m_1$. The belt is not slipping over the two pulleys. What is the angular acceleration of 1? I've already solved the problem writing down the rotational equations for the two pulleys with the constraint $\omega _1 r_1 =\omega _2 r_2$ given by the non-slipping condition. What I got is:$$\alpha _1=2\dfrac{\frac{\Gamma _1}{r_1}+\frac{\Gamma _2}{r_2}}{r_1(m_1+m_2)},$$ which is correct, according to my book. Then I've tried to attack the problem from a different point of view. Since the tensions in the belt are internal forces, the only torques producing a change in total angular momentum are the two torques applied to the two axis of the pulleys. At first my idea was to write:$$\dot L = \Gamma _1+ \Gamma _2,$$since the angular momentum here is independent from the pole, the solve for $\dot \omega _1$ ($\dot L=k\dot \omega_ 1$ for some constant). However, the two values $\Gamma _1$ and $\Gamma _2$ are the torques referred to the center of the respective pulleys and this approach doesn't work. In fact the derivative of $L$ is really:$$\dot L=\dot L_1+\dot L_2 = \Gamma _1+(T_1-T_2)r_1+\Gamma_2+(T_2-T_1) r_2,$$where $T_1,T_2$ are the two tensions in the belt and the angular momentums are respect the center of mass of 1 and 2. So the sentence that I wrote above _"Since the tensions... internal forces..."_ seems to be wrong. Could you please point out the wrong parts of my reasonings? Also, how could I write the two torques $\Gamma _1$ and $\Gamma _2$ respect to a different pole? Finally, is there a different way from the (working) one I wrote above to solve this problem?


Bohm, David (1980), Wholeness and the Implicate Order, London: Routledge, ISBN 0-7100-0971-2 As we can see from the book above, David created the hypothesis of the holopraphic universe in 1980. And we can see from the words below, the holographic universe of string theory is created in 1995. Using this approach he has argued that near a black hole, quantum fields could be described by a theory in a lower dimension.[Stephens, C. R.; 't Hooft, G.; Whiting, B. F. (1994). "Black hole evaporation without information loss". Classical and Quantum Gravity 11 (3): 621] This led to introduction of the holographic principle by him and Leonard Susskind.[Susskind, L. (1995). "The world as a hologram". Journal of Mathematical Physics 36 (11): 6377–6371.] So Whether the holographic universe of string theory prove the hypothesis of the holographic universe created by David Bohm or not? Or what is the link or differences between them? I have read the book of "The Elegant Universe" and the book of "The Fabric of the Cosmos:Spaca,Time, and the Texture of Reality", but Brian Greene doesn't mention about David Bohm. Thank you in advance!


title: How could flux can be a vector and a scalar?

text: Here is the General mathematical definition of Flux on Wikipedia: > The frequent symbol is $j$, and a definition for scalar flux of physical > quantity $q$ is the limit: $$j=\lim\limits_{A\to > 0}\cfrac{I}{A}=\cfrac{\mathrm{d}I}{\mathrm{d}A}$$ where: > $$I=\lim\limits_{\Delta t\to 0}\cfrac{\Delta q}{\Delta > t}=\cfrac{\mathrm{d}q}{\mathrm{d}t}$$ is the flow of quantity $q$ per unit > time $t$, and $A$ is the area through which theh quantity flows.For vector > flux, the surface intergal of $j$ over surface S, followed by an integral > over the time duration $t_1$ to $t_2$, gives the total amount of the > property flowing through the surface in that time $(t_2 − t_1)$: > $$q=\int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot\hat{\mathbf{n}}\, > \mathrm{d}A\mathrm{d}t$$ Although I have already know the relationship between $\mathrm{d}A$ and $\mathrm{d}\mathbf{A}$. However, what's the relationship between $j$ and $\mathbf{j}$? What's the definition of $\mathbf{j}$? I ask this question because of this: problem. I don't solve my problem from the two answers. I am not sure about the definition of heat flux. It is necessary for you to write rigorous description to this question.


title: How dangerous is firing a bullet straight up in to the air?

text: Question's title says it all how dangerous are the bullets which are fired up straight in to the air? I often find myself in this discussion where there are people who argues that it is harmless and some say it can be lethal at times. I want to clear my understanding about it to settle this issue once in for all.


title: Why does diffraction depend on wavelength?

text: We've all heard of the diffraction of radio waves over a mountain and the diffraction of water waves through a gap, but why does this effect depend on wavelength? I'm looking for as simple answer as possible - if it's a bit hand-wavey that might be OK. Another thing that troubles me is when I read the effect of diffraction depends on the size of the wavelength compared to the gap/ obstacle. But, how do you define the size of the obstacle if it's an edge, e.g. the tip of a mountain, the edge of a wall, the edge of a razor blade?


title: Connection between first and second quantization

text: This is my question: In a book on many body quantum theory I came across equality between antisymmetrized many-particle state vector which, as you know, includes sum over permutations of product states of one particle states AND collection of creation operators acting upon vacuum state. Now, I know that this product of creation operators is defined to be   anti- symmetric but how can you get it to look like a sum, which is implied when antisymmetrization operator acts on product space? Is it i mistake? Did they maybe wanted to write a product of field operators? Thank you.


title: Latent heat of vaporization

text: The molecular weight of water is 18.015 gram. The number of moles of water in one liter (1000 gram) will be: $3.34\times 10^{25}$ molecules (in 1kg). We know that latent heat of vaporization of water is $L_v= 2.26\times 10^6$ then the amount of heat required to rise one molecule of water will be $1.47\times 10^{-19}$ joule: Does this calculation is correct according to macroscopic and microscopic physics?


title: Deriving the action and the Lagrangian for a free point particle in Special Relativity

text: My question relates to > Landau & Lifshitz, _Classical Theory of Field,_ Chapter 2: _Relativistic > Mechanics,_ Paragraph 8: _The principle of least action._ As stated there, to determine the action integral for a free material particle, we note that this integral must not depend on our choice of reference system, that is, must be invariant under Lorenz transformations. Then it follows that it must depend on a scalar. Furthermore, it is clear that the integrand must be a differential of first order. But the only scalar of this kind that one can construct for a free particle is the interval $ds$, or $ads$, where a is some constant. So for a free particle the action must have the form $$ S = -a\int_a^b ds\,. $$ where $\int_a^b$ is an integral along the world line of a particle between the two particular events of the arrival of the particle at the initial position and at the final position at definite times $t1$ and $t2$, i.e. between two given world points; and $a$ is some constant characterizing the particle. For me this statements are inaccurate. (Maybe it's because I have few knowledge from maths and physics. However.)   1. Why should the action be invariant under Lorentz transformations? Is this a postulate or it's known from experiments. If this invariance follows from special theory of relativity, than how? Why the action should have the same value in all inertial frames? The analogue of the action in non-relativistic Newtonian mechanics is not invariant under Galilean transformations, if I am not mistaken. See e.g. this Phys.SE post.   2. It is stated "But the only scalar of this kind that one can construct for a free particle is the interval." Why? Can't the Lagrangian be for example $$ S = -a\int_a^b x^i x_i ds\,, $$ which is also invariant. The derivation of the Lagrangian for a non-relativistic free point particle was more detailed I think. See e.g. this Phys.SE post. Does the relativistic case need some more detalization?


title: Understanding Tensors

text: I don't seem to be able to visualize tensors. I am reading The Morgan Kauffman Game Physics Engine Development and he uses tensors to represent aerodynamics but he doesn't explain them so I am not really able to visualize them. Please explain in very simple ways. I just want to understand the basics.


title: Dirac operator Feynman propagator

text: Is it true that the following identity holds for the Feynman prescription Dirac propagator: $$ S_F(x) \stackrel{?}{=} \gamma^0[S_F(-x)]^\dagger\gamma^0 $$ where $S_F$ is defined as the Green's function: $$ (i\gamma^\mu\partial_\mu-m)S_F(x-y)=i\delta(x-y) $$ This is somewhat related to a previous question of mine: Green's function for adjoint Dirac Equation If the statement is true, how do you prove it?


title: Size of universe after inflation?

text: Wikipedia states the period of inflation was from $10^{-36}$sec to around $10^{-33}$sec or $10^{-32}$sec after Big Bang, but it doesn't say what the size of the universe was when inflation ended. Just saw a Brian Greene show on the Multiverse and I thought I heard him say size was galactic scales when inflation ended. However I've also read size was about a basketball. Are there multiple theories with different resulting sizes? Does 'size' even mean anything in this context?


Can a $TEM_{00}$ beam incident on a confocal cavity along its optical axis excite modes other than $TEM_{00}$ in the cavity? If not, is the FSR of the confocal cavity $\frac{c}{2L}$ in this case, instead of the oft-quoted $FSR=\frac{c}{4L}$?


title: Epsilon Tensor in FeynCalc

text: A few days ago I started to use the Mathematica package FeynCalc and one thing confuses me: Assume we have a four-vector $p_\mu$ and we contract it with the epsilon tensor. FeynCalc produces $\varepsilon^{\mu\nu\rho\sigma} p_\sigma = \varepsilon^{\mu\nu\rho p}$...what does the momentum as an index mean? Furthermore, the FeynCalc documentation says "Eps[a, b, c, d] is the head of the totally antisymmetric $\varepsilon$ (Levi-Civita) tensor. The a,b,... may have head LorentzIndex, Momentum or Integer." What is an epsilon tensor with a momentum index? I never saw this before.


title: Notations for statistical / systematic / numeric errors?

text: I constantly see the notation $$ 5.143(13) $$ for specifying that a value was measures / calculated to be 5.143 with an estimated error of 0.013. I have come to wonder though, just how commonly accepted the notation and variants are. After all, if it is commonly accepted, why have I found some literature explicitly stating what it means (though it might just have been distinguishing systematic and statistical errors). Additionally I'm not sure it would still work for something like $$ 513(30) \equiv 513 \pm 30 $$ or $$ 51.3(30) \equiv 51.3 \pm 3.0 $$ Is this notation a commonly accepted convention, that can be reasonably expected still to be understood by people reading a paper written now in 50 years? Is it understood in cases like those above?


title: Symplectic integrators of the pendulum equation?

text: In particular, a symplectic integrator to solve: $$\ddot{\theta} + \dfrac{g}{l} \sin(\theta) = 0.$$ I'm currently using velocity verlet - by realizing that $$\ddot{\theta} = -\nabla (-cos(\theta)) = A(\theta(t)),$$ ie. letting $$x = \theta$$ $$v = d \theta/dt$$ $$a = d^2 \theta /dt^2.$$ Is it safe to apply Verlet integration to generalized coordinates? In particular, does this hold true for a generalized coordinate theta: $$\theta_{t+dt} \approx \theta_t + \dot{\theta}_t dt + \frac{1}{2} \ddot{\theta}_t (dt)^2?$$


title: What are field quanta?

text: Just assume that I understand that a field in quantum field theory is an operator-valued distribution. For simplicity, forget about the distribution and think about a function $\varphi:M \rightarrow L(H)$ that assigns an operator to each point of spacetime. Can someone explain to me what (mathematically speaking) physicists mean by "quanta" of this field? EDIT: (follow up question): if one fixes an inertial system and a point $t_0$ in time (say 4:00pm), then there is "space" and a map $a^\dagger:\mathbb R^3 \rightarrow L(H) \rightarrow H$ $(x,y,z)\mapsto a^\dagger(t_0,x,y,z) \mapsto a^\dagger(t_0,x,y,z) |0>$ , which creates "quanta". Would it be correct to think about "particles at the point (x,y,z)" then one thinks about this map?


title: Does a form of energy exists without any medium?

text: Sun contains fire,is it?Or some pure form of energy?If space has no air how can fire exist there,it may exist cause a burning piece of magnesium keep on burning in vaccum but,to start fire,oxygen is must.


title: What is planetary surface temperature given constant sub-surface temperature?

text: If a planet of radius R1 has a constant sub-surface temperature T0 at R0 < R1, what is the long-term equilibrium surface temperature T1? Say we assume constant thermal diffusivity of the planet material, surface emissivity 1.0, no atmosphere, and no incoming radiation. I figure the temperature profile is harmonic, i.e. T=a+b/R, for constants a and b, and that we can use (R0,T0) to eliminate one of these constants. Is there enough information to obtain the other constant?


title: (Un)countability in QFT

text: I am a mathematician self-studying physics, and a currently working on QFT with Srednicki's book. One thing that bothers me is that for a scalar field (in the Hamiltonian version) there is a harmonic oscillator at every point of momentum space. These all commute with each other, which means that they can be observed simultaneously. This is an uncountable number of harmonic oscillators. If a classical field is nonzero, then it is nonzero at uncountably many places. So it seems to me that a generic measurement of the field will yield uncountably many points with non-zero energy, corresponding to uncountably many bosons. However, QFT seems to treat a field as having a countable number of particles. Each Feynmann diagram only involves a finite number of particles, and the LSZ reduction formula deals with a finite number of incoming/outgoing particles. So my question is, **Will a generic field state have countably many particles or uncountably many particles, and how does this reduce to a classical field in the classical limit?**


title: The contribution to mass from the dynamical breaking of chiral symmetry

text: The claim is often made that the discovery of the Higgs boson will give us information about the origin of mass. However, the bare masses of the up and down quarks are only around 5 MeV, quite a bit smaller than their "constituent" or "dynamical" mass of around 300 MeV. (Remember that a neutron, for example, is one up and two down quarks and has a total mass of 939 MeV.) What then, is the reasoning behind the claim that the Higgs will address the origin of mass when by far the majority of the mass of the neutron (and proton) is related instead to the dynamical breaking of chiral symmetry?


title: Why do regular lattice elements heat up faster?

text: For example iron. A metal spoon heats up much quicker than a wooden/plastic one. Why?