How does the internuclear repulsion vary in Hydrogenic atom collision?

Hydrogen fusion requires two hydrogen nuclei to get close enough (typically a few fm) to fuse. Much of the problem of creating a fusion reactor is overcoming the Coulomb repulsion between a pair of nuclei - the millions of degrees for Maxwellian distributions, the Bremstrahlung losses for inertial confinement. If we could align the paths of two neutral Hydrogen **atoms** (of whichever isotopes), what would the repulsion look like between them as they approach collision? Obviously at long range there is negligable force as both are neutral. But as they approach each other, what happens to the electron distribution? Intuitively, I expect a bonding cloud to form between the nuclei, and antibonding clouds beyond them. This would presumably attract at first until reaching the usual Hydrogen covalent bond length, after which the internuclear repulsion would increasingly dominate. But how does that compare to bare ionic collision? How much lower is the potential barrier? Obviously if it was significantly lower and we could somehow engineer the collision to achieve fusion, the cross section would be larger than ionic fusion, but how much? Or would the barrier be just as high over the final few femtometers?


The quantized energy level E depends on which power of n?

A particle in one dimension moves under the influence of a potential $V(x)= ax^6$, where $a$ is a real constant. For large $n$, what is the form of the dependence of the energy $E$ on $n$?


What interpretive difference is there between defining a function with or without a differential as a postfactor?

I have thought about this and looked for answers for a long time now, but I do not have any name or label for this problem, which is the reason for the long title of this question. I have come across this many times, but I have never found any rationale for this, i.e. a clear interpretation or explanation, so here are a few examples of what I mean. I suspect that they have different answers. I) In e.g. Shapiro and Teukolsy's book _Black Holes, White Dwarfs and Neutron Stars_ , (1983), the Salpeter birthrate function for stars is defined as $\psi_s\mathrm{d}\left(\frac{M}{M_\odot}\right) = 2\times10^{-12}\left(\frac{M}{M_\odot}\right)^{-2.35}\mathrm{d}\left(\frac{M}{M_\odot}\right)\mathrm{\ stars\ pc}^{-3}\mathrm{\ yr}^{-1}$ Questions: why does it matter that those differentials are included? Why make the statement look messier by multiplying both sides of the equation with something that obviously cancels out? If it is just to show what $\psi_s$ depends on, then it makes mre sense to me to define the function like so: $\psi_s(M) = 2\times10^{-12}\left(\frac{M}{M_\odot}\right)^{-2.35}\mathrm{\ stars\ pc}^{-3}\mathrm{\ yr}^{-1}$ What is (or may be) the answer? II) When defining the equation of continuity for brownian motion particles in 1D space, I have a lecture note that reasons in the following way: Given the setup                        |  n(x,t) |              |         |         J(x,t) ---->     ----> J(x + dx, t)              |         |              |         |         ---------------------->              x        x+dx      Where $n(x,t)$ is the number density of particles between positions $x$ and $x + \mathrm{d}x$ and $J(x, t)$ is the flux of these particles at given position $x$ and time $t$. The particle number is conserved, and they are assumed identical and non-interacting with each other. In the notes that I have, the following equation is readily displayed as though it was crystal clear why it was written inthis way: $n(x, t + \mathrm{d}t)\mathrm{d}x - n(x, t)\mathrm{d}x = - (J(x + \mathrm{d}x, t)\mathrm{d}t - J(x, t)\mathrm{d}t)$ or (multiplying the minus into the parenthesis) $n(x, t + \mathrm{d}t)\mathrm{d}x - n(x, t)\mathrm{d}x = J(x, t)\mathrm{d}t - J(x + \mathrm{d}x, t)\mathrm{d}t$ Question: I can not myself justify why it makes sense to multiply with the diferentials as they are here. What is the significance, or rather the line of thought that lies behind the way this equation turns out? I am unsure of my own answer which is only to look at the dimensions of the two functions themselves and then conclude what unit the factor should have on each side in order for the dimensions to make sense. Just to complete the derivation, rewriting the equation by replacing the functions $n(x, t + \mathrm{d}t)$ and $J(x + \mathrm{d}x, t)$ with their Taylor expansion (to first order) gives $n(x, t + \mathrm{d}t) = n(x, t) + \frac{\partial n}{\partial t}\mathrm{d}t$, $J(x + \mathrm{d}x, t) = J(x, t) + \frac{\partial J}{\partial x}\mathrm{d}x$, which, when inserted into the original equation gives $\frac{\partial n}{\partial t}\mathrm{d}t\mathrm{d}x = -\frac{\partial J}{\partial x}\mathrm{d}x\mathrm{d}t$ so producing, again, a shared factor ($\mathrm{d}t\mathrm{d}x$) on both sides of the equation which can just be divided away (right?), and we end with the coninuity equation $\frac{\partial n}{\partial t} = -\frac{\partial J}{\partial x}$ I hope this question makes sense or at least allows you to suggest a label for this kind of problem that I can use to find more information, Or better, help by answering the wuestion here. Cheers.


ODE envolving current

I have to do this exercise: > 9\. The current $I(t)$ at time $t$ flowing in an electric circuit obeys the > differential equation > > $$I''(t) + RI'(t) + I(t) = \sin \omega t$$ > > where $R$ and $\omega$ are positive constants. The solution can be expressed > in the form $I(t) = F(t) + A\sin(\omega t + a)$, where $F(t)\to 0$ as $t\to > +\infty$, and $A$ and $\alpha$ are constants depending on $R$ and $\omega$, > with $A > 0$. If there is a value of $\omega$ which makes $A$ as large as > possible, then $\omega/(2\pi)$ is called a _resonance frequency_ of the > circuit. > > (a) Find all resonance frequencies when $R = 1$.   >  (b) Find those values of $R$ for which the circuit will have a resonance > frequency. I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega t)+B\cos(\omega t)$. I got the first and second derivates $$I_{p}'=A\omega\cos(\omega t)-B\omega\sin(\omega t)$$ $$I_{p}''=-A\omega^{2}\sin(\omega t)-B\omega^{2}\cos(\omega t)$$ then, when I substitute in the equation $$I''(t)+I'(t)+I(t)=\sin(\omega t)$$ I got the system on $(A,B)$: $$\left\\{\begin{array}{l} -A\omega^{2}-B\omega+A=1 \\\ -B\omega^{2}+A\omega+B=0 \end{array}\right.$$ so $$A=\frac{(1-\omega^{2})}{(1-\omega^{2})^{2}+\omega^{2}}$$ I calculated $A$ because it follows $\sin(\omega t)$ and with this I can find all resonance frequencies. Well, to find them, I gotta find the values of $\omega$ which makes $A$ as large as possible. To $A\to+\infty$, so, its denominator must goes to 0: $$(1-\omega^{2})^{2}+\omega^{2}\to0$$ But that's not possible. The answer is $\displaystyle\frac{1}{2\pi\sqrt{2}}$. Comparing with the exercise, I know that $\omega=1/\sqrt{2}$. If the denominator was $(1-\omega^{2})^{2}-\omega^{2}$, I got the result, but I don't know what's wrong here. **EDIT:** I found after $$I_{p}(t)=\frac{1}{\sqrt{(1-\omega^{2})^{2}+\omega^{2}}}\sin(\omega t-\alpha)$$ where $\alpha=\arctan(\omega/(1-\omega^{2})$. So, $$A=\frac{1}{\sqrt{(1-\omega^{2})^{2}+\omega^{2}}}$$ and it's large just if its denominator is small... but how?!


Determining the limits of an integral

In Griffiths' _Introduction to Electrodynamics_ , Problem 1.28 (the triangular prism question) is especially challenging for me. I do not know how the limits of x are **0** to ( **1-y** ). My concern is the latter limit, **1-y**. Can you explain systemically how this is arrived at? On the face of it, I'm inclined to say the limits of x are **0** and **1**. I can see it's the diagonal across the x- and y-axes that's presenting the extra challenge, but I do not know how to go about establishing that indeed the limit is **1-y**.


What would happen in a collision of an antimatter singularty and a matter singularity?

Would energy be released if a black hole made out of antimatter and another of matter were to collide?


Is it part of special relativity that mass possessing energy is more dense?

I was reading http://www.edge.org/3rd_culture/hillis/hillis_p2.html and it says that a charged battery weighs more than a dead one or a rotating object weighs more than a stationary one (i.e. mass containing energy weighs more than mass that doesn't). Is this true in special relativity? Because the energy would be confined to the area of the original mass, wouldn't it become more dense if it is true? Also, if the mass we are speaking of does become more gravity-sensitive on energy gain, does it actually gain physical mass (protons, electrons and neutrons)??


Drag force on a cone

I was asked to calculate the drag force on a cone with velocity 10 m/s , everything was okay until i needed to calculate the cross sectional area , the radius of the base was 0.5 m , radius of the top 0.0005 m , given that the cone falls top first, which one should i use ? should i get some kind of an average?


Is $k_B \rightarrow 0$ the classical limit of stat. mech., as $\hbar \rightarrow 0$ is in QM?

I hear very often among my peers and seniors that just as how $\hbar\rightarrow0$ takes me to classical mechanics from quantum mechanics, $k_B\rightarrow0$ will take me to classical thermodynamics from statistical mechanics. As nice as this sounds, my gut feeling and intuition tells me this is not the right analogy. I think $N\rightarrow\infty$, the ensemble size of statistical mechanical system, is closer to $\hbar\rightarrow0$ in that respect. Is this correct? If so, what role does $k_B$ play then?


Asymptotic Invariants in General Relativity

I was trying to understand Witten's proof of the Positive Energy Theorem in General Relativity by reading the original argument given by Witten. I am comfortable with the overall argument, but I would like to understand the following statement, made on the second paragraph of page 394 in the previous link: > "The only invariants that can be formed from the $1/r$ term in the metric > tensor are the total energy and the total momentum." These invariants (which I will call "$1/r$-built" for short) refer to an asymptotically flat initial value (spacelike smooth) 3-surface in spacetime. Is there an obvious reason why this should be true? Looking at the definition of the ADM-energy and momentum, it seems plausible because the only data we have are the first and second fundamental form, and it should be possible to write any invariant as a combination of these, and consequently, any $1/r$-built invariant as a function of ADM energy and momentum. However, this reasoning is too hand-wavy, so I wonder if there is a clear cut explanation. My interest in this fact is that although it is not really a logical step in the proof, if true it is probably one of the best ways to motivate a spinorial proof (I was thinking of something along the lines of "if we can construct a manifestly non-negative $1/r$-built invariant by means of the asymptotic behavior of spinors, then it should be possible to prove that energy is non- negative by writing this invariant as a function of ADM energy and momentum"). From a purely physical perspective, if it were true that ADM energy and momentum suffice to specify the system (to order $1/r$), then they should be the only independent invariants. However, this suggests that an asymptotic observer who knew the total energy and momentum could reconstruct the metric up to order $1/r$. I was thinking if it is not possible to construct a counterexample by defining an axisymmetric spacetime in which the Killing field of azimuthal symmetry is build up only with $1/r$ terms and showing that it is asymptotically distinguishable from its non-rotating counterpart. In this spirit, I think it is worth asking a broader version of my question: > What kinds of "physically interesting" boundary terms can appear in a > Hamiltonian formulation for an asymptotically flat spacetime manifold in > General Relativity? Thanks in advance.


Is nature symmetric between particles and antiparticles?

Is nature symmetric with respect to presence of particles? Do we have an antiparticle for every particle thought of? Are there any proven examples where we don't have an antiparticle? And what about antiparticle of a photon (we know it can also behave as a particle)?


Microscopy types and techniques?

I didn't get the basic difference in between the different types of microscopy. for example there are several different microscopy techniques are available such as Bright field, Dark field, Confocal, STED, electron microscopy etc. Is only the illumination system varies in between different techniques or are there any other parameters?


Spinning bucket of water in zero gravity

Everyone knows how the surface of a spinning bucket of water would look like on earth - parabolic. But what if we turned off gravity (for instance by doing the experiment in a freely falling lift)? Would the surface be still parabolic? I'll explain my confusion in more detail. The velocity of the spinning bucket is transferred to the water by means of frictional forces arising in the boundary between the bucket and water. But these frictional forces exist no matter whether there is gravity or not. So if I consider the whole bulk of water inside the bucket as a single system, this frictional force would give it a positive torque. Thus the water has to rotate. For the sustained rotation of water, a centripetal force has to exist. In normal gravity, the water surface changes its shape into a paraboloid so that there is a net force on any particle directed inward. But in free fall, there is no pressure on a particle inside the liquid. Thus the only force that can supply the centripetal acceleration is inter-molecular force between the particles which is weak to sustain huge velocities. So what exactly happens?


Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i L^i,L_i] = L_i [L^i,L_i]$. So I could not understand this term with upper index. What is $L^i$ ?? Any clues?


Zeroth Law of Thermodynamics, temperature, and ordering

In my thermodynamics course (and in other places on the internet) it is asserted that the Zeroth Law of Thermodynamics can be used to define the concept of temperature. One statement of the Zeroth Law I have seen states that the relation _thermal equilibrium_ on two closed systems brought into diathermal contact defined is in fact an equivalence relation. The argument continues by saying that if we call the equivalence classes so defined _isotherms_ , then we can assign arbitrary numbers to these isotherms and these numbers are what we call _temperature_. We now have a set of numbers assigned to a set of equivalence relations. But what I **don't** see is how this numerical assignment bears any relation to physical temperature. Where is order defined? For example, if one of the classes gets the number "1", another gets the number "2", and a third "3", what in the above derivation shows that the isotherm "2" comes between the isotherm "1" and the isotherm "3"? Maybe more is needed than just the Zeroth Law. If so, what is necessary in order to complete the argument?


Work done by a frictional force on a cylinder

> _A cylinder of mass $m$ and radius $R$, is rolled on surface with > coefficient of kinetic friction $\mu_{k}$ about the axis passing through the > center and parallel to the surface, with initial angular velocity > $\omega_{0}$; the work done by frictional force from start until it begins > to roll w/o slipping is to be found out._ It is easy to see that $a=\mu_{k}g,\alpha=-2\mu_{k}g/R,t=\frac{R\omega_{0}}{3\mu_{k}g},d = \frac{1}{2}at^2 $ are acceleration, angular acceleration, time from beginning to the time of rolling w/o slipping and distance covered. I am somehow missing why I am getting $$W=\int F.ds+\int\tau.d\theta=\mu_{k}mg*d+\mu_{k}mgR*\frac{d}{R}\neq\triangle K.E.=-\frac{1}{6}mR^{2}\omega_{0}^{2}.$$ Please shed some light.


Field due to current in a wire

Suppose a current flows in a straight cylindrical wire so that an electric field $\textbf{E}$ is maintained in the wire. Will there be an electric field just outside the wire..?


Find the acceleration based on V² x d graph

How could I find the acceleration of a body knowing only that its position and velocity satisfy $v^2 = f(x)$, where $f(x)$ is a known function of $x$ (position)?


Particle colliders: why do they need an accelerator chain

Particle colliders like the LHC or the Tevatron use a complex accelerator chain to have particles at a given energy before being accelerated. For example:   * The CERN accelerator complex to inject in the LHC: protons are first accelerated by a LINAC, then by a "booster" synchrotron, then by a larger synchrotron, and finally by a very large synchrotron where they reach 450 GeV/c.   * As was discussed in link text , the chain for the Tevatron They both require "injector" accelerator(s). ![alt text](http://i.stack.imgur.com/PzikJ.gif) My question is simple: why ? Why can't we have a single accelerator with a source and the possibility to accelerate the particles to the top energy. * * * Note: this is also a seeding question, but I am curious how complete and clear an explanation to that can be provided.


Source term of the Einstein field equation

My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's "simplified account of the Einstein field equation of general relativity did need a qualification that he did not quite give." Feynman's intuitive discussion rests on relating the "radius excess" of a sphere to a constant times the enclosed gravitational mass $M$: for a sphere of measured radius $r_{\mathrm{meas}}$ and surface area $A$ enclosing matter with average mass density $\rho$ smoothly distributed throughout the sphere, $$\sqrt{\frac{A}{4π}}−r_{\mathrm{meas}}=\frac{G}{3c^2}\cdot M,$$ wherein $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $M=4π\rho r^3/3$. I don't know what $r$ is supposed to be, but it's presumably $\sqrt{\frac{A}{4\pi}}$. Feynman gratifyingly points out that $\frac{G}{3c^2}\approx 2.5\times 10^{−28}$ meters per kilogram (for Earth, this corresponds to a radius excess of about $1.5$ mm). Feynman is also careful to point out that this is a statement about average curvature. Penrose's criticism is: "the 'active' mass which is the source of gravity is not simply the same as the energy (according to Einstein's $E=mc^2$); instead, this source is the energy plus the sum of the pressures". Damned if I know what that means -- whose pressure on what? So, taking into account Penrose's criticism but maintaining Feynman's intuitive style, what is the active mass $M$?


Cross-section in relativistic limit: Fermi's golden rule still valid?

In order to calculate the cross-section of an interaction process the following formula is often used for first approximations: $$ \sigma = \frac {2\pi} {\hbar\,v_i} \left| M_{fi}\right|^2\varrho\left(E_f\right)\,V $$ $$ M_{fi} = \langle\psi_f|H_{int}|\psi_i\rangle $$ Very often plane waves are assumed for the final state and therefore the density of states is given by $$ \varrho\left(E_f\right) = \frac{\mathrm d n\left(E_f\right)}{\mathrm d E_f} = \frac{4\pi {p_f}^2}{\left(2\pi\hbar\right)^3}\frac V {v_f} $$ I understand the derivation of this equation in the context of the non relativistic Schrödinger equation. But why can I continue to use this formula in the relativistic limit: $v_i, v_f \to c\,,\quad p_f\approx E_f/c$. Very often books simply use this equation with matrix element derived from some relativistic theory, e.g. coupling factors and propagators from the Dirac equation or Electroweak interaction. How is this justified? ### Specific concerns:   * Is Fermi's golden rule still valid in the relativistic limit?   * Doesn't the density of final states has to be adapted in the relativistic limit?


How does pressure factor into mechanics?

We just started learning about pressure in our school, and i am a bit confused. I tried to google extensively, but i didn't really find much. In mechanics, so far in our school we've been taught about Newton's laws of motion and some related stuff, and suddenly we started learning about pressure. I asked this same question to my physics teacher, but he failed to answer. How does pressure actually factor into mechanics? We studied so far that to cause motion, you require a force. Then we study about pressure, and the examples are ones like pressure of gas in a contained beaker and then lowering/raising it's volume. Suddenly in the applications section however there are things like Paper pins have a low surface area at the end to maximize pressure, and that you can't walk on sand easily because the sand depresses under pressure. I understand the math behind it, but what does the math actually mean? I mean, a force is being applied. It should cause motion, but apparantly if the surface area of the end is small then pressure goes up? The force here remained constant(?) so how does pressure factor into the mechanics? Why did that just happen? I realize i may be way off topic and asking a stupid question, but i am having some trouble comprehending how pressure works. I understand the examples of things like pressure in a container, atm. pressure e.t.c but how does pressure transmit in a case like this? My apoligies if this is a stupid question.


Dark and bright areas around atoms in a scanning tunnelling microscope image

Recently IBM created world’s smallest ever animation on an atomic scale video. Researchers made the animation using a scanning tunnelling microscope to move thousands of carbon monoxide molecules to show a boy dancing, throwing a ball and bouncing on a trampoline. My question is, why in this video we see a pattern of dark and bright circles around each molecule? What do they represent?


How we approach RLC circult from RLGC model?

In the text, it introduces a practical model to investigate a transmission line (like BNC cable), it considers the transmission line has resistive $R$, inductance $L$, conductance $G$ and capacitance $C$. The model is illustrated as follow ![enter image description here](http://i.stack.imgur.com/Iaxe3.jpg) It is easy to derive the (telegraph) equations and figure out the impedance Z to be $$Z = \sqrt{\frac{R+iX_L}{G+i/X_C}}$$ where $i$ is the imaginary unit, $\omega$ is the angular frequency, $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance. And in other section, it introduce a RC circuit and RLC circuit, in which, the impedance are $$Z_{RC} = \sqrt{R^2 + X_C^2}, \qquad Z_{RLC}=\sqrt{R^2 + (X_L - X_C)^2}$$ It is pretty confusing because from RLGC model, if we make the electrical conductance $G$ to zero and consider no inductance ($L=0$), so the circuit becomes RC circuit, but from the first equation for the impedance given by the RLGC model, the impedance should be $$Z = \sqrt{-iRX_C}$$ Why are they not the same? How to approach RC and RLC case from RLGC model?


Thought experiment with entangled electrons

Suppose we start out by having two entangled electrons. We separate them by some distance and we put one electron inside a thin loop of wire connected to an extremely sensitive voltage measuring device at lab 1 and the second electron at lab 2. At this point, no measurement is made. Both electron's spin's are undetermined. Therefore... -The direction of the spin is undetermined. -We do not know if the electron's spin, $s=\pm\hbar$. -We do not know its magnetic moment,$ m = (-gu_bS)/\hbar$. $S = \frac{h}{2\pi}\sqrt{s(s+1)}$, $g$ is g factor, $u_b$ is Bohr Magneton. -We do not know the Magnetization, $M = (N/V)m$, where $N$ is number of magnetic moments and $V$ is the volume of the system in question. -We do not know the magnetic field, $B = \mu_0(H + M)$, where $\mu_0$ is vacuum permeability, $H = M/X$, where $X$ is the magnetic susceptibility. The magnetic field of the first electron at lab 1 is undetermined because no measurement has been made, therefore no magnetic field can possibly be present, $B = 0$. (Please correct if wrong) Now, we measure the second electron's spin by sending through a Stern-Gerlach device and having that electron hit a screen to record its spin value, $+$ or $-$, at lab 2. Regardless of whether or not the second electron's spin is up or down, we know that the first electron's spin is now determined. This means that the magnetic field at lab 1 has been determined and therefore a magnetic field must be present. Since there is a change in magnetic field from 0 to some non-zero value, there must be a change in voltage from the law of induction, $V = -\frac{d}{dt}\left(BNA\cos\theta\right)$, where $A$ is the area the magnetic flux is going through, and $N$ is the number of coiled wire. This suggest that there is a measurable effect, although extremely small, at lab 1 due to the entanglement breaking in lab 2. My question is, is this theoretically correct? If yes, then I would suggest this as a method of communication by the following. By creating a large ensemble of these entangled electrons, A-A', B-B', C-C', D-D', where A is an electron at lab 1, entangled to a second electron, A', at lab 2, etc. For example, by choosing to measure A' and C' and leaving B' and D' alone at lab 2, we create a measurable effect at lab 1 for the electrons' A and C, voltage is changed. Thus, this would constitute a sent message as (1 0 1 0). Where 1 would be a voltage change and 0 would be no voltage change. This would of course be a one time messaging system, but it still does not negate the fact that it would be able to send a message via entanglement by this specific scheme. This is true only if my scheme is logically and theoretically correct. I am open to scrutiny and correction. Please help me determine if my scheme is wrong. Thank you. :D


Do all the forces become one?

1. Were the forces of nature combined in one unifying force at the time of the Big Bang?   2. By which symmetry is this unification governed?   3. Are there any evidence for such unification of forces?   4. Has ever been published Theory or experiment in this issue? (Even original researches or unpublished theories. Anything that you can start with.)


What information about a meteor's trajectory, size, or height can be derived from a single location?

If one sees a meteor, is there any way to get even a rough approximation of its height, entry angle, size, or other characteristic _without triangulation_ from another position? If it appeared as a point source and got uniformly brighter, you'd know to take a step aside. And if it appeared on one horizon, traveled overhead, and disappeared over the other, you'd be able to say "Well, _that_ was a shallow angle of attack." But short of those scenarios (which are, probably for the best, rare), is there anything?


Landauer's principle vs Rayleigh–Jeans law

Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of $kTln2$? If it is so, can we also argue that this energy should be distributed over all the modes of the cavity? Furthermore, can it also be argued that this contradicts with the Rayleigh–Jeans law which says the total energy should be infinite?


Direction of friction on rollers with plank resting on top

In a question, its given that a plank is resting on 2 rollers like: ![enter image description here](http://i.stack.imgur.com/afMB1.png) Then the explanation, as part of computing the forces, for the direction of friction on the rollers look like: ![enter image description here](http://i.stack.imgur.com/KcJHQ.png) Why are the direction this way? Since the rollers are turning clockwise (plank is moving right), I expected the opposite? Explanation on the directions look like: (see line 5 counting from bottom) ![enter image description here](http://i.stack.imgur.com/K5lpx.png) Although, after the computation, the direction of $f_b$ is pointing to the right as its negative.


Question about moments?

Take a look at this diagram: http://www.chegg.com/homework-help/questions-and- answers/7-pipe-assembly-supports-vertical-loads-shown-determine-components- reaction-ball-socket-jo-q2260091 If you calculate moment around point B, you can ignore the forces of the tension and only the forces at the ball and socket and the weights come into play. Therefore for moment at B to be zero, the Y and X components of the ball and socket have to be zero. However, if the x and y components at the ball and socket are at 0, theres nothing to cancel out the X and Y components of the tension forces, meaning the forces are at unbalanced. What am I missing? **Please answer MY question not the question at the link**


Nanotechnologies: current problems and general limitations

I have some questions about Nanotechnologies. I'm not physicist and my knowledges about the topic are limited by the "Engines of Creation" book by Eric Drexler. So I think that my questions are related mostely to popular science.   1. As far as I know the molecular mechanisms like nanoassemblers and nanofactories exist in nature(I mean alive cells). Why have they not been created artificially by now?   2. In case that they have been artificially created yet, do they have any limitations in developing of macro objects? For instance, macromechanisms which exist in nature like animals are grow very slowly(months and years) and the initial program(i.e. DNA encryption) doesn't allow to make accurate copies. I mean that the twins who are growing from the same genetic material have a significant difference in adulthood(i.e. different phenotypes). Is this phenomena the result of some fundamental low of nature like Heisenberg uncertainty principle or any other that can't be resolved by modern technological methods?   3. What are the top well known laboratories and organisations in the world which are engaged to resolve such kind of problems? Thanks in advance.


Tracing out an observable vs integrating over unitaries

Let $O$ be an observable on a Hilbert space $\mathcal{H}$, and let $B$ be a subset of the spins composing $\mathcal{H}$, and let $\bar{B}$ be its complement. Now define $\displaystyle O_B = \frac{1}{\operatorname{Tr}_{\bar{B}}\mathbf{1}_{\bar{B}}} \operatorname{Tr}_{\bar{B}}(O) \otimes \mathbf{1}_{\bar{B}}$. Is this quantity equal to $\displaystyle \int d\mu(U) U O U^\dagger$? The integral is taken over the set of unitary operators acting on $\bar{B}$ and $\mu$ is the Haar measure of $U$. If so, why is this the case? What physics course/book/reference introduces these sorts of integrals? Note: this question came up from trying to understand the following paper: http://arxiv.org/abs/quant-ph/0603121


Why is the speed of light considered as a fundamental constant if its speed changes with medium resulting in refraction?

I know that the speed of light, the universal constant of gravitation and the Planck's constant are considered to be the three fundamental constants of the universe. But, why is speed of light considered as a fundamental constant? The speed of light changes from medium to medium for satisfying Fermat's principle then, how can it be a constant?


Gauss's Law :To find the Electric Field for a Non-Conducting Sphere

While determining the electric field in a Non-Conducting Sphere using Gauss's law,why the positive charges are considered inside the surface,but in determining the electric field in a conducting Sphere,why the positive charges are considered outside the surface? And,why if any point charge is inside a sphere,the the net electric field is considered zero?


Autocorrelation of sound in liquids vs gas

This is just a curiosity and you need to bear with me as my math skills were always sub-par and so is my English academic language. My specialty is Electronics but I have always been a programmer. Years ago I was working on my diploma paper that dealt with locating leaks using auto-correlation.   Basically you have several sound sensors along a pipe which record the sound wave. If a leak appears then you get a peak in the autocorrelation function. Then you perform a 'triangulation' from two or more sound sensors and you can locate the leak pretty accurately. Anyway, my task was mostly to help a PHD student with transposing her algorithms into Matlab.      We have succeeded doing this with air but as soon as we switched to water I could not use any kind of windowing and/or transformation to get relevant peaks. At some point I just hunted blindly for some Matlab functions that would give me a relevant peak somewhere that would correlate with the distance to the leak, but failed.      I cannot to this day understand why we were not able to succeed, though I do assume that is has to do with water's turbulence. The setup was a running tap routed through metal pipes (about 10cm diameter), with about 5 very sensitive sound sensors (going to about 100kHz) placed about 3 meters apart and some taps along the way that simulate leaks. Everything was placed in a phonic- insulated basement, signals were truncated taken to stabilize against footsteps/vibrations so measurements were pretty good.      The multiple question is: what could have been done to achieve the goal or at least get closer to it? Is it really achievable (probably not in real life where you have trucks running above)?   Some variables that could influence but for which I lack the knowledge to explain: pipe diameter, water debit rate, tap/leak rate and diameter, choice of liquid, pipe material, sound spectrum and possible frequency filtering.


Lebesgue integration

I know this question is probably not adequate to this SE either, but let me explain my situation: I'm civil engineering's college, so, there isn't a SE for civil engineering, and my doubts about integration in engineering are pratically pure physics. So, as I said, I'm graduating, but I'm from Brazil, education here is the same thing as nothing (belive me, it really sucks, I see people come out of physics's college without knowing who Maxwell was..), and I read my calculus books and see they all use Riemann integrals (of course, they don't say that..); but, in my searchs, I see a lot of Lebesgue integration, especially concerning problems of calculating the center of mass of an object, in continuum mechanics, etc. So, here's my question:   1. Lebesgue integration isn't the most easy thing in the world, I don't have a lot time, and education here sucks, should I spent my time studying Lebesgue integration instead of Riemann's?    2. Which one do you guys use more in physics with applications in engineering?; of course, the last one it's a little trouble, because engineering just "borrow" from physics. But, in general, which compensates more?


Are there any exceptions to Newton's laws?

So I am studying the laws of Newton and I'm wondering, are there any deficiencies related to the laws? I mean, somewhere where I can't use them or anything?


Normalization of Momentum Eigenfunctions: the number of particles

After finding the eigenfunctions $u_p(x)=Ce^{ipx/\hbar}$ of the momentum operator just like in this UCSD lecture notes, one seeks to normalize them, so one first tries: $$\int\limits_{-\infty}^{\infty} dx \, |C|^2 e^{-ipx/\hbar} e^{ipx/\hbar} = \int\limits_{-\infty}^{\infty} dx \, |C|^2 \rightarrow \infty $$ which diverges unless $C =0$. Then it is shown that $u_p(x)=\frac{1}{\sqrt{2\pi \hbar}}e^{ipx/\hbar}$ satisfies the normalization condition $\langle p'|p\rangle=\delta(p-p')$ Why does the UCSD page say that the first solutions (the divergent ones) "are not normalizable to one particle"? How does the development that follows relate to many particles? Does this have something to do with $\int\limits_{-\infty}^{\infty}dp \delta(p-p') =1$? Then all the $p'$ are the other particles? I have no source to show for this, but how would $\frac{1}{2\pi \hbar}$ particles per unit of length and unit of momentum relate? Is it correct to say that $2\pi \hbar$ is the expected number of times one repeats the measurement of the momentum? Then where would the 'per unit of length' part come from?


Why does cold water weigh more than hot water in a fixed volume?

Take a bucket of hot water and the other bucket of cold water. Why does the bucket full of cold water weigh more than bucket full of hot water?


Convergence of periodic single fermion operators

First, a quick remark: I'm a mathematician, now working on some problems coming from physics (in particular Ising models on quasiperiodic chains). A few things I find rather mysterious. I would appreciate your help. For the purpose of generality, let's consider the following Ising model on a chain of $N$ nodes. $$H_N = - \sum_{i = 1}^N J_i\sigma_i^{(x)}\sigma_{i+1}^{(x)} - \sum_{i = 1}^N\sigma_i^{(z)},$$ with $J_i$ depending on the node $i$ (we assume no particular order for generality), and $\sigma_i^{(x),(z)}$ the Pauli matrices. By Jordan-Wigner, we can consider the corresponding Fermionic operator given by $$\widehat{H}_N = \sum_{i,j}\left[c_i^{\dagger}A_{ij}c_j + \frac{1}{2}\left(c_i^{\dagger}B_{ij}c_j^{\dagger}+ H.c.\right)\right],$$ where $c_i$, $1\leq i \leq N$ are anticommuting Fermionic operators and $\left\\{A_{ij}\right\\},\left\\{B_{ij}\right\\}$, $1\leq i, j \leq N$ are the elements of appropriately chosen matrices $A, B$, which depend on $\left\\{J_i\right\\}_{1\leq i \leq N}$. We use periodic boundary conditions. Now we can extend $\widehat{H}_N$ to a lattice of infinite size, by gluing the unit cell of size $N$ infinitely many times. Let us call this new extension $\tilde{H}_N$. Now the questions: 1) What is $H.c.$? 2) I am interested in the thermodynamic limit $N\rightarrow\infty$. Is it obvious whether the sequence of operators $\left\\{\tilde{H}_N\right\\}$ converges, say in strong operator topology, to some well-defined operator $\tilde{H}$ as $N\rightarrow\infty$? Let me motivate the second question: For a certain sequence $\left\\{J_i\right\\}$, constructed deterministically with certain properties (so-called _quasi-periodic sequence_ ), I believe I can say something about what Physicists call the "energy-spectrum in the thermodynamic limit". I'm interested to know whether this energy spectrum is the spectrum (in the usual functional-analytic sense) of some operator $\tilde{H}$. Thanks for any help!


Gauge invariant but not gauge covariant regularization

I'm not sure if someone's already asked this before, but I was wondering, in field theory,   1. when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In particular, in Wikipedia, the regulator of Pauli-Villars is said to be as such.    2. Moreover, as a consequence of not being gauge covariant, the Wikipedia article says that this regulator can't be used in QCD. How to see the link between not being gauge covariant and QCD here? And, why can one use it in QED then?


Can one construct a new operator in terms of the powers of another operator?

Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation value of the operator $A$. Next we derive the general formula for the higher moments of $A$ (i.e. the expectation value of $A^n$ for $n=2,3,4\ldots $). In this situation, is it permitted to regard each of the $A^n$ for $n=1,2,3,\ldots$ as a proper operator by itself? For example, should every $A^n$ have a positive variance and other statistical properties (as long as we restrict ourselves to the state $\Psi$)? Can one make linear combinations of different powers to construct a new operator, e.g. $B = A + A^2$? Is it allowed to construct new operators acting on $\Psi$, that are defined in terms of their series expansion in $A^n$? For example, $D = \exp(CA)$ where $C$ is a constant?


periodicity of diffraction gratings

Imagine we have a diffraction grating consisting of N slits (being N very large) with a separation of "d" from slit to slit. Now, we could regard this grating as a diffraction grating constructed by a repeated double slit, in which case the periodicity would be 2d. I realize this can't be legitimate since the theoretical positions of the maximums would change and this can't be (since the grating is doesn't change no matter how we regard it). Nonetheless, dealing with examples of defectuous grating exercises I have seen examples that regard gratings as repetitions of double slits or triple slits. So, my question is the following, why is it possible in some cases and not in others? I mean, why can't we regard the typical grating mentioned above as a repetition of double slits (or triples or whatever)? (my guess is that maybe we always have to choose the $\textit{minimum}$ periodicity but I am not sure)


Radio antenna producing waves in the visible spectrum

If a radio could produce waves in the visible light spectrum, what would the result be? This is a thought experiment that I've pondered for a few years now. I realize there are a few/many real-world constraints, but if we lifted these constraints for the sake of thought, what could we expect? Personally, I don't see why we wouldn't observe visible light emitting from the antenna, disregarding any light from Blackbody Radiation.


Physics of weird "boing" sound in racquetball courts?

While playing racquetball, I frequently hear a very prominent "boing" sound (or more formally, a chirp). For example, you can hear it in this video when the ball hits the front wall. Does anyone know what the origin of this sound is, and why the pitch rises? Here is the spectrogram from the above video: ![enter image description here](http://i.stack.imgur.com/G2tdWm.png) A careful examination shows that there are at least four linear chirps, which I've highlighted below. If you really listen carefully, all four of these are audible. (However I can only distinguish between the two high frequency chirps when the audio is played at half speed.) ![enter image description here](http://i.stack.imgur.com/lzyGMm.png)


Space-time Topologies?

When it comes to questions of existence of bounds for PDE's and such, one must often make some assumptions regarding the topology of the space-time to use well known theorems. My question is two-pronged: i) I've often read, on Wikipedia (http://en.wikipedia.org/wiki/Spacetime) for example, that space-time is paracompact. I am aware of the mathematical definition and this seems counter-intuitive to me. Since the form of the stress-energy tensor entering Einstein's equations need only satisfy the conservation of energy $\nabla_{\mu} T^{\mu \nu} = 0$ condition from the Bianchi identities, how does one show this result? Can someone give me a reference for a proof? ii) In (electro)-vacuum, what other topologies are permitted globally? I have seen some papers finding solutions that resemble black holes (in the sense they have singularities, event horizons...) but are topologically not 2-spheres at their cross section (seemingly in violation of Hawking's theorem, see e.g, http://arxiv.org/abs/hep-th/9808032, but perhaps the fact the space- time is asymptotically anti de-Sitter is why there is no violation). What space-times are compact? The spheres $S^{n}$ are all compact, but presumably even the Schwarzschild space-time is not globally? I think one can only have a compact space-time if its Euler characteristic $\chi = 0$, can this be translated into a demand on $T^{\mu \nu}$? I am particularly interested in compact space-times, for one can then apply the Yamabe problem. Thanks!


Calculating Ground State Energy in 1D Potential

Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground- state energy $E_0$ via the Schrödinger equation. I'm completely stuck on this one. I've set up the time-independent Schrödinger equation, but it can't be solved without using special functions. I don't see how I can calculate the energy without solving the schrodinger equation. Any hints?


If electromagnetic fields give charge to particles, do photons carry charge?

As I understand these two statements:   1. An electromagnetic field gives particles charge   2. A photon is a quantum of electromagnetic field It must mean that a photon carries charge. But I guess it isn't true. Why not?


What could this very dark planet be made of?

I was reading about the planet TrES-2b which is less reflective than charcoal. What could possibly be its composition?


What is the correct way to handle significant figures when calculating compound uncertainties?

When processing experimental data, and calculating an uncertainty value in multiple steps, should intermediary uncertainties be used to a certain number of significant figures or kept to the full precision allowed by the calculator/computer? I have a number of measured values and their associated uncertainties, and I have processed this data to give me a final value and uncertainty without ever limiting significant figures in either the values or uncertainties at each step. Is this the correct way to go about this?


Sandstone getting soaked with water

I have seen someone putting a sandstone in water. With only about 10% of the stone sitting in the water. One could see the stone getting soaked with water. So there must be a force, which lets the water climb up through the stone against gravity. What is that force? Or is there some other effect present?


A question about the Thomson experiment

Recently, I was studying about Thomson's experiment with cathode rays. My textbook shows it like this. ![enter image description here](http://i.stack.imgur.com/GrPY3.jpg) It says: > When only electric field is applied, the electrons deviate from their path > and hit the cathode ray tube at point A. Similarly when only magnetic field > is applied, electron strikes the cathode ray tube at point C. But if we apply Fleming's Right Hand Rule, then we get the direction of force in the upward direction, so the rays should deviate towards A but they deviate towards C. I think it is because Fleming's Left Hand Rule is defined for conventional current(flow of positive charges) and what we are dealing with are negative charges. Is that correct?


How large can planets or moons appear?

In many artistic impressions or movies there are pictures or scenes where the sky is filled with an enormous moon (as seen from a planet) or vice versa. I wonder if there is an upper limit to the apparent size when viewed with the naked eye (no tele lens). Since the Roche Limit forbids celestial bodies coming too close to each other, there certainly is a limit to the apparent size. ![Artistic image of a large planet seen from a moon](http://i.stack.imgur.com/B3mEm.jpg)


Estimation of polarization perturbative term

I'm studying a diagrammatic approach to degenerate electron gas. Now I need to prove that the energy contribution, with an arbitrary potential, of polarization at first perturbative order given by the diagram: ![Diagramma](http://i.imgur.com/pECT8IX.png) is null. I expanded it as an integral: $$ \Pi_{(1)b}^*(q)=\lim_{p^o\to 0^+}\lambda\int d^4k\int d^4p\text{ }(G^{(0)}(k))^2e^{ip^0\eta}G^{(0)}(p)U_0(k-p)G^{(0)}(k-q) $$ $$ E^{(1)}_b\propto\int d^4q\text{ }U_0(q)\Pi^*_{(1)b}(q) $$ I've made several tries, but I can't prove it. Have you got any ideas? Thanks in advance.


Does a FTS work on the same principle as a michelson (amplitude division) interferometer?

As far as I can tell within an Fourier Transform Spectrometer the spectral information is gained from changing the path length along one arm, this sounds very similar to a michelson interferometer but using two apertures instead of one. So are there any underlying differences between the two?


What is the approximate electrical conductivity $\sigma$ of graphene in S/m or S/cm?

I am trying to find an approximate value of the electrical conductivity $\sigma$ of graphene in units of S/m or S/cm. This table on Wikipedia gives $\sigma$ values for a variety of materials (including references), but I do not see graphene. In the classic 2004 paper on graphene by Novoselov and Geim (Novoselov et al., _Science_ **2004** , _306_ , 666: available here), I see a plot of $\sigma \text{ (m} \Omega^{-1})$ versus $V_g \text{ (V)}$, where $V_g$ is the gate voltage: ![Electrical conductivity](http://i.stack.imgur.com/0H2pT.gif) Based on that plot, what can we say about the electrical conductivity $\sigma$? One confusing thing about the above plot is that it is in $\text{m} \Omega^{-1}$. But $1 \text{ S} = 1 \text{ } \Omega^{-1}$, so in the plot is given in units proportional to S, not S/m. Why is this?


Which BICEP2 r value should be compared to Planck's r<0.11?

The BICEP2 paper reports a tensor/scalar ratio $r = 0.20_{-0.05}^{+0.07}$, but then says: Subtracting the various dust models and re-deriving the r constraint still results in high significance of detection. For the model which is perhaps the most likely to be close to reality (DDM2 cross) the maximum likelihood value shifts to $r = 0.16_{-0.05}^{+0.06}$ Which value is proper to compare with Planck's $r<0.11$?


Is there evidence of a larger universe?

While mapping out the position, distance and movement of galaxies and quasars at the limits of the observable universe, have astronomers ever observed the motion of these object due to gravitational sources that may reside outside the limits of the observable universe? If so, would this break the principle of information travelling faster than the speed of light? If not, are these objects in fact the furthest objects there are?


How can we be sure that liquid iron/titanium droplets can form around particulates on hot Jupiters - just as they do on Earth and Venus?

I know that we have some solar system analogs. But is there a limit to it?


Does 'electricity' have mass? Is 'electricity' tangible?

**Background:** I'm in a legal academic discussion about the status of electronic 'goods' and whether they qualify as 'goods' in the same way a chair and a pen do. In this context (and specifically at the exact circumstance under discussion), it matters if electricity is 'tangible'. So far, most authors have blindly assumed electricity to be a flow of electrons, making a literal analogy with water, making statements such as:   * Information is stored in capacitors in the form of electrons. When a capacitor is filled more than 50% with electrons, it's considered to be 'on' (a bit with value '1').   * The information represented by a certain current (or rather, a series of on/off currents) has mass, because it consists of the electrons that flow through the wire.   * A virtual object is tangible because it exists in memory in the form of electrons that are there (or not) in a certain pattern. Now I have a background in informatics but only a basic knowledge of electricity, and as good as none on the fundamental (physics) level. However I still feel this representation is wrong, and that you can't just say that information in a RAM chip has mass because it consists of electrons that are or are not in the capacitors on that chip. I have found hints in that direction on sites such as http://amasci.com/miscon/eleca.html#made, but I can't quite make out what 'electricity' _is_ and how it relates to _current_ and _potential_ and other words that are used interchangeably in these discussions but which are, I think, different things. So my questions are (all just different angles of looking at the same underlying concept):   * What is 'electricity', really, on a fundamental level; but explained in terms a layman can understand? Is there an analogy with other things that is accurate, unlike the 'flowing water' analogy, which is sufficient for high school level but is a simplification? (at least, I think...)   * Do 'electricity', an 'electrical current' and an 'electrical charge' have mass, apart from the object they are embodies in? Does the mass of a copper wire change when you put a current through it, because of the electrons coming in and going out?   * How do electrons fit into this? Is electricity composed of a bunch of electrons that flow through a mass? I think not, reading the link I gave before, but I don't quite understand what their role is.   * Most authors blindly assume that electricity is merely a flow of electrons through mass. In how far and under what assumptions is this correct?


To what extent is the "minimal substitution" or "minimal coupling" for the EM vector potential valid?

In all text books (and papers for that matter) about QFT and the classical limit of relativistic equations, one comes across the "minimal substitution" to introduce the magnetic potential into the equation (Schrödinger/Dirac/Klein-Gordon) through: $$ \hat{p}^2 \rightarrow (\hat{p} - e \hat{A})^2$$ The wording suggests that this is an approximation for small? electromagnetic fields (or at least not strongly coupled). I understand this was chosen such that the "classical" Lorentz force is retrieved from the Hamiltonian, but not why exactly this form and not another that leads to the same result.   1. To what extent is this "approximation" valid?   2. How can one improve this "minimal substitution"? Is there a more general expression, intuitively something like a series of the vector potential?


Classical scattering of two particles by a Yukawa potential

A point-like particle $A$, coming from minus spatial infinity, heads at another one, $B$, with an impact parameter of $b$. Initial momenta are $p_A$ and $p_B=0$. They repel each other via a Yukawa potential $$\ \ V(r_A,r_B)= +g^2\frac{e^{-\frac{|r_A-r_B|}{\lambda}}}{|r_A-r_B|} \ge 0.$$ > How does the angle of the incoming particle $A$ change, relative to how it > was before?


How does current flow from the emitter, through the base and to the collector in a NPN transistor?

So, I understand that for a NPN transistor to work the emitter-base junction needs to be forward biased and the collector-base junction needs to be reverse biased. I understand how current flows from the emitter to the base when forward biased, but I can't seem to wrap my head around how it flows from the emitter, through the base, and into the collector. From what I understand about the depletion region of a PN junction diode, when forward biased, the electrons in the N-type material are repulsed by the negative charge of the supply (say a battery) and the holes in the P-type material are repulsed by the positive charge from the supply, thus narrowing the depletion region and allowing electrons to flow from the N region to the P region. I also understand that the holes and electrons are attracted to the source charges when reverse biased, widening the depletion region and blocking any electrons from flowing. What I don't understand is what happens to the depletion region of the collector-base junction that allows electrons to flow from the emitter to the collector even though the collector-base junction is reverse biased. Take a simple common emitter circuit for example. I apparently am not cool enough to post pictures yet so you'll have to use your imagination. So pretend we have the base connected to the positive terminal on a 1v battery (Vbb). The collector is connected to the positive terminal on a 10v battery (Vcc). The emitter and both negative terminals are connected to ground. Also pretend that there is some kind of current limiting. How does current get from the emitter through to the collector? I can see how they are at different potentials, but I don't understand how this overcomes the reverse biasing of the collector- base junction. Can anyone tell me what actually happens at the junction to allow current flow and how changing Vbb will affect that current?


Bogoliubov transformation with a slight twist

Given a Hamiltonian of the form $H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix} \begin{pmatrix}\omega_0 & \Omega f_k \\\ \Omega f_k^* & \omega_0\end{pmatrix} \begin{pmatrix}a_k \\\ b_k\end{pmatrix}$ where $a_k$ and $b_k$ are bosonic annihilation operators, $\omega_0$ and $\Omega$ are real constants and $f_k$ is a complex constant. How does one diagonalise this with a Bogoliubov transformation? I've seen an excellent answer to a similar question here, but I'm not quite sure how it translates to this example. Any hints or pointers much appreciated.


What is the symmetry associated with electric charge conservation

Is there a kind of symmetry that yields the conservation of charges? and if so , how it works for both type of charges? Electrical Charges


Feynman Rules in Momentum space

What's the difference between Feynman rules in momentum space for $\phi^3$ theory and for $\phi^4$ theory? I know it's only a slight difference and perhaps found in the vertex factor? But for some reason I can't seem to find this difference anywhere.


Acceleration by spherical particles (micron-scale) by an external force

I am looking for an expression for the velocity of a micron sized (1 - 10 micron diameter) sized particles under accelerating forces. I have aerosols in mind. This is what I have in mind The resisting force acting on the particle is $F_D = 3 \pi \mu Vd$ so its (mechanical) mobility is $\frac{V}{F_D} = \frac{1}{3 \pi \mu d}$ This expresses velocity of a particle per unit force. Does that mean that if the same force acts on a bunch of different sized particles, the smaller particles should move faster due to their smaller size and the dependance is $\frac{1}{d}$? Is this problem perhaps more complicated than this (full F=ma treatment)


``integrated vertex operators" in 1-loop open/closed bosonic string amplitude

This question is in reference to the first ~15 minutes of this String Theory lecture by Prof.Shiraz Minwalla, http://theory.tifr.res.in/Videos/strings28_24sep08.mp4 Can one give a reference (paper/review/book) for this (to my mind very technical) issue being explained here? I have seen Shiraz mention this issue in the previous 2-3 lectures too and its one of those rare concepts for which I could not understand his explanation. I can't relate as to what in Polchinski's book corresponds to this. (though this course claims to follow that book). * * * The final answer that Shiraz wants to get comes as two integrals (one for open and one for the closed case) on the top of the LHS of the board at 15:32 of the video. He claims something about being able to rewrite the moduli space integral in such a way that the b-c 0-mode ghost expectation value decouples from the "integrated vertex operators". I can't understand the point that he is making here about why since the ghost 0-mode is a constant one can evaluate it any any point and that somehow leads to some simplification etc. The initial few minutes of motivation part of the lecture is also not clear to me as to what is he trying to do - about the independence or not of the holomorphic and anti-holomorphic part of the b-ghosts and their boundary values on the worldsheet.


Explain these graphs of rotation and velocity of pucks on air hockey board

I've been tasked to do a simple experiment on the elasticity of collisions. For this experiment I used two "pucks" (very light circular metal pieces of certain height but hollow) and a table that works like air hockey tables (decreases friction by blowing air from underneath). One puck was placed on this board and the other one was shot into it. Each puck had two reflective markers on them, one in the center and one on the edge. The positions of these markers were logged by two cameras shooting infrared light. I am now trying to understand this position data. This is the data I have (I'm using Mathematica):               m1Vel = Differences /@ {m11x, m11y};     m2Vel = Differences /@ {m21x, m21y};     m12Vel = Differences /@ {m12x, m12y};     m22Vel = Differences /@ {m22x, m22y};     m1DeltaX = m11x - m12x;     m1DeltaY = m11y - m12y;     m2DeltaX = m21x - m22x;      m2DeltaY = m21y - m22y;     angularVel[dy_, dx_] := Differences@ArcTan[dy/dx]     vectorNorm2[list_] := Sqrt[list[[1]]^2 + list[[2]]^2];      Using this to plot position data for the pucks M1 and M2:               ListLinePlot[{m11x, m11y, m21x, m21y},       PlotLegend -> {"M1 X", "M1 Y", "M2 X", "M2 Y"}, LegendSize -> 0.5,       LegendPosition -> {1.1, 0}]      ![Position data](http://i.stack.imgur.com/0a7e3.png) And then approximate the velocity for each puck, for the marker in the center:               ListLinePlot[{vectorNorm2[m1Vel], vectorNorm2[m2Vel],        vectorNorm2[m1Vel] + vectorNorm2[m2Vel]}, PlotRange -> Full,       PlotLegend -> {"M1 v", "M2 v", "M1+M2 v"}, LegendSize -> 0.5,       LegendPosition -> {1.1, 0}]      ![Velocity for center marker](http://i.stack.imgur.com/r1jRV.png) And for the marker on the edge: ![Velocity for edge marker](http://i.stack.imgur.com/FDS8P.png) And finally the rotation of each puck, using the approximation that the angle from a horizontal line is $\mathrm{arctan}(\frac{\Delta x}{\Delta y})$ and the angle velocity therefore the difference between the angle at one point and the angle at the next point, as seen in the function angularVel above.               ListLinePlot[{MovingAverage[angularVel[m1DeltaY, m1DeltaX], 10]^2,        MovingAverage[angularVel[m2DeltaY, m2DeltaX], 10]^2},       PlotRange -> Full, PlotLegend -> {"M1 w", "M2 w"}, LegendSize -> 0.5,       LegendPosition -> {1.1, 0}]      ![Rotation of each puck](http://i.stack.imgur.com/bdVLN.png) **Alright, so what's the matter?**   1. I was expecting both of the velocity graphs to look like the first. Since the kinetic energy is proportional to the velocity squared, it is unacceptable to me that it goes up and down in the second velocity graph. It should decrease in a monoton manner. The first collision is with the other puck, but there are collisions after that with the walls.   2. The rotational energy is proportional to the angular velocity. I get that if in the collision with a wall some energy is transferred from translation to rotation and that the rotational energy therefore does not decrease in a monoton manner, but these really sharp peaks (even sharper without the moving average) I cannot understand. Since I'm studying the elasticity I really need only to understand what happens in the first collision. But I feel like a fraud if I write something up about that, neglecting the rest of the graph with all of its peculiarities. **If you had to explain these things in a report, what would you write?**


Time ordering and time derivative in path integral formalism and operator formalism

In operator formalism, for example a 2-point time-ordered Green's function is defined as $\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{op}=\theta(x_1-x_2)\phi(x_1)\phi(x_2)+\theta(x_2-x_1)\phi(x_2)\phi(x_1),$ where the subscript "op" refers to operator formalism. Now if one is to take a time derivative of it, the result will be $\frac{\partial}{\partial x_1^0}\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{op}=\langle\mathcal{T}{\frac{\partial \phi(x_1)}{\partial x_1^0}}\phi(x_2)\rangle_{op}+\delta(x_1^0-x_2^0)[\phi(x_1),\phi(x_2)]$, the delta function comes from differentiating the theta functions. This means time derivative does not commute with time ordering. If we consider path integral formalism, the time-ordered Green's function is defined as $\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{pi}=\int\mathcal{D}\phi\phi(x_1)\phi(x_2)e^{iS(\phi)}$. Of course $\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{op}=\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{pi},$ as is proved in any QFT textbook. However in path integral case time derivative commutes with time ordering, because we don't have anything like a theta function thus $\frac{\partial}{\partial x_1^0}\int\mathcal{D}\phi\phi(x_1)\phi(x_2)e^{iS(\phi)}=\int\mathcal{D}\phi\frac{\partial}{\partial x_1^0}\phi(x_1)\phi(x_2)e^{iS(\phi)}$. I did a bit googling and found out that for the path integral case the time-ordered product is called "$\mathcal{T^*}$ product" and operator case just "$\mathcal{T}$ product". I am not that interested in what is causing the difference(still explanations on this are welcomed), because I can already vaguely see it's due to some sort of ambiguity in defining the product of fields at equal time. The question that interests me is, which is the right one to use when calculating Feynman diagrams? I did find a case where both give the same result, i.e. scalar QED(c.f. Itzykson & Zuber, section 6-1-4), but is it always the case? If these two formulations are not effectively equivalent, then it seems every time we write down something like $\langle\partial_0\phi\cdots\rangle$, we have to specify whether it's in the sense of the path integral definition or operator definition. **EDIT:** As much as I enjoy user1504's answer, after thinking and reading a bit more I don't think analytic continuation is all the mystery. In Peskin&Schroeder chap 9.6 they manage to use path integral to get a result equivalent to operator approach, without any reference to analytic continuation. It goes like this : Consider a T-product for free KG field $\langle T\\{\phi(x)\phi(x_1)\\}\rangle=\int\mathcal{D}\phi\phi(x)\phi(x_1)e^{iS(\phi)}$. Apply Dyson-Schwinger equation, we get $\int\mathcal{D}\phi(\partial^2+m^2)\phi(x)\phi(x_1)e^{iS}=-i\delta^4(x-x_1)$, then they just assume the $\partial^2$ commute with path integration(which is already weird according to our discussion) and they conclude $(\partial^2+m^2)\int\mathcal{D}\phi\phi(x)\phi(x_1)e^{iS}=(\partial^2+m^2)\langle T\\{\phi(x)\phi(x_1)\\}\rangle=-i\delta^4(x-x_1)$. This is just the right result given by operator approach, in which $\delta(x^0-x_1^0)$ comes from $\theta$ function. Given my limited knowledge on the issue, this consistency looks almost a miracle to me. What is so wicked behind these maths? **Response to @drake** :If $a$ is a positive infinitesimal, then $\int \dot A(t) B(t) \,e^{iS}\equiv\int D\phi\, {A(t+a)-A(t)\over a}B(t)\,e^{iS}=\frac{1}{a}\langle T\\{A(t+a)B(t)\\}\rangle-\frac{1}{a}\langle A(t)B(t)\rangle$, notice the second term has an ordering ambiguity from path integral(say $A=\dot{\phi},B=\phi$), and we can make it in any order we want by choosing an appropriate time discretization, c.f. Ron Maimon's post cited by drake. Keeping this in mind we proceed: $\frac{1}{a}\langle T\\{A(t+a)B(t)\\}\rangle-\frac{1}{a}\langle A(t)B(t)\rangle\\\=\frac{1}{a}\theta(a)\langle A(t+a)B(t)\rangle+\frac{1}{a}\theta(-a)\langle B(t)A(t+a)\rangle-\frac{1}{a}\langle A(t)B(t)\rangle\\\=\frac{1}{a}\theta(a)\langle A(t+a)B(t)\rangle+\frac{1}{a}[1-\theta(a)]\langle B(t)A(t+a)\rangle-\frac{1}{a}\langle A(t)B(t)\rangle\\\=\frac{\theta(a)}{a}\langle [A(t+a),B(t)]\rangle+\frac{1}{a}[\langle B(t)A(t+a)\rangle-\langle A(t)B(t)\rangle]$ Now taking advantage of ordering ambiguity of the last term to make it $\langle B(t)A(t)\rangle$(this amounts to defining A using backward discretization, say $A=\dot{\phi}(t)=\frac{\phi(t+\epsilon^-)-\phi(t)}{\epsilon^-}$), then the finally: $\frac{\theta(a)}{a}\langle [A(t+a),B(t)]\rangle+\frac{1}{a}\langle B(t)[A(t+a)-A(t)\rangle]\to \frac{1}{2a}\langle [A(t),B(t)]\rangle+\langle B(t)\dot{A}(t)\rangle$.(Here again a very dubious step, to get $\frac{1}{2a}$ we need to assume $\theta(a\to 0^+)=\theta(0)=\frac{1}{2}$, but this is really not true because $\theta$ is discontinuous) However on the other hand, since $a$ was defined to be a postive infinitesimal, at the very beginning we could've written $\frac{1}{a}\langle T\\{A(t+a)B(t)\\}\rangle-\frac{1}{a}\langle A(t)B(t)\rangle=\frac{1}{a}\langle A(t+a)B(t)\rangle-\frac{1}{a}\langle A(t)B(t)\rangle$, then all the above derivation doesn't work. I'm sure there are more paradoxes if we keep doing these manipulations.


Irreducible Representations of SO(n) tensors

My interest is purely in $\text{SO}(n)$ tensors and how one works out their irrep decomposition. For instance, for rank 2 tensors we simply split into an antisymmetric part, a traceless symmetric part and the trace. Is there a more general, recursive procedure for higher rank tensors? Even if not, what is the usual method when trying to do it for say rank 3 or 4? Any pointers to the literature would be more than helpful.


In Newtonian pressure, what type of function is force?

This is pressure in Newtonian mechanics: $$P=\frac {dF}{dA}.$$ What does this mean? (Doesn't it mean that force is a function of area?) What type of function is force?


How are excess charges distributed over non-spherical conductors?

My textbook gives the following explanation on how excess charges are spread over conductors: > The excess charge on an isolated conductor moves entirely to the conductor's > surface. However, unless the conductor is spherical, the charge does not > distribute itself uniformly. I was studying for a test and there was the following question, regarding the thin-wall conducting cylindrical shell below (which is coaxial to the rod inside it): ![enter image description here](http://i.stack.imgur.com/jiLmD.png) $Q_1 > 0, Q_2 < 0$ > What is the charge on the interior and exterior surface of the shell? At first I thought the excess charge on the interior surface would be zero, since all the excess charge would move to the outer surface of the shell. But according to my textbook, it is as follows: > We consider a cylindrical Gaussian surface whose radius places it within the > shell itself. The electric field is zero at all points on the surface since > any field within a conducting material would lead to current flow (and thus > to a situation other than the electrostatic ones being considered here), so > the total electric flux through the Gaussian surface is zero and the net > charge within it is zero (by Gauss's law). Since the central rod has charge > $Q_1$, the inner surface of the shell must have charge $Q_{in} = -Q_1 = > -3.40 \times 10^{-12}\: \mathrm{C}$. > > Since the shell is known to have total charge $Q_2 = -2.00\: Q_1$ it must > have charge $Q_{out} = Q_2 - Q_{in} = - Q_1 = -3.40 \times 10^{-12}\: > \mathrm{C}$ on its outer surface. So there is $-Q_1$ excess charge on the inner surface because said charges are being attracted by the rod? What if there was no rod, how would the excess charge be distributed over the shell's surface? Can I tell how the excess charge will be distributed over the surface of any non-spherical conductor, or only in special cases such as the above?


Can the lightning be captured and used as power source?

I would like to update my knowledge in this area, that is really out-of-dated and stopped somewhere like ten years ago. I asked the very same question on my physics lecture at my studies and got the answer, that although some tests and experiments were made (by French scientists?), there is no material, either natural or man-made, that would suffer direct lightning hit and therefore could be used to "capture" and store energy from lightning, for future use or processing of any kind. Can someone tell me, how does it looks now? Has anything changed during last ten years in this area?


What does the speed of light have to do with mass and energy?

I've been thinking about Einstein recently, and I came back to the famous equation e=mc^2. Why does this work? It's saying the difference between mass and energy happens to be the speed of light squared? I'm sure there's a larger explanation behind it, but the simplicity of the equation seems improbable.


Reference request for supersymmetry

I have a good background in linear algebra, Topology and Differential geometry. I would like to understand the concept of supersymmetry in theoretical physics, and hopefully read Witten's landmark paper _Supersymmetry and Morse theory_ sometime later. Please suggest suitable references. Thanks


Does the LSZ reduction method prove gauge-independence in massless gauge theories?

I've been working my way through L. Baulieu's excellent paper [ _Perturbative gauge theories_ , Physics Reports, Volume 129, Issue 1, December 1985, Pages 1-74]. Towards the end, he goes on to prove that renormalized Yang-Mills theories that are BRST invariant exhibit gauge independence and unitary. At the end of the proof he states: > _One should note that we have implicitly assumed in this demonstration the > existence of the S-matrix, i.e., the applicability of LSZ reduction formula > [...] the proof is only valid when all gauge bosons are massive. Whenever > massless gauge bosons are present in theory, all our arguments become > formal._ In the case of QCD, the gluons are massless, so does the LSZ formula not apply to QCD? Is there a formal argument that proves gauge-independence as he suggests?


Good Source to Understand Angular Momentum

I am looking for a good source to understand angular momentum. I know the basics but I am looking for a sound in-depth knowledge like directions of angular momentum, when it is not parallel to angular velocity, differences between normal angular momentum and spin angular momentum, how one is intrinsic property and other is not, etc. Can you please help? Any book, any video, any website etc.? Thanks for your help.


Is it expected tha all stellar black holes will be spinning near the maximum allowed $\omega$-velocity?

Using a bit of classical reasoning I'm imagining black hole formation to be much like an ice skater pulling in her arms: ![skater pulls in her arms to increase angular velocity](http://i.stack.imgur.com/9fiOU.png) Now, the size difference between a star and its black hole can't even be effectively captured in an image. The black hole for our sun would be much less than a pixel in this image: ![size comparison of our sun and planets](http://i.stack.imgur.com/XoKDr.png) That suggests to me that even a very slowly rotating stars would have much more angular momentum than could be supported by their resulting black holes. I haven't done the calculation because I don't really understand the Kerr metric but even with a bunch of classical hand-waving I'd think that just about every black hole formed in a stellar-collapse would be spinning maximally. So my question is, do we expect nearly all black holes to be spinning maximally? If so, (roughly) how much angular momentum is lost because the star had much more than the black hole could support? And, how is all of this extra angular momentum shed during collapse? Is it just in the form of tons of matter being ejected until the the angular momentum is low enough to allow for the formation of a Kerr black hole?


Doubt about states of matter

What does exactly means "Small molecules may appear as solid, liquid, and gaseous phases without losing their molecular integrity"?, I can't image how just a molecule can be a gas, liquid or solid. Or in fact the previous phrase refers to the state of aggregations of those small molecules? My source is this article: B Wunderlich, A classification of molecules, phases, and transitions as recognized by thermal analysis, Thermochimica Acta, Volumes 340–341, 14 December 1999, Pages 37-52 If you could suggest me a reference that extend the previous one to more modern states of matter like Bose-Einstein condensates, i would really appreciate it.


Microscopic explanation of optical activity

The origin of linear birefringence in crystal can be easily explained by the symmetry of the crystal. However, it seems it is hard to be applied in circular birefringence (i.e. optical activity), since there is no macroscopic symmetry in sugar solutions. So I wonder how we explain the levorotatory and the dextrorotatory in atom level POV?


Temperature of thermally isolated space region

If we thermally isolate a region in space, say using a hypothetical material of $0$ conductivity, and measure the region's temperature, will it be 2.7K?


Why does water gulp out of a water bottle with a narrow opening instead of a steady flow?

For example, take a water bottle. Fill it with water and then turn it upside down. Instead of flowing steadily downward, it gulps down in parts. Why?


$Q$-value in nuclear reaction $^{9}Be\left(\gamma,n\right)^{8}Be$

I have two questions. **First** , I understand that in a nuclear reaction $$Q:=K_{after}-K_{before}\equiv E_{0,before}-E_{0,after} \qquad (1)$$ where $K$ is the total kinetic energy, and $E_0$ is the total rest energy. My question is, in the reaction $^{9}Be\left(\gamma,n\right)^{8}Be$, where should I put the energy of the $\gamma$-photon? I mean, conservation of energy says $$Mc^{2}+h\nu=mc^{2}+M'c^{2}+K_{n}+K' \qquad(2)$$ where $M$=mass of $^{9}Be$, $m$=mass of the neutron, $M'$=mass of $^{8}Be$, and $K_{n}$, $K'$ are the kinetic energies of the neutron and of the $^{8}Be$, respectively. Then, using (1) (with the rest energies version) I wrote $$Q=(M-m-M')c^2 \qquad (3)$$ but this is equivalent to: (using (2)) $$Q=K_{n}+K'-h\nu \qquad (4)$$ So, my **1st question** is: is (4) the correct expression for $Q$? **Second** , I want to calculate the kinetic energies $K_n$ and $K'$. In order to do this, I'm thinking to use (4) (if it's correct) with $K_{n}=\frac{1}{2}mv^{2}$ and $K'=\frac{1}{2}M'V^{2}$, and the conservation of linear momentum $$\frac{h}{\lambda}=mv+M'V \qquad (5)$$ Assuming that the neutron and the $^{8}Be$ continue moving in the direction of the $\gamma$-photon. So, my **2nd question** is: can I make this last assumption? Or in other words, is (5) a correct expression for the linear momentum conservation?


Transforming a sound wave into a literal light wave. Is it possible?

Literally transforming sound into an actual light wave seems almost impossible. But transforming the sound wave into a light wave while containing a single mass? I know that sound can contain a mass of solid, liquid or a gas type. but does light have a mass at all? Is it possible to literally transform sound into light while containing a single mass? I have a experimental project. I want to know if it's possible in the slightest way? let me figure out how, just for the fun of the research.


Is light and sound waves interconvertible?

Light is produced when an air bubble underwater is collapsed with sound wave. Why does it happen so?


Where did Schrödinger solve the radiating problem of Bohr's model?

One of the problems with Bohr's theory to describe the hydrogen atom, was that the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus. Now Schrödinger describes the electron as a wave function. His theory is able to describe all atoms (in contrast to Bohr's model), but how is the radiation problem solved? I understand that the wave has not an exact position in the time anymore. But the electron still "moves", so it has an acceleration anyway (because of vibrations or so). Why is in this theory the electron not radiating anymore? And if it is, why doesn't the atom collapse?


Is the harmonic oscillator potential unique in having equally spaced discrete energy levels?

I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and quadratic in others, but that's only trivially different. I am not referring to equally spaced as a limiting behavior either, I mean truly integer spaced. Any ideas? If not, is there a proof for its uniqueness?


can be exist the negative mass?

I'm not sure about this but I guess there must be negative masses in the universe because of the symmetry. If the gravity is one of the main forces in nature it must has negatives mass to be able to uniform with other forces. I think the m^2 in the relative energy E^2=P^2c^2+(m^2)c^4 refer to it and it must corrected that E=|m|c^2


If space and time are equivalent, what's Spin in time dimension

This troubles me: We are talking about time and space being equivalent, but still only consider Spin in the $x$, $y$ or $z$-direction. What's Spin in time dimension? Is it distinction between particles and antiparticles?


Writing wave functions with spin of a system of particles

Suppose I have 2 fermions in a potential $V(x)$. Both particles are moving in one dimension: the $x$ axis. Then, neglecting the interaction between the particles, the spatial wave function of the system would be of the form $$\psi_{n_{1}}(x_{1})\psi_{n_{2}}(x_{2}) $$ Now, if I'm considering particles with spin 1/2, the notation $\alpha(1)$ indicates that the particle 1 has spin up, and $\beta(2)$ denotes the particle 2 having spin down. Now, I want to write the complete wave function, a function of the form $$\psi_{n_{1}n_{2}s_{1}s_{2}}(x_{1},x_{2},s_{1},s_{2})=\psi_{n_{1}}(x_{1})\psi_{n_{2}}(x_{2})F(\alpha,\beta)$$ where $F(\alpha,\beta)$ is a function of the spin of the system. To this end, I have that the only physically possible functions $F(\alpha,\beta)$ are: Symmetric: $\chi_{\alpha}:=\alpha(1)\alpha(2),\quad\chi_{\beta}:=\beta(1)\beta(2),\quad\chi_{+}:=\frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2)+\alpha(2)\beta(1)\right]$ Antisymmetric: $\chi_{-}:=\frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2)-\alpha(2)\beta(1)\right]$ In order to write down the complete wave function with spin, I understand I have to consider the energy levels. For example the ground state: $\psi_{1}(x_{1})\psi_{1}(x_{2})$. If $\psi_{1}(x_{1})\psi_{1}(x_{2})$ is symmetric ( _as I understand it is_ ), then I must multiply this function times the antisymmetric function $\chi_{-}$ (in order to get an antisymmetric wave function, for two fermions). If $\psi_{1}(x_{1})\psi_{1}(x_{2})$ is antisymmetric (and I understand this is impossible, since the ground state is not degenerate), then I'd have 3 wave functions, obtained by multiplying $\psi_{1}(x_{1})\psi_{1}(x_{2})$ times $\chi_{\alpha}$, $\chi_{\beta}$ and $\chi_{+}$. Now, for the 1st excited level, say $\psi_{2}(x_{1})\psi_{1}(x_{2})$, **my question is** , what happens when this function is not symmetric neither antisymmetric? I mean, I could built a symmetric $$f_S=\frac{1}{\sqrt{2}}\left[\psi_{2}(x_{1})\psi_{1}(x_{2})+\psi_{2}(x_{2})\psi_{1}(x_{1})\right]$$ or an antisymmetric $$f_A=\frac{1}{\sqrt{2}}\left[\psi_{2}(x_{1})\psi_{1}(x_{2})-\psi_{2}(x_{2})\psi_{1}(x_{1})\right]$$ wave function. But which one of these must I choose? Or, must I calculate the resultant complete wave functions with both? Then, when I count the states with the 1st excited energy, I'd have 4 instead of 1 or 3.


Which way does the scale tip?

I found the problem described in the attached picture on the internet. In the comment sections there were two opposing solutions. So it made me wonder which of those would be the actual solution. So basically the question would be the following. Assume we would have two identical beakers, filled with the same amount of the same liquid, lets say water. In the left beaker a ping pong ball would be attached to the bottom of the beaker with a string and above the right beaker a steel ball of the same size (volume) as the ping pong ball would be hung by a string, submerging the steel ball in the water as shown in the picture. If both beakers would be put on to a scale, what side would tip? According to the internet either of the following answers was believed to be the solution.   1. The left side would tip down, because the ping pong ball and the cord add mass to the left side, since they are actually connected to the system.   2. The right side would tip down, because of buoyancy of the water on the steel ball pushing the steel ball up and the scale down. Now what would the solution be according to physics? ![enter image description here](http://i.stack.imgur.com/Mj3Y0.jpg)


How do moving charges produce magnetic fields?

I'm tutoring high school students. I've always taught them that: > A charged particle moving without acceleration produces an electric _as well > as a magnetic field_. It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving?


Why are magnetic fields only produced by moving charges?

Why do charged particles only produce magnetic fields while in motion?


Is a magnetic field just a moving charge?

I am trying to understand electromagnetism. I would like to pare myself down to the minimum number of necessary concepts. From my physics education (high school only), I understand that a moving charge **creates** a magnetic field. The implication is that magnetism is a discrete concept that I also need to understand. However, is this view actually confusing: is it the case that a magnetic field **is** a moving charge? Either way, a link to a primer accessible to someone with a engineering degree level maths would be wonderful. If it is the case magnetism **is** a moving charge, then an introduction following this approach would be wonderful. Thank you!


Why don't stationary electric charges possess a magnetic field?

Why don't stationary electric charges posses magnetic field, while moving charges do?


No magnetic field from a static charge - Is there a simple physical argument to show why?

For a charge moving in an electric field $\vec E$, its equation of motion is given by the electric part of the Lorentz force $$\frac d {dt}\gamma m \vec v = e\vec E$$This comes from the conservation of relativistic energy in a static electric field. But a magnetic field would still make this conservation law true since the magnetic force is always orthogonal to the velocity of the charge and therefore doesn't change its energy. Is there a simply physical argument that shows why a static charge doesn't create a magnetic field?


Why doesn't mass of bob affect time period?

Please correct me if I'm going wrong -   By the gravitation formula: $F = \frac{G m_1 m_2}{r^2} $, So if the mass of a bob is greater then the torque on it should increase because the Force increased now by a very small magnitude. So it should go faster and thus the Time period should be lesser. But my Physics book says that Time period is only affected by effective length and $g$ Why doesn't mass of bob affect it?


What is meant by a local Lagrangian density? How will a non-local Lagrangian look like and what is the problem that we do not consider such Lagrangian densities?


How can earthquakes shift the earth's axis?

One often comes across news articles that claim that an earthquake shifted the earth's axis. > > http://news.google.com/?q=earthquake%20shifted%20OR%20shifts%20earth%27s%20axis If you ignore the influence of other celestial bodies, an internal event like an earthquake surely can't change the direction of the angular momentum of the Earth (unless stuff is ejected out of Earth), since angular momentum has to be conserved in the absence of an external torque. So the axis has to remain fixed. Am I missing something? Or are geologists trying to say that the resulting movement of tectonic plates causes a change in the point of intersection of the axis (which remains the same) and the plates that include the poles, so that it _seems_ as if the axis has shifted? **EDIT** Some articles mention the value of the shift in the axis and also the change in the length of the day. If, as Ted Bunn's answer indicates below, the shift in the axis isn't actually real but is because of the movement of tectonic plates with respect to the axis, shouldn't the shift be different at the north and south poles? How are the shifts and the change in day-length calculated?


Stringy corrections to Friedmann equation

Does anyone know a reference or a paper which discusses string theory correction to Friedmann equations?


Can one make a ball rotate around a vertical axis using only a combination of horizontal axis rotations?

This is a nice problem that I would like to share. **Problem:** In a public garden, there a statue consisting of a spherical stone and a stone cup. The ball is 1 meter in diameter and weighs at least a ton. The cup is an upside-down hemispherical shell, and the ball sits in this shell and fits it almost exactly. Water is pumped into the bottom of shell so that a thin film exists between shell and the ball. The result is a ball that is free to rotate with negligible friction. You only have access to the ball near the top, so while you can push it to make it turn around any horizontal axis, you can't get enough of a grip to make it turn around the vertical axis. Can you impart a net angular momentum around the vertical axis anyway, so that the balls spins around the vertical axis? **Source:** _Vector Calculus, Linear Algebra, and Differential Forms_ by Hubbard and Hubbard.


Difference between RPA and generalized RPA

The random phase approximation (RPA) is an approximation method in condensed matter physics and in nuclear physics. What is the difference between RPA and generalized RPA?


String Theory and Fourier Analysis

Me and my friend, both many years from learning string theory, had a recent debate about it anyway. He said he already partially discounts it because after learning waves, he believes any function, and thus any kind of physical process can be created through a superposition of an infinite sum of waves. Since string theory to our understanding is made up of primarily oscillating strings, he argues that it is a mathematical necessity for string theory to create equations for the physical world, regardless of whether the theory actually is "true" or not. My understanding is that Fourier analysis can only create arbitrary periodic functions, can it in principal create any real world field or particle, etc.? Besides this, I argued that this might be a pointless question anyway, as if we can create a perfect model of the universe with string theory, then the question of whether the universe is really made of oscillating strings is more of a philosophical question that may not mean anything at all. Through reading Brian Green's books I know that M-theory at least incorporates more than just strings, but I have no clue as to how things such as branes work, can his argument even apply to these other fundamental objects in string theory? So my question is, is Fourier Analysis essentially what String Theory is? And if so, does that make it any less of a true physical theory of the universe?


Which electromagnetic radiation is faster in water, microwaves or light?

Well I've been asked this question, but I haven't been able to come with an answer yet using books and some web searches. The point is as the title says, to answer the question with the whole phenomenon explanation if possible.


How can helicity be conserved but chirality not?

I read in a book that for $\beta$-decay the electrons have always been found to have an expectation value for their helicity of $h=-v/c$. Then ist is said in the book, that it follows from this fact that such electrons are in a left-handed chiral state which is characteristic for the weak interaction. In another article I read that the chiral state of an electron is not conserved in time. The electron will soon evolve a component with a right- handed chiral state and it will be a mixture of right- and lef-handed chiral states. Suppose after the decay one electron moves like a free particle.   When it evolves a right-handed chiral component in addition to the left-handed component it starts off with, how can its helicity be conserved?


Twist of null Killing fields

I have a (hopefully) quick question: is it possible to have a null Killing field $\xi ^ \mu$ such that the twist 1-form $\omega_{\mu} = \epsilon_{\mu\nu\alpha\beta}\xi^\nu \nabla^\alpha \xi^\beta \neq 0$ but the exterior derivative $(d \omega)_{\mu\nu} = 2\nabla_{[\mu}\omega_{\nu]} = 0$? Or does $(d \omega)_{\mu\nu} = 0$ always imply $\omega_{\mu} = 0$ for a null Killing field?


Photoelasticity and existence of stress

Shall we consider photoelasticity a non-mathematical, or purely visual, proof of the existence of stresses in mechanical structures undergoing external loading?


Can we introduce the mass of a quantum field as an interaction?

At a free massless Lagrangian \begin{equation} L_0 = \frac 1 2 ( \partial \psi)^2 ,\end{equation} add an interaction term \begin{equation} L_I = \frac 1 2 m^2 \psi^2\end{equation} where m is small enough to converge its perturbation calculation. From these setup, is it possible to gain the same results as of the free massive Lagrangian? Please point me a book description or some internet site. Thanks.


Info request on studying QIT/QIS or QM with a Computer Science background

I've been considering a career change for a long time and recently discovered the Two-Slit Experiment, which, to put it frankly, blew my mind. I then started some hefty reading and investigation into all things Quantum (Bell, Campbell, entanglement, QIT), which has led me here. I have 20 years experience in IT working as a programmer, having graduated with a Bachelor in Computer Science back in the early 90s. As mentioned, I have been considering a career change for some time (having become quite burned out in this industry), and have found something that has piqued my interest more than anything else (tho I did consider Astrophysics a couple of years ago). I am not yet sure which area of "Quantum" I will be most drawn to, possibly Quantum Information Theory/Science. I am aware that there is a significant amount of Maths and Physics pre- requisites involved. It seems likely, given that I have not done any Maths or Physics study probably since I was doing my CS undergrad, that I will need to start over with an undergrad in Science/Physics in order to get the fundamentals. So my questions are:   1. What is the ideal path of education to get to QIT/QIS or QM?   2. Is the path to QIT/QIS also via the Physics route? Can I leverage my CS background?   3. Resources (books, online courses, etc) that would help with the transition from CS-type thinking to Physics/Maths-type thinking. Apologies if this encompasses elements too broad or off-topic, I'm trying to get a better understanding on what is to come following this path. Thanks in advance.


Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?

or a phenomenon where we can only measure the standard deviation ($\sigma_w$) of a variable $w$ and not the mean $\overline{w}$


Electric and Water Flow Analogy

I’ve been studying electricity lately and it has been quite hard to understand the terminology, and picturing it in my mind. After struggling to understand what each property of electric currents meant, I found an analogy that related electric current to water flow. I am not sure if this analogy is 100% accurate, but it sure has helped me understand the complex (at least for young, starter scientist like me) topic of electricity. This far I have managed to relate in the analogy that: \-----Electricity-------------------------Water------------ Current (Amps)(A)-------Flow rate (Volume/Time) Resistance (Ohms)(Ω)----pipe size (Inches) Voltage (Volts)(V)------Water pressure (psi) But I am still missing properties like charge. What properties am I missing? How could it be related to water flow? If it can’t be related, can you please give me a short definition and explanation to better grasp the concept? Please include units in both sides of the analogy. Also, please keep the answer short and simple for us, the inexpert scientist (including me) that struggle to grasp the concept. **Thanks a bunch for your expert help** It is kind of discussed in these articles: Flow of water and flow of electrons, how this analogy works? Power in hydraulic analogy But I couldn’t find (or understand) the answer provided in them.


Why does it seem that the potential difference dependence of capacitance and total energy stored in a parallel-plate capacitor are contradictory?

Consider a parallel-plate capacitor. Charge is stored physically on electrodes ("plates") which are flat and parallel to one another. If one electrode has charge $+Q$ and the other electrode has charge $-Q$, and $V$ is the potential difference between the electrodes, then the capacitance $C$ is $$C = \frac{Q}{V}$$ (This definition of $C$ is given in, for example, _Introduction to Electrodynamics_ by David J. Griffiths.) But, now, let's think about the energy stored in the electric field between the electrodes of this parallel-plate capacitor. As stated in Griffiths on page 105, "How much work $W$ does it take to charge the capacitor up to a final amount $Q$?" It turns out that $W$ is $$W = \frac{1}{2} CV^2$$ So: (i) the capacitor's capacitance $C$ goes like $\frac{1}{V}$; and (ii) the energy $W$ stored in the electric field goes like $V^2$. **Are statements (i) and (ii) at odds with one another?** I am sure that they _cannot_ be. But conceptually I am having difficulty. We desire high capacitance -- we want to put as much charge on the electrodes as possible, because if we accomplish this, then I think that will increase the energy density of the system. But is what I just said true? If we manage to increase $Q$, then by $V = \frac{Q}{C}$, the potential difference $V$ between the plates will also increase. This, I think, is why capacitor electrodes are separated by a material (such as a polarizable dielectric material like a slab of plastic); otherwise $V$ will become too large and the breakdown voltage will be reached, generating a spark. But, now, the equation $W = \frac{1}{2} CV^2$ (where I think that $W$ can be conceptualized as the energy stored in the electric field between the electrodes) seems to say that as $V$ increases, so does the energy $W$, quadratically. So, my question is, **do we want a capacitor to have a _large_ potential difference $V$ or a _small_ potential difference $V$?** If $V$ is large, then $W$ is large (which we want), but $C$ is small (which we do **_not_** want). Am I somehow thinking of two different potential differences $V$ and confusing them?


Coupling of vector gauge and a massive tensor field

I was reviewing the paper-Coupling of a vector gauge field to a massive tensor field In the calculation I found the term $ 2\mu^2 \varepsilon^{ijk} \dfrac{\partial_j}{\partial^2}B_k\dot{B}$ which needed to be integrated by parts. How can I do this? Please help me. _Note_ : Here $B_k$ is the space component of an antisymmetric tensor field and $\dot{B}$ is the time derivative of $B_k$. ( _Edit: corrected the index_ )


Why is parasitic capacitance in inductor said to be in parallel?

Internal resistance of inductance (or other devices) are said to be in series. But parasitic capacitance is said to be in parallel (in case of an inductor). Why is that so? What determines whether an internal property is in series or parallel?