Why does soda fizz when it meets ice?

I read that the soda's temperature allows it to contain its $\text{CO}_2$: colder means it can retain its bubbles better. So, why does soda fizz when it meets ice? If anything, because the soda is made colder, it should retain its gas better, as per above. The ice is taking away energy from the soda, thus freezing. Won't heating, not freezing, encourage the gas to escape, or is it not an energy matter?


Non Zero correlation function (for large separations) in one particle state?

So i computed the following equal time correlation function for a one particle state. The vacuum correlations give the function $$\langle \phi(\vec x)\phi(\vec y)\rangle_0=D(\vec x-\vec y)\\\ =\frac{e^{-m|\vec x-\vec y|}}{4\pi |\vec x-\vec y|}$$ while the one particle correlation function looks like: $$ \langle\vec q| \phi(\vec x)\phi(\vec y)|\vec q\rangle=2\omega_q\langle0|a_q \phi(\vec x)\phi(\vec y)a^{\dagger}_q|0\rangle\\\\\propto 2cos[\vec q\cdot(\vec x-\vec y)]+D(\vec x-\vec y)\\\=2cos[\vec q\cdot(\vec x-\vec y)]+\frac{e^{-m|\vec x-\vec y|}}{4\pi |\vec x-\vec y|}$$ I recognize that since this is a spacelike separation that it can't be observed, the commutator of this type vanishes since the terms are invariant under $\vec x\rightarrow \vec y$. Since there is no elapsed time there is also no valid interpretation as particle propagation. it just seems odd to me that the field has non vanishing correlations for large spatial separations. I was wondering if there is any consequence to this that people know about. Thanks, Selman


What is the status of massless photons traveling through a medium?

Photons in vacuum have no proper time, and they are not considered as observers and not as reference frame. But what about photons travelling through matter? Their velocity is lower than light speed, thus the proper time formula (multiplication with reciprocal gamma) would yield a proper time, but do we consider that they have a proper time/ that they are observers/ that they may be reference frames?


Tensor contra- and covariance concept

Imagine we have $p$-contravariant and $q$-covariant tensor, such that $$ t~=~t^{i_1, \dots, i_p} _{j_1, \dots, j_q} e_{i_1}\otimes \dots \otimes e_{i_p} \otimes e^{j_1} \otimes \dots \otimes e^{j_q}~\in ~T \left( p,q, L\right),$$ where $e_{i_l}$ are basis vectors of $L$ (arbitrary linear space), $e^{j_k}$ its dual basis in $L^{*}$. My question is following: what is the physical meaning of vector and its dual basises? I mean, for example, how do I interpret 2 times contravariant and 1 time covariant tensor in physics.


Quantum Cascade Laser

I want to learn about quantum cascade laser in a six month time. Can anyone please suggest me any reading materials or video lectures. I have background in calculus and linear algebra. I am learning quantum mechanics now.


Infinite Round Square-Well zeroes of Spherical Bessel function

Consider a potential well where the potential is zero within a spherical well of radius a centered on the origin. Taking $a$ as our length scale and $\hslash^2/(2ma^2)$ as our energy scale, we gain the dimensionless form $$ -\biggl(\frac{\partial^2}{\partial r'^2}+\frac{2}{r'}\frac{\partial}{\partial r'}+\frac{1}{r'^2}[\frac{1}{sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial}{\partial\theta})\frac{\partial^2}{\partial\phi^2}]\biggr)\psi(r',\theta,\phi)=(E'-U')\psi(r',\theta,\phi) $$ Seeking a solution where the dependence on $r'$ factors from the dependence . on angles $$ -\biggl(\frac{\partial^2}{\partial r'^2}+\frac{2}{r'}\frac{\partial}{\partial r'}-\frac{l(l+1)}{r'^2}\biggr)R(r')=(E'-U')R(r') $$ We substitute: $ \rho=\sqrt{E'-U'}r'=k'r' $ where $E'>U'$ Dividing the above equation by $E'-U'$: $$ \biggl(-\frac{\partial^2}{\partial \rho^2}-\frac{2}{\rho}\frac{\partial}{\partial \rho}+\frac{l(l+1)}{\rho^2}\biggr)R(\rho)=R(\rho) $$ The case where U is infinite and so the wavefunction must be zero for $r>a$, i.e., $r'>1$. Thus we must have: $j_l(k'r')=0$ for $r'=1$ That is $k'$ must be a zero of $j_l$. The energy of the wavefunction can then be calculated from: $E'=k'^2$. Here is the overall solution: $$ \psi_{n_rlm}(r',\theta ,\phi)=\frac{\sqrt{2}}{j_{l+1}(k'_{n_r,l})}j_l(k'_{n_r,l}r')Y_{lm}(\theta , \phi) $$ where $j_{l}(k'_{n_r,l})=0$ If i want to plot the radial part of the function I need $k'_{n_r,l}$. should I use `fzero` in $Matlab$ for example to find theme first? Thanks for any help to draw it.               jnu=sqrt(pi./(2*x)).*besselj(nu+1/2,x);


Why do electrons move towards regions of lower potential?

Can someone explain why electrons move from high to low potential regions in quantum mechanics? By the way, what is the very definition of this potential? Moreover, how does this generalize to other particles?


Force on a point charge q inside a cavity in an uncharged conductor

This is problem 2.40 from Introduction to Electrodynamics by D. J. Griffiths: > A point charge $q$ is inside a cavity (not necessarily spherical or anything > similarly regular) in an uncharged conductor. Is the force on $q$ > necessarily zero ? What my understanding tells me is that if the charge on the inner surface of the conductor is just enough to cancel the field from the point charge from every direction, then the force from any two opposite pieces on the inner surface will produce equal but opposite forces. Thus, the net force is zero. Is my reasoning correct?


What is better than time-dependent perturbation theory if the pointer states aren't energy eigenstates?

Time-dependent perturbation theory works excellently if the interaction is weak and the pointer states are approximately energy eigenstates. However, what if the pointer states are not remotely energy eigenstates? Then, the pointer states themselves evolve in time nontrivially by more than just a phase rotation.


III-V semiconductor compounds - band gap approximations

I'm looking for the most common and accurate approximation functions for III-V semiconductor compounds. So far, I found two (InGaAs as the sample compound):   1. The simplest one: $E(Ga_{1-x}In_xAs)=(1-x)E_{GaAs}+x \cdot E_{InAs}$   2. Binomial: $E(Ga_{1-x}In_xAs)=(1-x)E_{GaAs}+x \cdot E_{InAs} - b \cdot x \cdot (1-x)$ For InGaAs, b=1.44 Are there more? I mean the methods for which the parameters can be found in articles, not measured by myself. When using the first one is sufficient? Do the scientists bother to calculate next corrections? Are they useful?


Projectile motion without air resistance

Does the mass of projectile affect its trajectory when it is projected in no air resistance environment and why?


Is there a general H-theorem?

In statistical mechanics, Boltzmann showed that for dilute gases the H-function increases. I remember from a lecture that there is no general H-theorem, e.g. for non-dilute gases or in the quantum case but there seem to be also claims to the contrary. So what is the status of the (generalised) H-theorem?


Costant volume gas thermometer

I have two constant volume gas thermometers, one with an ideal gas $pv=RT$ and the other following the Van der Waals equation $(p+\frac{a}{v^2})(v-b)=RT$. They are calibrated using the freezing point and the boiling point of water. Will they show the same temperatures?


Can ship/boat propellers be placed, with adequate protection, alongside the fuselage instead of at the back?

On small private aircraft the engine is placed at the front of the fuselage, or on the wings E.g. Stationair, Otter On combat aircraft the engine/s is placed at the end of the fuselage E.g. Raptor, Typhoon On larger aircraft constructed to carry cargo/passengers the engines are placed on the wings E.g. DC3, A380 The positioning probably has to do with the amount of load to be lifted into the air, and manouverability desired... which seems sensible. Yet most water-going craft (boats, power-boats, merchantmen, naval craft) have their propellers/push-pull placed at the back of the fuselage. What is the advantage in placing the driving component for boats in this position in addition to protecting them against damage? Can the propellers on a boat, with adequate protection say- a tubular enclosure, placed alongside the fuselage not drive the watercraft with equal efficiency and power?


Suppose that the position of some object is goiven by the expression: x = 6.8t + 9.0t^2. Determine the object's acceleration

x is in meters t is in seconds Answer = 18 m/s/s How do I solve this problem? Thank you for your time.


Average of $\sin^2 \omega t$, $\cos^2 \omega t$ for long times

I don't understand how the averages mentioned in excerpt below work out. The question is regarding calculating the power spectrum of an oscillating classical dipole. IS averaging over long periods, for example, For $\sin^2 \omega_o t$ given by $$<\sin^2 \omega_o t> = \frac{\int t \sin^2 \omega_o t}{\int \sin^2 \omega_o t} =?? \frac{1}{2}$$ Also, how is the time important in doing this average. What I mean is, how is it justified to replace the function with its average value for times longer than its time period, how does this make sense physically? Thanks ![enter image description here](http://i.stack.imgur.com/kbBqN.jpg)


Schrödinger function: Separable wave function with even potential function of x

I have done the Problem 2.1 in Griffiths' quantum mechanics, and it seems not making sense to me. What if the wave function isn't symmetric at all? Then obviously the proof doesn't work. The solution confuses me. > If $V(x)=V(-x)$ then the "position" wave function can always be taken to be > either even or odd.


The Heisenberg uncertainty principle: Interpreting $\Delta p$, $\Delta t$, etc

(1) I have a textbook question that states the following: An electron has a speed of 500 m/s with an accuracy of 0.004%. Calculate the certainty with which we can locate the position of the electron. The solution starts with $$\frac{\Delta p}{p}=\frac{0.004}{100}$$ I don't understand how the question is being interpreted to give the above expression. (2) The very next question says: The average lifetime of an excited atomic state is $10^{-9}$s. If the spectral line associated with the decay of this state is 6000 Angstrom, estimate the width of the line. The solution begins with $\Delta t = 10^{-9}$s. Again, I don't understand how this is arrived at. I'd expect it to be $t=10^{-9}$s. * * * Can somebody give me an intuitive understanding of uncertainty (perhaps in words) that I can apply to these two problems? I know the statistical definition, but I haven't found it very helpful in this context.


Measuring nucleons using electron beams

sorry if the question is too elementary. From: The Britannica Guide to Particle Physics: > The sizes of atoms, nuclei, and nucleons are measured by firing a beam of > electrons at an appropriate target. The higher the energy of the electrons, > the farther they penetrate before being deflected by the electric charges > within the atom. For example, a beam with an energy of a few hundred > electron volts (eV) scatters from the electrons in a target atom. The way in > which the beam is scattered (electron scattering) can then be studied to > determine the general distribution of the atomic electrons. > At energies of a few hundred megaelectron volts (MeV; 106 eV), electrons in > the beam are little affected by atomic electrons. Instead, they penetrate > the atom and are scattered by the positive nucleus. Therefore, if such a > beam is fired at liquid hydrogen, whose atoms contain only single protons in > their nuclei, the pattern of scattered electrons reveals the size of the > proton. At energies greater than a gigaelectron volt (GeV; 109 eV), the > electrons penetrate within the protons and neutrons, and their scattering > patterns reveal an inner structure. Thus, protons and neutrons are no more > indivisible than atoms are. Indeed, they contain still smaller particles, > which are called quarks. I'm confused by this. I understand that the increasing velocity of the electron particles in the beam can overcome repulsive force of the electron cloud in an atom. Thus detecting size of the atom and even particular electrons. But what happens to beam particles in MeV range? At higher velocities, when they do reach nucleus, doesn't the positive charge then attract beam particles? How does it repel them? Is it that they have so much speed that they on near miss of a proton deflect or is something else happening here? What happens on GeV range? Do electron beam particles pass through proton to allow detection of quarks? Or is there happening something completely differentTM, like total annihilation of collided particles and recreation of particles from energy. Thanks for your time...


Will a relativistic rocket be destroyed by incident photons?

I have read, that as a rocket approaches light speed, objects in front of it become blue shifted and also intensity of light increases. Does this mean that at some point any rocket will be destroyed, just because the blue-shift will cause impacting photons to become gamma rays of arbitrary high energy - so high in fact that no material could withstand it?


Would a high-explosive in a vacuum be less harmful?

Putting aside shrapnel effects, I believe that high-explosives cause damage by producing a shockwave. How do shockwaves work in space? I've managed to convince myself that a high-explosive shockwave would work "about the same" by producing a shell of high-speed gas that would arrive at the surface with pretty much the same amount of energy that you'd get in air. Is that right?


How charge distribution takes place when a battery is connected to a conductor?

When one terminal of a battery say of 1.5 volt connected to a short length wire, few electrons get transferred from battery terminal to the wire raising the potential of the wire also to 1.5 volt. We know that raising the potential of a wire requires transferring certain numbers of electrons. Now consider the same battery is connected to very long wire of some thousand kilometer length. Will the battery be able to raise the potential of the wire?


Does it matter in which direction I travel in relativity theory?

I'm not a physicist, but I'm still very interested in the relativity theory, especially in how the twin paradox is explained. Actually, it does not make sense to me and I hope you can answer my following question to help me understand how it works: Does it matter in which direction i travel in the relativity theory? When I first read about the theory, I thought, that it does not matter if A moves away from B, or B moves away from A, because every party will see the effect of "time going slower" for the other pary, meaning A sees B aging slower and B sees A aging slower. But regarding the twin paradox, when A leaves the earth with insane speed, and returns years later, A has aged less than the people on earth. If this is actually true, then my previous assumption is wrong and it **does** matter if A moves away from B or the other way round, because the effect could be turned around, so that the earth ages less than A, if the earth moved away from A instead. If this is true, then this brings up a next question: How can we know if B moves away from A or A moves away from B? Since the earth, our solar system, and even our galaxy is already traveling in the universe at some speed in a specific direction, we could actually stop moving by flying in the exact opposite direction at the same speed. **but** would this be considered as the earth moving away or as we moving away? I really hope you can help me with that :) **Update for @Alfred Centauri's answer** `the stay-at-home twin does not change direction while the travelling twin does.` When I look at it from only the travelers perspective, then yes. But when I look at it from the earth's perspective, then the earth indeed changes its direction, while the traveler does not. Why is that changing direction thing that important? As I could see in Will's answer, the change of speed in universe seems to be the critical difference. In the time where the traveler is still on the earth, both the traveler and the earth move at the same speed in the universe, which means they age exactly the same. As soon as the traveler starts to accelerate or decelerate (move faster or slower than the earth) he **slows** his _own_ time, no matter if he accelerates or decelerates. If this is true, this means that not only the relative speed of the objects influences the time. Also the _change of speed_ in the universe does. But this opens another question: What if 2 travelers move to the exact same position in the universe, but traveler A goes there at twice the speed? Are the two travelers the same age when they meet at the destination? Or is one traveler older because he moved faster?


Application of Calculus in Physics

Why do we apply Calculus in Physics when most of the quantities are not continuous and are not symmetrical at all levels of magnification? Aren't most, if not all, forms of Matter and Energy discrete? We talk about differential elements of energy, charge, fields, current, liquids and so many other quantities which don't or can't exist and we derive results from such assumptions. How is this right?


Shell model of an odd-odd nucleus: $^6$Li

Lithium-6 isotope has an approximate magnetic momentum of $0.88\ \mu_N$ in its fundamental nuclear state. I'm trying to find its angular momentum and parity. I found in a standard table: $I=1^+$ and while I get the parity part, I don't understand the rest. What I have done is filling the shells separately for protons and neutrons and finding that both have their outer particle in the $1 P_{3/2}$ state. Sum of these gives possible total angular momenta to be either $0$, $1$, $2$ or $3$. So my question is: how do you choose between these to find the real value $I=1$?


Quantum cryptography: encryptions

I am studying quantum cryptography and I have a very basic question. Suppose A and B share a secret key k, where k=0 or 1. A wants to send one qubit to B. What A does is, if k=1, she 'flips' the qubit (i.e. applies the Pauli X matrix), otherwise she leaves the qubit as it is. She sends this to B. Now B, knowing the value of k, either flips it or leaves it as it is. I am wondering why this protocol is insecure. How can an eavesdropper E cause problems?


Time reversal operator symmetry of dirac lagrangian

I want to prove time reversal symmetry of Dirac Lagrangian, I have some problems with calculations. I start with \begin{eqnarray} T\psi T = U \psi \end{eqnarray} \begin{eqnarray} T\bar{\psi } T = \psi^\dagger U^\dagger\gamma_0 \end{eqnarray} and I take time reversal transformation of the first part of Dirac Lagrangian \begin{eqnarray} T\bar{\psi } \gamma_aD^a\psi T = {\psi }^\dagger U^\dagger\gamma_0 \gamma_aUD_a\psi \end{eqnarray} I find, in order to parity invariant of the last part of Dirac Lagrangian \begin{eqnarray} U^\dagger\gamma_a U = \gamma^a \end{eqnarray} this relation must satisfy. But it is not true for the part \begin{eqnarray} T\bar{\psi } \psi T = {\psi }^\dagger U^\dagger\gamma_0 U \psi \end{eqnarray} from here I find \begin{eqnarray} U^\dagger\gamma_0 U = \gamma_0 \end{eqnarray} This is not consistent with $U^\dagger\gamma_a U = \gamma^a $ for my convention. Here I used (-+++) metric and $TV^aT =V_a$, $\bar{\psi}= \psi\gamma_0$. Can you show me my fault.


Lennard Jones liquid vapour phase transition

I am trying to generate a phase diagram of pressure versus temperature in a NPT ensemble using molecular dynamics method. I am using the Lennard Jones potential with 108 particles and Berendsen thermostat and barostat to fix the parameter. In order to observe phase transition I plot the radial distribution function (RDF) and also density versus temperature graph for various fixed temperature. However it is hard to differentiate liquid and gaseous phase from RDF. Another peculiar thing is that there should be a discontinuity in density as the transition temperature is reached, which I observe only for solid liquid transition but not for the liquid vapor transition. The change in density should be huge for liquid vapor transition. Any idea on what possibly goes wrong? Periodic boundary   cut off radius = 2.5   Truncated and shifted lennard jones potential   number of particle = 108   number of steps = 2000 timestep = 0.004   Berendsen barostat of scale coeefficient of 0.1 is used to scale the size of simulation box The density versus temperature graph for pressure of 1.0MPa:   http://i.stack.imgur.com/x0d0p.jpg The solid liquid transition can be seen but not the liquid vapor transition.


How to interpret band structures

I'm currently taking a Solid State Physics class, and is currently reading about the quantum mechanical description of solids. I then came across the following figure: ![Band structure for Al](http://i.stack.imgur.com/9ydGp.png) It's supposed to be the band structure for aluminium. My question is basically: How do I interpret these band structures ? I can't even see why they cut of the region from $X$ to $\Gamma$ at the dotted line in the first figure, and not in the second :/ Why is the last bit of $X$ to $\Gamma$ that you see in figure 2, not important in figure 1 ? I understand that, the $X$, $\Gamma$, $K$, $L$ and $W$ are points of symmetry on the brillioun zone. But what does it mean for the electron(s) ? Only that they have higher energies at some points ? I'm just having some trouble understanding what this band structure tells me, and what I'm able to see/do with it ? So can anyone maybe explain it, maybe even easily, what I actually have to understand when I see this ? Thanks in advance.


what happens during collision at molecular level?

When two massive bodies collide we treat them simply by their mass interactions with momentum and energy conservations. Down at the molecular level it must be more like a field particle collision for the particles coming in close contact, why dont we consider all these effects ? Are they included in our equations already ?


How come squeezing a water bottle makes the water come out?

This seems natural, but I can't wrap my head around it when I think about it. When I squeeze an open bottle filled with water, the water will spill out. When I squeeze a bottle, the material collapses where I squeeze it, but expands in other areas, resulting in a constant volume. If the volume is constant, then I would think that the water shouldn't spill out. If I were to guess, there is something related to the pressure my hand is creating inside the bottle, but I'm not entirely sure.


Rate of radiation of infra-red ray from a hot body?

I have the equation $\frac{dE}{dt}=a(\tau^4-\tau_s^4)$ however I can't find anywhere what E is supposed to be. It is not pointed out anywhere in the text and what is confusing is that in engineering E was always Young's modulus. Though the pages heading is Radiation of infra-red rays. Is E the radiation of infra-red rays? Because I have another equation down the bottom saying that the equation $\frac{dT}{dt}=-k(T-T_s)$ was derived from it. I can't see the relation. It just looks like they have changed the letters.. without any meaning.


Charge or Charges on the oil droplets?

Due to my poor understanding of various paper and other material provided by google on Millikan's oil drop experiment I ask the following question here.   1. As the drops were charged by either friction or x-ray radiation or both, how did they ensure that they found out the charge of a **single** electron? For all we can say it just as well may have been 2 or more electrons!    2. Also even after the drops were charged by lets say a single electron (in excess) how could one be sure that it did not form a dipole? Why assume spherical symmetry?


What is the motivation behind $\tau=\mu \frac{\mathrm{d}u}{\mathrm{d}y}$ for Newtonian fluid?

Where does the motivation come from? $\tau$ is the shear stress, $u$ is the velocity and $\mu$ is the shear viscosity. EDIT: Since I wrote the question on phone I wasn't clear enough about what I was actually wondering about (see comments below); thus let me clarify. I was simply wondering where did the derivative term come from and not why the expression is linear in the derivative term or why it is the first derivative and not e.g. the second and so on. The diagram in the answer I accepted shows exactly how the gradient comes into the picture.


Potential of the pseudo Nambu-Goldstone boson

Pseudo Nambu-Goldstone bosons appear in breaking approximate symmetries, for instance, the QCD axions. Many literatures just write the action of those pseudoscalars without any explanation. In general, how is the effective action, especially the potential, of these Pseudo Nambu-Goldstone bosons obtained?


How can gas from compressed air can "take" heat from surrounding environment?

I have recently been reading about why a can of compressed air gets cold when the air it contains is discharged. From what I understand the change from a liquid to a gas requires energy and therefore heat is drawn from the surrounding environment. What confuses me is how the phase change sucks up heat? If energy is required to complete the phase change wouldn't that mean that the phase change would not take place unless heat was added? I guess I am confused as to how the phase change extracts the heat from the surrounding environment. If I wanted to compress a gas into a liquid I would have to use force or my own energy to accomplish this, yet when going from liquid to gas the energy is just taken from the surrounding environment. This is confusing to me.


Non-zero charge density due to Lorentz contraction in current carrying wires

In trying to answer this question I came across the following problem. The original question relates to the idea that what looks like a magnetic field in one reference frame, ends up as an electrostatic field in another frame through special relativity. I'm going to imagine that the wire is just two blobs, one of positive charge, the other of negative charge (same magnitude of charge density), and I can choose to move them relative to one another along the length of the wire (Basically I'm ignoring all of the real physics of currents in wires to get a pristine situation). We have a test charge; we're going to describe the world in it's rest frame. Case 1 Both the positive and negative charge densities are at rest, in the test particle's frame. Obviously there is no electric force on the test charge, since the two charge distributions cancel each other out. There is no magnetic force either Case 2 Positive charge density is moving with velocity +v, negative charge density is moving with velocity −v. By symmetry, any relativistic effects on the the charge density must affect both densities in the same way; therefore no electric force. But we do have moving charges, so we'd see a magnetic force. Case 3 Positive charge is stationary relative to our test charge; negative charges are moving relative the positive charge and our test charge; length contraction yields higher negative charge density, and an electrostatic force on the test particle. Case (2) would seem to imply that we only see pure magnetic fields in a special symmetric inertial frame -- that frame where the effects of length contraction serve to make the two charge densities exactly the same. However, Case (3) is the normal lab measurement situation. I'm not sure if anyone has tried to put an upper bound on the magnitudes of any electrostatic forces related to running a real current through a real wire. If I run a computations based on a numerical example for drift velocity I get a result that the "extra" linear charge density for a 1mm diameter copper wire is <<1 one electron per meter; a deviation that would be very hard to measure experimentally. If this logic holds we'd have:   1. Theoretically, lab measurements of real wires do have some electrostatic forces involved, although these forces are way too small to have significant effects (and are probably smaller than thermal effects),   2. And we have to go to a reference frame where both charge densities are moving in order to see pure magnetic effects; I guess that this could be true, but I've never heard anyone refer to it in the application of special relativity to classical E&M; is there a flaw in my logic?


Materials with different gravitomagnetic permeability?

If you start with general relativity, and assume small perturbations around a nearly flat metric, it is possible to obtain linearized equations of gravity that look a lot like Maxwell's equations, also called GEM equations. One of them reads: $$\nabla \times \mathbf{B}_\text{g} = 4 \left( -\frac{4 \pi G}{c^2} \mathbf{J}_\text{g} + \frac{1}{c^2} \frac{\partial \mathbf{E}_\text{g}} {\partial t} \right)$$ Where $\mathbf{E}_g$ is the (static) gravitational field, $\mathbf{B}_g$ is a gravitational analogon to the magnetic field that encodes some general- relativistic effects. You see that it is coupled to the mass current density $\mathbf{J}_g$, which has the effect that rotating or moving masses have a distinct (attractive or repulsive) gravitational effect. For example, two massive rings rotating in the same direction along a common axis will experience a small additional repulsion (or a slightly reduced net gravitational attraction). These effects are typically very small, but have been observed, e.g. by Gravity Probe B. Now compare with the corresponding electromagnetic equation: $$ \nabla \times \mathbf{B} = \frac{1}{\epsilon_0 c^2} \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$$ The factor in front of the electric current $\mathbf{J}$ is the magnetic permeability $\mu_0 = 1/\epsilon_0 c^2$. In a medium, it becomes $\mu_r \mu_0$ instead. This relative permeability $\mu_r$ is material dependent and source of a lot of interesting phenomena and technical applications, including transformators. You can simularly identify the gravitomagnetic permeability $\mu_0^g = 4\pi G/c^2$. (Don't know if I should put the minus sign in the constant or not.) **Now I'm wondering if there is a corresponding relative $\mu_r^g$. Are there materials that show a different GM permeability? Maybe event time-dependent, or manipulable through external fields?** From a microscopic persepective, it seems this would be somehow related to the spin or angular momentum of particles in the material. The effect is probably tiny, but if one would be able to vary it on very short time scales, it could maybe produce measurable effects. If found a mention of this idea in a 1963 paper (there's a PDF on the net if you Google the title):   * Robert L. Forward: _Guidelines to Antigravity_ \- American Journal of Physics, March 1963, Volume 31, Issue 3, pp. 166 Unfortunately, more recent publications on this topic seem to be of dubious veracity (going in the direction of Podkletnov et al.).


Superconductors and magnetic field

I am working on a project, and I have a question: Concerning superconductors considered as diamagnetic material, what is the equation that describes the repulsive force to a diamagnetic material in a magnetic field?


Does magnetised material lose weight?

I've had argument on a forum about this. People think, that: if the same object is magnetised, it has less energy. so by $E = mc^2$ it weighs less than an object that is not magnetised (hence spin of its particles is chaotic). I've thought about experiment :) Let's say we have square of 4 protons and another square of 4 antiprotons. If we magnetise each square and annihilate them, do we get the same energy that we would get if only the antiprotons were magnetised?


Conversion of mass to energy in chemical/nuclear reactions

Is mass converted into energy in exothermic chemical / nuclear reactions? My (A Level) knowledge of chemistry suggests that this isn't the case. In a simple burning reaction, e.g. $C+O_2\to \text{CO}_2$, energy is released by the CO bonds forming; the atoms lose potential energy when they pull themselves towards each other, in the same way that a falling object converts GPE to KE. There are the same number of protons, electrons etc. in both the reactants and products. I would have assumed that this reasoning extends to nuclear fission/fusion as well, but one physics textbook repeatedly references very small amounts of mass being converted into energy in nuclear reactions. So I just wanted to know if I was wrong about either of these types of reactions, and if so, what mass is lost exactly?


Time dilation $t'=(1-\frac{v}{c})t$ instead of $t'=\frac{1}{\gamma} t$

Suppose we have a train moving. When the origin of train's frame coincides with the origin of observers frame; the the time is set to zero. At that very instant, a photon is emitted from train towards the direction train is moving. After time $t$ measured by observer at rest the photon will be at a distance $ct$ and the train at a distance $vt$; but the photon is at a distance $ ct- vt$ from the train, now the time that the passenger on train measures for the photon to reach that point is the distance divided by the velocity of light ( which of course is $c$) so time measured by observer on train is $t'=\frac{ct-vt}{c}= (1-\frac{v}{c})t$ . where $t$ is time measured by observer at rest. Where is the mistake in this reasoning?


How do "holographic plates" work?

I asked a question about laser stage lighting over at Audio Video Production, and received an excellent answer that explained that laser clusters are generated from a single beam via something called a "holographic plate". I'd imagine they're made up of a crystal-like structure of materials with different refractive indices, but I'm not exactly sure. If that's the case, surely such tricks with refraction would cause component colours to be refracted at different angles, but this doesn't seem to be the case. How do these really work? **Update:** There seems to be some confusion. Cluster laser lighting units seem to be able to produce arbitrary numbers of output beams, and have some rudimentary control their direction, all from a single laser diode. I'm not asking about the principles of holograms, but rather about the mechanism by which a single laser beam passing through the plate results in multiple distinct beams coming out of the other side.


Force through quantum mechanics

In classical physics force is: $$F=\frac {dp}{dt}$$ How about quantum mechanics? In Old Quantum Mechanics momentum is: $p=\hbar \cdot k$ so force will be: $$F=\hbar \frac {dk}{dt}$$ what does $\frac {dk}{dt}$ mean?


Is there anything physically infinite?

I can't think of a single thing that could be infinite. Because the universe is expanding, isn't it? But there is an ever-changing barrier, so why could there be anything infinite, both hypothetically and realistically?


Why quantum hall effect has chiral edge state?

The most popular explaination may be the following: in magnetic field, electrons move in cycolotron orbits, such cycolotron orbits ensure electrons to move in one direction at the edge. That is why the edge state is chiral. But I think this picture is too classical. Could anyone provide me with other explaination about this question? Does it has relation with time-reversal broken, chern number, or something else with topology? Thanks in advance~


Why elastic materials are discribed by tensors?

I am starting to read about elasticity of thin surfaces and I don't understand why tensors play such a major part? What are the tensors describing about the material? And just to clarify - Is there some calculation often done with 2nd order tensors (describing 3d materials) that is common for elastic materials?


Can low-gravity planets sustain a breathable atmosphere?

If astronauts could deliver a large quantity of breathable air to somewhere with lower gravity, such as Earth's moon, would the air form an atmosphere, or would it float away and disappear? Is there a minimum amount of gravity necessary to trap a breathable atmosphere on a planet?


saturation property of nuclear forces ? and its relation binding energy per nucleon constantcy?

Please explain why binding energy per nucleon is constant(pratically) for atomic number, A, larger than 30 and less than 170, and explain the saturation property of nuclear forces with analogy that is easy to understand?


Given expectation values for E and B, can you find an associated state?

When we quantize the electromagnetic field, we develop the concept of the field operator $A(\vec{r},t)$ and the simultaneous eigenstates of momentum and the free field Hamiltonian (i.e., each eigenstate is given by specifying the number of photons with momentum $k$ and polarization $\mu$). We can then construct the operators for the electric and magnetic fields, and we can calculate their expectation values for an arbitrary state. Now, suppose the expectation value of the electric field is $E(\vec{r},t)$ and the magnetic field is $B(\vec{r},t)$. Assuming $E$ and $B$ obey Maxwell's Equations, can we construct a state that has these expectation values? Is it unique, or could there be multiple states with the same expectation value for $E(\vec{r},t)$ and $B(\vec{r},t)$? What if the expectation values are time independent (i.e., static fields $E(\vec{r},t)=E(\vec{r},0)$ and $B(\vec{r},t)=B(\vec{r},0)$ for all $t$)?


How do you tell if a metric is curved?

I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me. To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - $M$ and $a$. $M$ is identified as some mass, and $a$ is identified as angular momentum per unit mass. He says that this metric reduces to flat space in the limit $M \rightarrow 0$, and is given by $$ds^2 = -dt^2 + \frac{r^2 + a^2 \cos^2\theta}{r^2 + a^2}dr^2 + \left(r^2 + a^2 \cos^2\theta \right)d\theta^2 + \left(r^2 + a^2 \right)\sin^2\theta d\phi^2 $$ and $r$, $\theta$ and $\phi$ are regular spherical polar co-ordinates. But I don't understand why this space is obviously flat. The Schwarzschild metric also contains terms involving $dt^2$, $dr^2$, $d\theta^2$ and $d\phi^2$ but is curved. I always thought that a metric with off diagonal elements implied a curved space, but clearly I was very wrong. **Question:** How do you tell if a metric is curved or not, from it's components?


How long will it take to temperatures of two solids to become equal?

If I have two solids with commmon area where they are touching each other, is there any way for calculating the time it will take to them to become equal? For example, I have two blocks of different materials with different temperatures. They touch each other on one side. Can I calculate the time until their temperatures become equal? ![enter image description here](http://i.stack.imgur.com/dxNsp.png) EDIT: According to your answer, I found this formula: $$ \Delta Q=\frac{\lambda S(T_2-T_1)}{L}\,\Delta t $$ Can I calculate $\lambda$ for my example as proportion of weights? $$ \lambda_{r} = {m_{1}\lambda_{1}\over m_{1} + m_{2}} + {m_{2}\lambda_{2}\over m_{1} + m_{2}} $$


How is quantum mechanics compatible with the speed of light limit?

Consider a free electron in space. Let us suppose we measure its position to be at point A with a high degree of accuracy at time 0. If I recall my QM correctly, as time passes the wave function spreads out, and there is a small but finite chance of finding it pretty much anywhere in the universe. Suppose it's measured one second later by a different observer more than one light second away and, although extremely unlikely, this observer discovers that electron. I.e. the electron appears to have traversed the intervening distance faster than light speed. What's going on here? I can think of several, not necessarily contradictory, possibilities:   1. I'm misremembering how wave functions work, and in particular the wave function has zero (not just very small) amplitude beyond the light speed cone.   2. Since we can't control this travel, no information is transmitted and therefore special relativity is preserved (similar to how non-local correlations from EPR type experiments don't transmit information)   3. Although the difference between positions is greater than could have been traversed by the electron traveling at c, had we measured the momentum instead, we would have always found it to be less than $m_e c$ and it's really the instantaneous momentum that special relativity restricts; not the distance divided by time.   4. My question is ill-posed, and somehow meaningless. Would anyone care to explain how this issue is resolved?


Irreversibility and the Fermi golden rule

When a quantum system is perturbatively coupled to a continuum of states, one uses the Fermi's golden rule to compute the rate of transition form an initial state to a set of states contained in an infinitesimal volume of phase space (the energy density factor). However, I am still puzzled by the fact that the continuum feature of the spectrum implies this non reversible evolution whereas a discrete one would imply Rabi oscillations and finite time reversibility. How does one makes the transition from one to the other?


Two masses attached by a spring

> Suppose there is a system of two masses $M_1$ and $M_2$ connected by a > spring with constant $K$ on a frictionless surface. What is the frequency of > this system's oscillation? So I have fixed a coordinate system with the first mass at position $x_1$ and the second at $x_2$ so that if the spring has length $l$ at equilibrium, we have: $$m_1 x_1'' = k \left([x_2-x_1]-l\right)$$ $$m_2 x_2'' = -k \left([x_2-x_1]-l\right)$$ but I don't know how to solve this to find frequency. Any hints? I suppose I could try to solve this system of differential equations, but I don't really know how.


Is my textbook in error about this geometric optics relation?

Given a centered optical system (having an axis of rotational symmetry), let $H_1$ be the intersection of the optical axis and the "principal object plane" (I'm studying in French and have no idea how to translate that... it's the plane whose image is a plane with a scaling factor of 1). Let $H_2$ be the intersection of the optical axis and the "principal focal plane", in other words the image of $H_1$. ![](http://i.stack.imgur.com/j9bQC.gif) My book states that the distances between those points and the focal points, $f_1=H_1F_1$ and $f_2=H_2F_2$, verify the following relations: $$n_1f_1=-n_2f_2$$ $$\frac{n_2}{f_2}=-\frac{n_1}{f_1}=V$$ Where V is the vergence. But these seem to contradict eachother. From relation 1, we have $$n_1=-\frac{n_2f_2}{f_1}$$ Substituting into the second relation: $$\frac{n_2}{f_2}=\frac{n_2f_2}{f_1^2}$$ $$1=\frac{f_2^2}{f_1^2}$$ $$f_1=\pm f_2$$ But the book then goes on to say that that's only true when $n_1=n_2$.


Breaking ocean waves on other planets possible?

I have always wondered if there were an ocean on another planet with stronger or weaker gravity, would it be possible to have breaking waves?


Free body diagram and Newton's second law

> Two bodies of mass $m_1=50 \text{kg}$ and $m_2 = 10 \text{kg}$ are connected > with a light rope through a pulley (no friction between the rope and the > pulley). $m_2$ moves on surface with angle $37^{\circ}$ with a friction > coefficient $\mu_k=0.5$. Force $\vec F$ acts on $m_1$. The system was moving > with a constant velocity for some time until the force stopped and then the > system started to accelerate. What's the acceleration of the system? ![Free body diagram](http://i.stack.imgur.com/cp4JM.png) I assumed that the system accelerates to the left, therefore the equation for the $m_1$ would be: $ \begin{cases} N_1=m_1 g\\\ T-N_1 \mu=m_1a \end{cases} $ Therefore $T=m_1(a+g \mu)$ For $m_2$ the equation would be: $ \begin{cases} N_2=m_2 g \cos 37^{\circ}\\\ m_2 g \sin 37^{\circ} - T - N_2 \mu=m_2a \end{cases} $ Therefore $m_2 g \sin 37^{\circ} -T- m_2 g \cos 37^{\circ} \mu = m_2a $ The solution would be: $100 \cdot 0.2024 - 50a -250=10a \\\ a \approx -3.829 \frac{\text{m}}{\text{s}^2}$ Which is wrong according to the answers. The answer should be $\approx -5.83\frac{\text{m}}{\text{s}^2}$ which you can get pretty close if instead of $10a$ in the last step it was $-10a$. So I'm pretty sure that my mistake here is with the signs (of the acceleration), but I can't figure out why I'm wrong.


Non-conservative electric fields due to changing magnetic flux?

What I read in several places, tells me that, the fact Coulomb's Law follows _inverse-square law and gives a force which is radial_ , implies that a static electric field must be _conservative_.(In short, Coulomb's law _is_ conservative!) But the electric field in presence of a time-changing magnetic flux (either time changing fields or a change of area of the loop, Motional EMF) is non-conservative, since closed-loop line integral of electric field is no longer zero but is $-\frac{d\phi}{dt}$ where $\phi$ is the magnetic flux through the surface bound by the loop along which the line integral is carried out. I have the following question:- Consider a situation for the sake of presenting my argument, where there are only steady currents and no accelerated charges, and the changing flux is due to a wire sliding over another U-shaped wire and thus changing the area bound by the loop formed by the U-shaped end and the moving wire.![enter image description here](http://i.stack.imgur.com/B4b5X.png) In this case, all the components and the charges (metallic free electrons and the positive kernels) are unaccelerated (i.e. on an average) and hence, there should be no radiation of electromagnetic waves. Therefore, whatever the electric field produced, which is responsible for the existence of current in the loop, must be produced by a particular non-accelerated charge distribution in the apparatus. But, whatever the field produced by any complicated arrangement of charges, through superposition must follow the inverse-square and the radial dependence of Coulomb's law and must be conservative, since all the superposed fields are conservative in themselves as the individual charges follow the conservative coulomb's law. > **So, how can a non-conservative configuration of the field, result from the > superposition of fields produced by an arrangement of individual charges > which follow a law (Coulomb's) which is conservative , assuming the absence > of electromagnetic waves?**


Gauss's Law in action

![enter image description here](http://i.stack.imgur.com/gJq0p.jpg) Need someone to tell me if I got this done correctly (a) Draw Gaussuian cylinder inside the black cylinder to find charge enclosed $Q_{en} = Q(\frac{r}{a})^2$ Apply Gauss's Law $E2\pi r \ell = \dfrac{Q\left ( \frac{r}{a} \right )^2}{\varepsilon_0}$ $E = \dfrac{Q\left ( \frac{r}{a} \right )^2}{2 \pi r \ell\varepsilon_0} = \dfrac{Qr}{2\pi \varepsilon_0 a^2\ell}$ $E = \dfrac{\rho r}{2 \varepsilon_0}$ (b) Basically same principle, but the total charge of the black cylinder $E = \dfrac{\rho a^2}{2r \varepsilon_0}$ (c) 0 under static equilibrium (d) The enclosed charge should be a sum of cylinders' charges. $Q_{en} = \rho V + \lambda \ell = \rho \pi a^2 \ell + \lambda \ell = \ell ( \rho \pi a^2 + \lambda)$ Hence $E2\pi r \ell = \dfrac{\ell ( \rho \pi a^2 + \lambda)}{\varepsilon_0} \implies E =\dfrac{ \rho \pi a^2 + \lambda}{2 \pi r \varepsilon_0} $ The term looks very awkward, so it's probably wrong. Thank you for reading


Determine radius, speed, radial and tangential component of velocity

![enter image description here](http://i.stack.imgur.com/Qdhv5.png) I need help mainly with the first question. If you can solve also the second problem,I'd greatly appreciate it.


gravitational convergence of light

light has a non-zero energy-stress tensor, so a flux of radiation will slightly affect curvature of spacetime > **Question:** assume a flux of radiation in the $z$ direction, in flat > Minkowski space it will propagate and spread slightly in the $x$ and $y$ > directions. But what happens if we take into account self-gravitational > effects of light onto itself? will the radiation flux self-gravitate and > eventually reconverge?


An EM problem of polarized sphere

Today, I'm doing an EM problem and my question is exactly the same as this site shows: http://www.physicsforums.com/showthread.php?t=133501 To reiterate my confusion, why we can't use that integral formula at the end V(r)=1/4 pi epislon...+1/4 pi epislon... This formula appears right in the same chapter of the problem. I know we can solve it by applying the Gauss Law, but why this formula won't work. I calculated it by using this formula, and I get a totally different answer.


Is changing angle of grooves in a Fresnel lens important for the lens to work?

I'm thinking of making a huge Fresnel lens out of a plastic sheet I found in my storage. As it's for fun, and I'm not very concerned about precision, I thought I could make it using a nail, rope and two blades positioned so as to form a sharp angle between their cutting edges (or simply form a "V" shape). From looking at a flat Fresnel lens condensation diagram and sketching my own "expanded" lens, I figured I would need to account for groove angle changing when cutting grooves to allow the resulting lens to focus light. But as I'm not very knowledgeable in optics, I don't know if it's important. ![enter image description here](http://i.stack.imgur.com/PwxAW.png) Will the lens still focus light in one spot if the grooves are cut at the same angle? If so, which angle should I choose then?


What causes drag in a fluid?

What causes resistance of an object to motion within a fluid like water? Please explain to me the molecular dynamics of the situation.


What happens when astronomical bodies made up of Dark Matter collapse or collide?

Does dark matter not produce heat or radiation after a collision etc?   Or, we detect radiations having no generators?


Bloch wave function orthonormality?

there is this text book that is giving me a hard time for a while now: It shows that Bloch wave functions can be written as $$\Psi_{n\vec{k}}\left(\vec{r}\right) = \frac{1}{\sqrt{V}}e^{i\vec k \vec r}u_{n\vec k}\left(\vec r\right),$$ which is fine to me. It also states that the Bloch factors $u_{n\vec k}\left(\vec r\right)$ may be orthonormalized on the (primitive) unit cell volume $V_{UC}$: $$\frac{1}{V_{UC}}\int_{V_{UC}}d^3r u^*_{n\vec k}\left(\vec r\right) u_{n'\vec k}\left(\vec r\right) = \delta_{n n'}$$. However, and here starts my problem, it then concludes that therefore, the Bloch functions $\Psi_{n\vec{k}}\left(\vec{r}\right)$ fulfill $$\int_V d^3r \Psi^*_{n\vec{k}}\left(\vec{r}\right) \Psi_{n'\vec{k'}}\left(\vec{r}\right)=$$ $$=\frac 1 V \sum_\vec{R} \int_{V_{UC}\left(\vec R\right)}d^3r e^{-i\vec k \left(\vec R + \vec r\right)} u^*_{n\vec k}\left(\vec R + \vec r\right) e^{i\vec {k'} \left(\vec R + \vec r\right)} u_{n'\vec{k'}}\left(\vec R + \vec r\right)=$$ $$=\frac 1 N \sum_{\vec R} e^{i\left(\vec{k'}-\vec{k}\right)\vec R} \frac{1}{V_{UC}}\int_{V_{UC}} d^3r u^*_{n\vec{k}}\left(\vec r\right) u_{n'\vec{k'}}\left(\vec r\right)=$$ $$=\delta_{\vec{k'}\vec{k}}\delta_{n'n}$$ with lattice vectors $\vec R$ and crystal volume $V = N V_{UC}$. **But I just don't get the last two lines.** I mean, the integral in the second last line actually should read $\int_{V_{UC}} d^3r e^{i\left(\vec{k'}-\vec{k}\right)\vec r}u^*_{n\vec{k}}\left(\vec r\right) u_{n'\vec{k'}}\left(\vec r\right)$, shouldn't it? And, if that is true and I didn't miss something important already, I can't understand how that would yield these two $\delta$s ... I'm almost sure I missed something, but I just desperately keep fail getting it, so any help would be greatly appreciated! **EDIT** Thank you very much for your reactions! However, it seems I failed to state my problem clear enough, so I figured it might be best to tell you what my approach so far was step by step so may be someone can see where I actually go wrong: Starting with $$\int_V d^3r \Psi^*_{n\vec{k}}\left(\vec{r}\right) \Psi_{n'\vec{k'}}\left(\vec{r}\right),$$ I partitioned the integration domain $V=NV_{CU}$, thus getting a sum of integrations over the unit cell volume, yielding $$\sum_{\vec R}\int_{V_{UC}} d^3r \Psi^*_{n\vec{k}}\left(\vec{r} + \vec R \right) \Psi_{n'\vec{k'}}\left(\vec{r} + \vec R\right),$$ which happens to be exactly the second line when exploiting $\Psi_{n\vec{k}}\left(\vec{r}\right) = \frac{1}{\sqrt{V}}e^{i\vec k \vec r}u_{n\vec k}\left(\vec r\right)$. I then proceeded using $\Psi\left( \vec r + \vec R\right)=e^{i\vec k \vec R}\Psi\left(\vec r\right)$. But this yields $$ \sum_{\vec R} e^{i\left(\vec{k'}-\vec{k}\right)\vec R} \int_{V_{UC}} d^3r \Psi^*_{n\vec{k}}\left(\vec r\right) \Psi_{n'\vec{k'}}\left(\vec r\right)$$ which disagrees with the third line in the book where the integral is over the Bloch factors $u_{n\vec k}\left(r\right)$ only. However even assuming this is just a typo (which I'm not so sure of ...), I would be confronted with the integral $\int_{V_{UC}} d^3r e^{i\left(\vec {k'} - \vec k\right)\vec r} u^*_{n\vec{k}}\left(\vec r\right) u_{n'\vec{k'}}\left(\vec r\right)$ and I can't see how those two $\delta$s would arise from that either. Thank you all again for your reactions and I hope I know actually stated my problem clearly.


Explanation for the minus sign in $\Omega_3$ in the Kappa symmetry of the Green - Schwarz formalism for F1 strings

> _Just so that there can be more higher - level physics questions here, let > me post this question + answer. Also because I'm a bit sad that there are > almost no questions on the Green-Schwarz formalism._ In the Green-Schwarz formalism, for F1 strings, we have the action $$S=S_1+S_2$$ Where $${S_1} = - T\int_{}^{} {\sqrt { - \det \left( {{\Pi _{\alpha \mu }}\Pi _\beta ^\mu } \right)} {{\text{d}}^2}\sigma } $$ and $S_2$ is the additional action term that arises through Kappa Symmetry. The transformations of kappa symmetry for F1 strings are intuitively related to that for D0 branes, etc. They are given by: $$\begin{gathered} \delta {X^\mu } = {{\bar \Theta }^A}{\gamma ^\mu } \\\ \delta {\Theta ^A} = - \delta {{\bar \Theta }^A}{\gamma ^\mu }{\Theta ^A} \\\ \end{gathered} $$ So that: $$\delta \Pi _\alpha ^\mu = - 2\delta {\bar \Theta ^A}{\gamma ^\mu }{\partial _\alpha }{\Theta ^A}$$ From this transformation, the variation in $S_1$ by kappa symmetry, is almost trivially (not that trivial, thiough...): $$\delta {S_1} = \frac{2}{\pi }\int_{}^{} {\sqrt { - \lambda } {\lambda ^{\alpha \beta }}\Pi _\alpha ^\mu \delta {{\bar \Theta }^A}{\gamma _\mu }{\partial _\beta }{\Theta ^A}{{\text{d}}^2}\sigma } $$ Here, $$\begin{gathered} \lambda = \det {\lambda ^{\alpha \beta }} \\\ {\lambda ^{\alpha \beta }} = {\Pi _{\alpha \mu }}\Pi _\beta ^\mu \\\ \end{gathered} $$ To determine $S_2$, howefver, is not all that trivial since the simple procedure I've mentioned here, for example, is just not practical by any means. We there fore use a 2-form, form $\Omega_2$, such that: $${S_2} = \int_{}^{} {{\Omega _2}} = \int {{\epsilon ^{\alpha \beta }}{\Omega _{\alpha \beta }}{{\text{d}}^2}\sigma } $$ We may also (formally) define a 3-form $\Omega_3=\mbox{d}\Omega_2,$ so that by Stokes' theorem, we have it that: $$ \int_M {{\Omega _2}} = \int_D^{} {{\Omega _3}} $$ $M$ would be the worldsheet whereas $D$ would be it's interior, ; $M=\partial D$. There are 3 supersymmetric 1-forms prominent to us namely, ${\text{d}}{\Theta ^1},{\text{d}}{\Theta ^2},{\Pi ^\mu }$. If ${\Omega _3}$ is to be supersymmetryic, which it better be, then, we shoul'd have it that ${\Omega _3}$ involves just these 3 1-forms. A very sensible choice of 3-form for $\Omega_3$ is: $${\Omega _3} = A\left( {{\text{d}}{{\bar \Theta }^1}{\gamma _\mu }{\text{d}}{\Theta ^1} -{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{d}}{{\bar \Theta }^2}{\gamma _\mu }{\text{d}}{\Theta ^2}} \right){\Pi ^\mu }$$ Now, when I first learnt this, I was pretty confused at the "$-$" si1gn, (probably because of the fact that I was too confused to bother reading the next line of bbs : ) ...). As I guess many others may also be confused at this, let me ask and answer the question: > Is this minus sign, by any chance, related to the fact that Type IIA String > Theory is chiral? Does this mean that the above expression only holds for > the Type IIA, and not for the Type IIB ? .


Why does change in speed of a wave make it refract?

When a light wave enters a medium with a higher refractive index (e.g. from air to standard glass) and its speed decreases, why does that make it refract/bend? I understand that wavelength decreases and frequency stays the same and therefore its speed decreases, but I can't find anywhere whatsoever why the speed decrease cause the wave to refract. So could someone please explain this?


Possibility for radiation in dark matter that is not interactive with regular matter?

**Definition:** Radiation in this case does not refer to electromagnetic radiation. It refers to any kind of emission of energy, even energy that does not interact with regular matter. Just like dark matter does not interact with electromagnetic radiation, could regular matter not interact with "dark matter radiation" (I'm not talking about the usual "really high wavelength radiation" kind)?


Will a stone thrown in space move forever?

If I throw a stone on space, in a place where gravity is equal zero, will the stone move forward forever, because no air, so no friction?


Working with a Routhian for a specific system

I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: $$L=(1/2)m(\dot{r}^{2}+r^{2}\dot{\phi}^{2}sin^{2}\theta_{0})-mgrcos(\theta_{0}),$$ I obtained the corresponding Routhian: $$R=(1/2)p_{\phi}\dot{\phi}-(1/2)m\dot{r}^{2}+mgrcos\theta_{0}.$$ I'm being asked to do the following: obtain the Hamilton equation of motion for $p_{\phi}$, show that it's constant in time and find its initial value, obtain the Lagrange equation of motion for $r$ from the Routhian, and obtain the Hamiltonian When I try to do the first part (obtain the Hamilton e.o.m.) I get $∂R/∂p_{\phi}=(1/2)\dot{\phi}$ which doesn't show that it's constant in time or that angular momentum is conserved, which I think is what we're trying to show here. What am I doing wrong? Also, how can I obtain the Lagrange e.o.m. for $r$ using the Routhian? Do I just take $$\frac{d}{dt}\frac{∂R}{∂\dot{r}}-\frac{∂R}{∂r}=0,$$ since $r$ appears in the Routhian and plug in $r=r_{0}$ to find the initial value? Similarly, how do I go about obtaining the Hamiltonian from the Routhian?


Buoyancy in gases

If buoyancy is caused by pressure gradient and if the pressure of a gas is same everywhere in its container, which means there's no difference of pressures on surfaces of the lighter-than-air object, how come we observe such an effect?


Shouldn't LHC have used $p\bar{p}$ collisions, instead of $pp$ collisions, to study baryogenesis?

Baryogenesis is the physical process(es) that produced baryon antibaryon asymmetry in the early universe. That means, the laws that governed the bigbang was baryon-antibaryon symmetric. On the other hand it is usually said that LHC is probing the early universe by creating similar conditions of the early universe. My question is, by using $p\bar{p}$ collisions, instead of $pp$ collisions, we are not exactly reproducing the early conditions of the universe because the conditions we created are not baryon antibaryon symmetric, are we?


Calculating angle of projectile motion

I realize that similar questions have been asked but I'd like to put my problem here and what I have to make sure everything is right. So my problem is: I've got a rock $2.5$ meters above the ground and I shoot it out of a slingshot. It hits another rock which is $3.6$ meters above the ground, $14$ meters away from it's initial position $x$. The initial velocity of the rock is $15.9$ m/s. So the formula I know is: $ \frac{1}{2} \cdot g \cdot t^2 - (v_o \cdot \sin ( \theta ) ) + y $ I also know: $x = (v_o \cdot \cos ( \theta ) ) \cdot t$ But these don't seem to help me solve my problem as I don't have $t$ nor $ \theta$.


R charge of the chiral multiplet in $2+1$ dimensions

These are two examples that I am puzzled by,   * One can see in this paper on page 16 that for ${\cal N} =2$ theory on $2+1$ the R-charge of the $\phi$ and the $\psi$ is determined to be $\frac{1}{2}$ and $-\frac{1}{2}$. How?  Further in this case since the supercharges are in a vectorial (self- conjugate!) representation of the R-symmetry group why is the R-charge of $\psi$ and $\psi^*$ different? Since post radial quantization $\bar{Q}=S$ it follows that $Q$ and $\bar{Q}$ transform in exactly the same way under $SO(2)$ and hence shouldn't the R-charges $\psi$ and $\bar{\psi}$ been the same?   * One can see in this paper at the bottom of page 7 a statement about how the scalar component of the ${\cal N}=3$ chiral multiplet in $2+1$ dimensions is in a $SU(2)_R$ triplet. So does this (what?) determine the R-charge of the scalar field and other components in the multiplet?  What is puzzling is why did the authors in the second paper think of the R-symmetry group as $SU(2)$ for ${\cal N}=3$ rather than the $SO(3)$ that would be gotten by definition? I would have thought that the R-charges of a field are the same as the eigenvalue under the Cartan subalgebra of the R-symmetry group of the supercharge(s) which acted on the lowest helicity state state of the multiplet. But isn't the R-charge of the lowest helicity state of a multiplet merely a convention? Can't the scalar field be always chosen to be uncharged (R=0) under the $SO({\cal N})$ R-symmetry of $2+1$?


Is boson sampling a problem in 'continuous variable' quantum information?

When people generally speak of quantum information in the context of continuous variables, what is generally meant is that observables, like position/momentum or the field quadratures of quantum optics, can take a continuous range of values and that information processing tasks are performed using operators on these variables. However, most work in this area restricts attention to a countable number of modes of the radiation field (in quantum optics). In this case, the state of the system can be described using a countably infinite number of variables, like the number eigenstates in the field. But this is still called 'continuous variable' quantum information because the relevant observables still take continuous values. Is that so? If so, is the boson sampling problem in the domain of continuous variables? I don't know what exactly the relevant variables are in this case. A preliminary reading seems to suggest to me that the observable is the photon number, since what is observed is the probability after measuring in some basis. This would now mean that in the sense I think of continuous variables, boson sampling is not a problem in this domain. Am I right/wrong? Which of my statements here are false?


Free particle propagation amplitude calculation

I have a quick calculational question. In Peskin and Schroeder, Chapter 2, they want to look at the amplitude for a particle to propagate between two arbitrary points, $x$ and $x_0$, in an arbitrary amount of time t given the Hamiltonian $\sqrt{p^2c^2+m^2c^4}$. To do this, we look at the inner product of the time-evolved particle that was at $x_0$ with the $x$ eigenket: $\langle \vec{x}|\exp(-i\hat{H}t/\hbar)|\vec{x_0}\rangle$, correct? If we insert two complete sets of momenta, the integral becomes (give or take factors of $\hbar$): $\frac 1 {(2\pi)^3}\int d^3p\; \exp(-it\sqrt{p^2c^2+m^2c^4}/\hbar)\exp(i\vec{p}\cdot(\vec{x}-\vec{x_0}))$ I'm not quite sure where to proceed from here... Peskin and Schroeder somehow reach: $\frac 1 {2\pi^2 |x-x_0|}\int_0^\infty dp\;p\sin(p|x-x_0|)\exp((-it\sqrt{p^2c^2+m^2c^4}/\hbar)$. To get here, however, it seems that you would have to assume that $p$ points in the same direction as $x-x_0$ to get spherical symmetry or something. Why can we do this? Aren't we summing over all possible momenta? Thanks!


Is the Apophis asteroid a concern?

I was watching the TED lecture _How to defend Earth from asteroids_ , and the speaker Phil Plait spoke of the possibility of this asteroid hitting Earth. He also said something about a Mayan legend. He says it's not a concern, is there a consensus on that? And what is the source for this Mayan legend?


So where is mistake in the formula of wave number (magnitude of wave vector)?

I have following form for wave vector $k_2=n_2 \omega/c_0$. Now because $\omega=2 \pi c/ \lambda$, then $k_2=n_2 \omega/c_0=\frac{n_2 2 \pi c_0}{c_0 \lambda}=\frac{n_2 2 \pi}{\lambda}$. But problem is that $k$ (is magnitude of wave vector~ wave number) and according to wikipedia $k=\frac{2 \pi}{\lambda}$. So where is mistake?


Building some measurement appratus that distinguish between two mixtures

We have a measurement $M$ that distinguishs between $\rho_1$ and $\rho_0$, if it has three possible answers 1,2,3 and whenever it answers something different than 3 it's correct. $M$ succeeds with probability $p$ if for every $\rho s$ above it answer the correct answer with probability at least $p$. Now I found the most efficient value of $p$ that $M$ succeeds to disntiguish between $|0\rangle\langle0|$ and $|+\rangle\langle+|$, but I am not sure how to contruct a three dimensional measurement that achieves this value of $p$. Any hints, or references? Thanks.


The quickest 6-3 play in baseball: to bounce or not to bounce?

What's the quickest way the get the ball (say from shortstop) into the first- baseman's glove, given some fixed initial (throwing) speed? Directly or with one bounce? I'm fairly sure that the answer depends on the initial throwing speed, but where's the cut-off point (roughly)? * * * PS 1: Numbers, _if needed_ : > Under the current rules, a major league baseball weighs between $5$ and > $5\frac1 4$ ounces ($142$ and $149\,\mathrm g$), and is $9$ to $9\frac1 4$ > inches ($229–235\,\mathrm{mm}$) in circumference ($2\frac7 8–3$ in or > $73–76\,\mathrm{mm}$ in diameter). The horizontal distance to be travelled is $36\,\mathrm m$. PS 2: The surface is "clay" (around first base). The ball is covered in leather. PS 3: Let's assume that throwing accuracy and catching abilities are not issues.


Visualization of 1D spin chain wave fuction

What are the known methods for visualizing quantum states of one-dimensional spin chains? They can be based either on their wave functions or density matrices. My particular interest is in plotting eigenstates of 1D spin chains (e.g. Heisenberg model) in a way in which correlations between particles are related to the self-similarity of the image. **As a side note** , I found a Wikipedia Commons image with its description on the topic (a thumbnail below). However, it is not supported neither by a peer reviewed publication nor by a decent preprint. The only text related is extremely drafty. However, I'm the most interested in published results, and even more when they relate properties of the image to the properties of the state. ![enter image description here](http://i.stack.imgur.com/MM0hN.png)


How is information related to energy in physics?

I recently attended a talk by Dr. Ravi Gomatam on 'quantum reality', where the speaker suggested, that conservation of energy is not a fundamental law, and is conditional, but the conservation of information is fundamental. What exactly is the meaning of information? Can it be quantified? How is it related to energy?


Neutron electric dipole moment and $T$ symmetry violation

Our textbook (and other sources I have found) says that non-zero electric dipole moment of neutron would violate $T$ symmetry. They prove this statement by first assuming $\boldsymbol{D}=\beta\boldsymbol{J}$, where $\boldsymbol{D}$ is the dipole moment, $\boldsymbol{J}$ is the angular momentum, and $\beta$ is a constant. But why? Why is $\boldsymbol{D}$ proportional to $\boldsymbol{J}$? Why is $\boldsymbol{D}$ related to $\boldsymbol{J}$ at all? And how can't this argument be applied to other composite particles such as atoms and molecules, thereby breaking T symmetry for most of the world?


Are lens spaces classified via a Weinberg angle?

I am thinking about Kaluza Klein theory in the 3 dimensional lens spaces. These have an isometry group SU(2)xU(1), generically, and in some way interpolate between the extreme cases of manifolds $S^2 \times S^1$ and $S^3$ (or, arguably depending of your parametrisation, across $S^2 \times S^1 ... S^3 ... S^1 \times S^2$). My question is, simply, if the application of Kaluza Klein construction forces different coupling constants to the two groups $SU(2)$ and $U(1)$, and then a kind of discrete Weinberg angle, or if they have always the same coupling. Note that when you build a lens space bundle over CP2 (which is equivalent to a $S^1$ bundle over $CP^2$ x $CP^1$, if I read correctly a remark of Atiyah reported by Kreck and Stolz), you get an isometry group of SU(3) times SU(2) times (1), so the same question could be asked here, in this more widely known setup. Of course that odd dimensional spaces do not carry chirality, but my question is just about having different couplings, that is all. It could be interesting, as a plus, to compare with coupling relationships coming from Non Commutative Geometry...


What happens when a ball stops bouncing?

If I were to drop a bouncy ball onto a surface, each successive bounce will be lower in height as energy is dissipated. Eventually, however, the ball will cease to bounce and will remain in contact with the ground. _What happens during that small moment of time that is the transition period?_ Here's my line of thought so far: Let's say, for example, that the ball could be modeled as an "ideal" bouncy ball. Each time it hits the floor, it rebounds with one-half of the impact velocity, the other half of which is absorbed and dissipated by various things. Will this ideal bouncy ball ever stop bouncing? The answer, as far as I can tell, is yes (well, _maybe_ ). Since each bounce has half of the initial velocity of the previous bounce, the time between each bounce also halves. The first bounce might take $1$ second, the second takes $0.5$ seconds, the third $0.25$ seconds, etc. This infinite series converges to a finite amount of time, but with an infinite number of bounces. Since it's obviously not possible for an actual ball to bounce an infinite number of times in finite time, it is clear that the above model doesn't work. Eventually, I decided that the problem must be in the fact that the model doesn't take into account the fact that ball spends time compressing and decompressing while in contact with the floor. Even once the ball touches the ground, the center of gravity continues to move downward as the ball compresses. Soon, the ball expands, and the center of gravity moves upward. Lets $h_1$ be the height of the center of gravity when the ball is uncompressed, and $h_2$ the height of the center of gravity of the ball when compressed, and $D = h_1 - h_2$. Eventually, the height of the bounce will be smaller than $D$, so that center of gravity simply vibrates between those two locations, and the ball never leaves the ground.


Maximum depth for bacteria on rogue planets

If microscopic life evolves on a planet which is subsequently ejected from orbiting its star, then the planet would cool down. How deep into the interior of the planet could organisms reach, assuming the mantle had cooled down enough? For example, on Earth there are rock-boring bacteria that have tunneled their way 3 km into the crust. The limiting factor for depth of bacteria on Earth seems to be temperature, but presumably if the mantle had cooled down then at some depth other factors would come into play such as pressure or rock hardness or composition.


Time and distance where velocity is a function of time

I'm creating a video game set in space in which spaceships travelling between places accelerate until they hit a maximum velocity. They then travel at that velocity until a time when they need to start decelerating so as to arrive at their destination and not overshoot. There is no relativity or friction or anything else 'fancy' involved For the algorithm I'm writing to do this, I need to know a) the time it will take a ship to reach the half way point in its journey (assuming it continues to accelerate forever), and b) the time it will take to reach maximum velocity, and c) the distance travelled at time t (assuming continuous acceleration). I can then use that to tell the ship at what time to start decelerating, depending on how far the whole journey is. However, to accelerate the ships, I multiply their current velocity by 1.5 every second (to decelerate, I divide by 1.5). i.e. v(t) = v(t-1) * 1.5 I can also assume Initial velocity (velocity at time 0) is a constant and always positive If anyone can point me in the right direction as to what formula to use (or even better, the pseudo-code to do it), I'd be most appreciative


Why did the asteroid belt between Mars and Jupiter form as it did?

I'm curious about why the asteroid belt wasn't pulled by Mars's or Jupiter's gravity or formed into either moons or planets. Why did it form into an asteroid belt instead?


List of experimental band gap

does anyone know where one can find a list, database, book of experimental bandgap values of semiconductors? Is there such a collection? Or do I need to scour papers one at a time to get the values?


LHC vacuum tube diameter

What's the diameter of the tube where the protons fly around? I dont mean the overall diameter (~1-10 m), instead the inner most ultra-high vacuum tube where the protons are. The inner pipe, the capillary, the channel.. I would guess maybe 1 mm?


Can I levitate an object without using an electromagnet?

I know that it's possible to make an object levitate using an electromagnet to hold it up. But is it also possible to do this with regular magnets? Is there a special kind of magnet I need in order to have one powerful enough to hold an object up? I'm asking because I have this idea in mind where I want to make a decorative item that is levitating an inch or so above its container. ![alt text](http://i.stack.imgur.com/30uUp.png) **A** is the object I want to levitate.   **B** is the container.   **C** shows the magnetic field that is repelling **A** and **B** from each other achieving the levitation. The size of this would be as small as a food plate, maybe even smaller. **B** is barely a pound or two in weight. Is that possible?


Dipole moment induced in a spherical particle

Consider a spherical metal particle made out of gold. If there is an external charge somewhere near the gold particle, is there a way to calculate the resulting dipole moment that is induced by the external charge? Or is there a way to calculate the induced dipole moment that results from an external dipole moment instead of an actual charge? Would be glad if you could give me some ideas or equations to tackle these two problems.


How to write the single electron spin-orbit coupling under an external magnetic field?

As we know, without the external magnetic field, the single electron spin- orbit coupling(SOC) has the form $\boldsymbol{\sigma}\cdot(\boldsymbol{\nabla} V\times \mathbf{p})$ up to a coefficient, where $V$ is the static electric potential. Now if we add an static external magnetic field $\mathbf{B}=\boldsymbol{\nabla}\times \mathbf{A}$, does the SOC preserve the above form $\boldsymbol{\sigma}\cdot(\boldsymbol{\nabla} V\times \mathbf{p})$ or change into $\boldsymbol{\sigma}\cdot(\boldsymbol{\nabla} V\times (\mathbf{p}-e\mathbf{A}/c))$? The notation $\mathbf{p}$ always represents $-i\hbar(\partial_x,\partial_y,\partial_z)$ in all of our formulas. This reminds me another question, may be worthless: Does this expression $\mathbf{r}\times(\mathbf{p}-e\mathbf{A}/c)$ make sense? How to write the orbital angular momentum under an external magnetic field? Thank you very much.


State of constant motion

Why does an object remains in its state of constant motion if there are no forces acting on that object? My understanding is that all the energy of the motion will be kept inside and a change in the speed needs a change in the amount of the energy stored by that object. But how is that energy stored in a form of speed? Does anything changes inside the object's structure if it is moving?


How much iron ware to make a Faraday cage

In a thunderstorm I was thinking the following: suppose I am rowing in a lake during a thunderstorm. How big a Faraday cage do I need to make to protect myself? If lightning strikes the cage, will it dissipate away through the water?


Where does the light go when I turn off the light switch?

As above... Surely it must go somewhere? My child asked me this question and it was a difficult one to answer.


Why is the string theory graviton spin-2?

In string theory, the first excited level of the bosonic string can be decomposed into irreducible representations of the transverse rotation group, $SO(D-2)$. We then claim that the symmetric traceless part (i.e. the **35** rep) is the spin-2 graviton - but isn't the label "spin-2" intrinsically 3+1 dimensional? I.e. it labels the representation under the little group $SU(2)$?


Acceleration vector - deceleration vs direction

If acceleration of something $= - 10 \text{ m s}^{-2}$ And forwards is define as north. Does that mean the object is getting slower (decelerating) or accelerating in the reverse direction (south) How can you tell the difference?


What happens when we cut objects?

What is the role of the molecular bonds in the process of cutting something? What is the role of the Pauli exclusion principle, responsible for the "hardness" of matter? Moreover, is all the energy produced by the break of bonds transformed into heat?


Book on optics in curved space-time

As evidenced from my earlier questions on vision and curved space, I am struggling a little bit with visual perception in curved space-time. I would like a book recommendation on optics and vision in curved space-time.   1. I find books on Gravitational lensing to be inadequate as Gravitational lensing itself deals mostly with specific cases in astronomy. It does not have the breadth and scope I require.   2. I want the book to discuss in detail how space (of low curvature) could affect visual perception in everyday life scenarios. Very high mathematical rigor is desired.   3. I would also really like the book to have pretty colors and pictures for aesthetic value. If you know any book that come close to satisfying my conditions, please do tell me. I will be very grateful if you can. Also, as per the books policy, it is also OK to provide an explanation for any subdiscipline of this topic instead, especially if that subdiscipline is not covered by most resources. But please do not make such an answer Community Wiki.


Why does a stack of transparencies appear white when a single sheet is clear?

My daughter and I found a box of old transparencies (clear plastic sheets) and something struck me as odd: even though each transparency is clear, the whole stack of transparencies looks whitish - why is that?


Chirality and helicity

I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf p)}{|\mathbf p|}\Psi . $$ I showed that both of them are lorentz invariant operators, they commutes with dirac free hamiltonian and their eigenvalues are $\hat {P}_{ch}\Psi (x) = \pm \Psi (x), \quad \hat {P}_{h}\Psi (x) = \pm \Psi (x)$. Is it enough for showing their identity? Or I also need to connect the given eigenvalue of $\hat {P}_{h}$ with corresponding eigenvalue of $\hat {P}_{ch}$?


Is there a wave function for anyons?

People talk about anyons a lot. But i have never seen an anyon wave function. I suspect that there is no such thing as a wave function for anyons. I mean, anyons are not generalizations of bosons or fermions. For bosons and fermions, one can have many-body hilbert spaces, but for anyons, there is no such thing.


What really happens with Time Dilation?

I know if you move your time moves slower than someone who is stationary, by Lorentz's transformation. However, I don't get how this happens. What does it mean when time moves slower? How does it affect my watch for example when I move close to the speed of light? Does an electron for example feel time-dilation?


decomposing forces

I have a question about something I'm not sure about. I will first explain the story. Image If you calculate the force of a truck going downhill and applying its brakes:   * The mass is $50000\text{ kg}$   * The deceleration is $7\ \mathrm{m / s^2}$   * And the angle is 7 degrees. $F_1$ is the componenent of the gravity that is parallel to the to surface of the ground ( the F// or FII). $$F_1 = 50000 *9.81* \sin(7^\circ) = 59777\text{ N (force downwards)}$$ The resulting force is $50000 \times 7 = 350000\text{ N}$ (using the deceleration value) (force downwards). $$F_\text{brake} = 59777 + 350000 = 409777\text{ N (force upwards)}$$ If your truck goes up instead of down this time and then applies its brakes, do you need to do this? $$F_\text{resulting} = F_\text{brake} + F_1 = 409777+59777 = 469554\text{ N}$$ Is it right to count $F_1$ twice, the first time in the F included in the brake and the second time separately?


Why does sweetness of coke change after freezing completely

I freeze my coke in a freezer completely to solid and then keep it out to melt and as it melts portion by portion I starts to drink, initially It will be very sweet and later it wont be sweet at all. why does this happen? We know that all sugar in coke will be dissolved (Solid solution) but how a major portion of this melts faster than water in the juice their by getting more sweet? How these sucrose can escape through the crystals formed by the ice and join in the water. I measured the sweetness (Brix) and found it varies.


Why are the physical sciences described perfectly by mathematics?


Could two identical stars revolve around each other in a common orbit if we only account for Newtonian physics?

Both a parent star and its planet revolve around the center of mass of the system, the reason we see stellar wobble. But if we take this to be true, which it is, there can be a configuration in which two identical stars revolve around their center of mass, in a common orbit. What I find astonishing in this case is that they will be revolving around something having no mass at all, in a shared orbit, like two runners trying to catch each other but never quite being able to do so. In that case, in a purely Newtonian system, the centripetal force must be provided by the gravitational attraction between the stars. Now if I assume both the stars to have a mass $m$, at a separation of $d$ from each other, revolving in the common orbit diametrically opposite to each other, $$\frac{mv^2}{r} = \frac{Gm^2}{d^2}\; {\rm where}\; r = \frac{d}{2}$$ solving for $v$, we get a velocity where a stable orbit is formed, $v = \sqrt{\frac{Gm}{2d}}$ Note: I have failed to find such an expression, I am not certain about the math. I did find a system described in such a way that two stars orbit a common point in separate ellipses though. Is my conventional wisdom correct, or is my derivation of this expression somehow intrinsically flawed? Does this expression already exist? And I suppose the odds of observing such a system are 'astronomical', but has something like it ever been observed?


What physical forces allows for electromagnetic induction

In electromagnetic induction, what force is actually doing the work? what physical force actually drives the electrons around the circuit? Let's say we have a coil and an increasing magnetic field through the loop. The electrons start to flow by faraday's law. Is there a very real electric field that is induced around the wire which does the work? because magnetic fields can most certainly not do work. Also, why do the charges even start moving in the first place? At each instant in time, there is some instantaneous value for the magnetic field. The magnetic field cannot cause the electrons to start moving because it doesn't affect stationary charges


Accretion disk physics - Stellar formation

I was going through the Wikipedia page for Accretion disks, and I couldn't comprehend what the meaning of this is: > "If matter is to fall inwards it must lose not only gravitational energy but > also lose angular momentum." What does that mean? Also, why are these accretion disks planar? I learned that the reason is conservation of angular momentum, but what is the physics behind it?


vacuum level alignment

When two materials are connected like the case in heterojunctions, we always firstly align the so called vacuum level[1], and then decide the relative position of other energy levels like conductance and valence bands. But in the wikipedia page it says " The vacuum level alignment approach may or may not hold due to details of the interface", so my question today is, how to judge whether the vacuum levels align or not? [1]http://en.wikipedia.org/wiki/Vacuum_level


How to relate photon's higher frequency to time dilation?

The usual explanation for photon's higher frequency in lower altitudes (higher gravity), when the photon is going downward towards a massive body, is that gravitational potential energy is converted into photon's energy which, by its turn, through $E=hf$ implies higher frequency. Because, since $c$ is constant, it doesn't make sense to turn that potential energy into kinetic energy. That, I understand. My question is, why is it affirmed that this implies that time runs slower in the lower altitude? I mean why is it such that this higher frequency is equivalent to slower running clocks?


Identifying an optics equation in a code refactoring project

I apologise if this a somewhat naive post. My knowledge of optics is limited, and I am learning on the job, so to speak. I'm a computational biologist, dabbling in a bit of physics. I'm currently reading Sonke Johnsen's "The Optics of Life" to try and get a better picture. I am involved in an academic project to refactor some obsolete code from a program created over 15 years ago, which models the resolution and sensitivity in reflective superposition compound eyes. The problem is that a lot of the code uses non-descriptive variables that are difficult for me to interpret. I am stuck with working out what this is doing:               DA = sqrt((ER * ER) - (AR * AR))    /* DA??? */     AC = atan(AR / DA) / CONV           /* AC??? */      This is then used to adjust the aperture diameter:               AD = (AC / 360.0) * EC              /* adjust aperture diameter */      In this code the variables definitions are as follows: > ER = eye radius > > AR = aperture radius > > CONV = pi / 180.0 (converts radians to degrees) > > AD = aperture diameter > > EC = eye circumference What might `DA` and `AC` be? Any help should be greatly appreciated :)


Variational method applied to brownian motion

It's possible apply the variational method to the brownian motion ? I mean, one of requisites on $y(t)$ is that it must be continuous and $\partial_t{y(t)}$ too, and in this case, $\partial_t{y(t)}$ is not.


Charge distribution inside a parallel plate capacitor

I'm having difficulty deriving a known expression for the magnitude of the Electric Field within a parallel plate capacitor with applied potential difference $V$ and an immobile charge distribution within the capacitor with uniform density $\rho$. The expected result is: > $$\frac{\rho(d-2x)}{2\varepsilon_{0}}-\frac{V}{d},$$ Where $d$ is the > distance between the plates and $x$ is the distance from the positively > charged plate. My initial attempt involved using Gauss' law in a Gaussian surface $\Sigma$ projecting out of the plate a distance $x$: $$\iint_{\partial \Sigma}\vec{E}\cdot\mathrm{d}\vec{A}=\frac{\rho}{\varepsilon_{0}}\iiint_{\Sigma}\:\mathrm{d}V+\frac{Q}{\varepsilon_{0}},$$ Where $Q$ is the magnitude of the charge on each plate. We therefore get: $$EA=\frac{Ax\rho}{\varepsilon_{0}}+\frac{Q}{\varepsilon_{0}}\implies E=\frac{\rho x}{\varepsilon_{0}}+\frac{Q}{\varepsilon_{0}A}=\frac{\rho x}{\varepsilon_{0}}+\frac{V}{d}$$ As $V=\frac{Q}{\varepsilon_{0}A}d$. However, this is not what we should get, so I'd appreciate any pointers in the right direction.


What is the fringe separation in Young's double slit experiment?

In the double slit experiment, a pattern of dark and bright bands appears on the screen. Is the fringe separation the distance between two bright bands? Question: > In a double slit experiment using light from a helium-neon laser, a student > obtained the following results: > > width of 10 fringes 10x = 1.5 cm > > separation of of slits a = 1.0 mm > > slit-to-screen distance D = 2.40 m > > Determine the wavelength of light. How would I find the fringe separation here? If I divide 1.5 by 10, I would get the width of 1 fringe. Is that the same thing as the fringe separation? A diagram showing what the fringe separation is would be really helpful. The textbook that I'm currently reading defines the fringe separation as the distance between the centres of adjacent bright fringes. How would I find out the fringe separation here in this question? Thank you in advance.


Number of conditions for a two-particle state to be decomposable

Suppose we have a general two-particle state $ \Phi (x_1, x_2 ) = \sum_{n_1,n_2} \phi_{n_1,n_2}(x_1,x_2)|n_1,n_2> $, where $n_1$ can be any of $n$ possible states, and $n_2$ can be any of $m$ states. If the state is decomposable then the coefficients $\phi_{1,2}$ can be decomposed into $\phi_{n_1} * \phi_{n_2} $. It seems to me that for the decomposable state there are $n + m$ independent coefficients ($n$ coefficients describing the state of partice 1, and $m$ coefficients describing the state of particle 2), and if the system is not decomposable, i.e., if there is entanglement between the two particles, then the number of independent coefficients is $n * m$ ($n$ possibilities for $n_1$ times $ m$ possibilities for $n_2$ in $\phi_{n_1,n_2}$). If this logic is correct, then the number of conditions to be fulfilled by a decomposable state is $nm - n - m$. However, according to the book I am studying the number of conditions is $nm - n - m +1 $. I wonder why there is an extra condition. I am not considering normalization, because the states are seen as rays in projective space, and furthermore both the decomposable and entangled states would have to fulfill the same normalization requirements so I guess there would be no difference.


When is the right ascension of the mean sun 0?

I understand that the right ascension of the mean sun changes (at least over a specified period) by a constant rate, but where is it zero? I had naively assumed that it would be zero at the most recent vernal equinox, but when I try to calculate the equation of time using this assumption and true sun positions, all my values are about 7.5 minutes larger than they should be. When (at what date and UT time) is the right ascension of the mean sun 0? And why?


Adding stuff to the path integral (Faddeev-Popov method)

I'm wondering about the Faddeev-Popov method described in Peskin Schroeder and also on page 7 in this link. What gives them the right to simply add the Gaussian $\omega$ and thus introduce the $\xi$ parameter? It seems so arbitrary to me. Are there any rigorous derivation of the Faddeev-Popov method?


Why is information in a volume proportional to it's surface area?

I read the wiki article on the holographic principle, but it never answered this question. Can anyone explain the math that leads to this conclusion?


How many celestial bodies could be in stable orbit at roughly the same distance from a star?

**How many planet-like celestial bodies** of roughly the same mass (say within 50%) could orbit a star at roughly (say within 10%) the **same distance from the star** and be in stable orbits. By stable I mean they would not require tethers or significant energy to stay there. I feel sure the answer is at least 2, Earth plus another body at Lagrangian Point L3. I have heard it's not 3 -- that putting Earth-size bodies at Earth's L4 and L5 would destabilize. Is that true? Please explain concisely and cite reputably. I'm asking in the interests of Earthling living space within the solar system, in the far future of course. (Or other stellar systems in the uber far future.) Thanks to @dgh for steering the terminology away from "planet" which is defined by the IAU as having cleared its neighborhood (i.e. orbital radius) of other bodies. ## Research Possibilities   * Klemerer Rosette (theoretically unlimited, but practically no more stable than a ball on a hill)   * Iridium Satellites (66 are active in stable, Low Earth Orbit, in six planes)


Why does electron-positron annihilation prefer to emit photons?

If gravitons are massless, and neutrinos nearly so, why aren't pairs of either of them normally expected products of electron-positron annihilations? Are they possible but simply unlikely, or is there actually some conserved quantity prohibiting their creation? Edit: I'm talking about the low energy limit, not in accelerator beam collisions.


Using the monte carlo method to compute the magnetic field of a curent carrying loop

I have written a program in cpp that computes the magnetic field at a point from a current carrying loop. It uses the biot savart law and the monte carlo technique to carry out the integral. The program functions as expected for points that are not too close to the loop but from looking at a plot of the vectors you can see strange errors close to the loop. I checked my results against http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/curloo.html#c3 for a point on the axis and http://www.netdenizen.com/emagnet/offaxis/iloopcalculator.htm for a point off the axis. My program agrees with these sources for points not too close to the loop. This is my code. http://pastebin.com/4BH9qEZz My main question is if there is anything inherent in the monte carlo technique that would cause this error?


Proving that Measurement increases von Neumann entropy

Let $V$ be a finite dimensional complex inner product space. Let $\mathcal{M}$ be the classical sample space of measurement outcomes that may occur in a given experiment, and $M_\mu$, $\mu \in \mathcal{M}$ are a family of measurement operators acting on the pure states of $V$ that satisfy $\sum\limits_{\mu}M_\mu^\dagger M_\mu = Id_V$. The state of the system before the measurement is given by the density operator $\rho$. The probability that the outcome of the experiment is $\mu$ is $P_\rho (\mu) = \operatorname{Tr} ( M_\mu^\dagger M_\mu \rho)$. The system is now in the mixed state \begin{equation} \rho' = \sum\limits_{\mu} P_\rho(\mu) \frac{M_\mu \rho M_\mu^\dagger}{P_\rho(\mu)} = \sum\limits_{\mu}M_\mu \rho M_\mu^\dagger \end{equation} Now I am asked to prove that $$ S(\rho') \geq S(\rho)$$ where $S$ denotes the _von Neumann_ entropy. The hint says, that i should use that $$ - {\rm Tr} \left[ \rho \log \tau \right] \geq - {\rm Tr} \left[ \rho \log \rho \right] $$ which I have proven already in a previous exercise. Unfortunately I do not have any idea how to start. I'm not even sure that the definitions I took from the lecture notes are correct. All I know is that I have to get to $$ - {\rm Tr} \left[ \sum \limits_{\mu}M_\mu \rho M_\mu^\dagger \log \rho' \right] = - {\rm Tr} \left[ \rho \log \rho' \right]$$ in order to apply the hint. I tried to write it in a basis where either $\rho$ or $\rho'$ are diagonal but I found no way of achieving the above, I'm still feeling uncomfortable with density and measurement operators and appreciate any hint. PS: I'm not sure whether this is more of a math.SE question, but given the physical meaning of the calculation I decided to post it here. I'm happy to move it if it is considered unsuitable for physics.SE.


Electromagnetic inertia due to advanced radiation?

The scalar potential $\phi$ and vector potential $A$ at a distance $r$ from a charge $q$ are given approximately by $$\phi = \frac{q}{r}$$ $$\mathbf{A} = \frac{q\mathbf v}{r}$$ where the constants have been suppressed. The corresponding electric and magnetic fields are given by $$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf A}{\partial t}$$ $$\mathbf{B} = \nabla \times \mathbf A$$ Now the gradient and curl terms fall off as $1/r^2$ so that at large distances from the charge $q$ we only have an electric field $E$ given approximately by $$\mathbf{E} = - \frac{\partial \mathbf A}{\partial t}\ \ \ \ \ \ \ \ \ (1)$$ where $$\mathbf{A} = \frac{q\mathbf v}{r}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ Thus at large distances an accelerating charge produces an electric field that is proportional to its acceleration. This is the standard (retarded) radiation field of an accelerated charge. But perhaps this is not the only solution to equation (1). Imagine that the charge is fixed with respect to an inertial frame and that the observer is moving with velocity $-v$. In the rest frame of the observer the charge is moving with velocity $v$. This implies that in the observer's rest frame there is an $A$-field. Now if the observer changes his velocity then in his comoving frame he experiences a changing $A$-field. My proposal is that Equation (1) still applies in this situation so that this "apparently" changing $A$-field induces an electric field in the observer's accelerating frame. This field causes any charges that are co-moving with the accelerating observer to feel a kind of electromagnetic inertial force. I realise that this is non-standard as Maxwell's equations are usually only applied in an inertial frame. My hypothesis is that the retarded solutions apply to an inertial frame whereas the previously unused advanced solutions apply to accelerated frames.


How far will the stone travel in 2.0 s?

A stone is thrown vertically down from the top of a 40 story building with an initial speed of 1 m/s. How far will the stone travel in 2.0 s? the answer is 22 m and my answer is 17.6 m I used this equation to solve the problem $x=v_0t-(1/2)gt^2$ = (1)(2)-(0.5)(9.8)(4)=-17.6 = 17.6m So, where is the mistake?


Electron shells in atoms: What causes them to exist as they do?

I have seen similar posts, but I haven't seen what seems to be a clear and direct answer. Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away? It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties. What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell? Thank you! Please be patient with me, new to this forum and just an amateur fan of physics.