title: How do transformers work?

text: A transformer is basically a primary inductor connected to a voltage $U_P$ which you want to transform. You also have an iron rod and a secondary inductor. So when there is a current $I_P$ the iron rods becomes magnetic. When you connect the primary inductor to AC, that means that you'll have a changing current, which causes a change in flux which causes induction. My question is, is $U_S$ just the induction voltage created by the iron rod?


title: THT (Tetrahydrothiophene) absorption spectrum

text: I am looking for the absorption spectrum of THT. What is the best way to find these types of exotic material's spectral characteristics?


title: Interference of independent electrons

text: In the Pfleegor-Mandel Experiment, photons from two separate lasers are interfered to produce an interference pattern. In the experiment, the rate of photons from individual lasers was so low that individual photons from each laser are actually interfering with … nothing! I'm fascinated to get some more insight on this. Has it been replicated with something other than photons? Electrons, for example?   * Is it possible that it is some property of the measuring system? For instance, it's well established that radio frequency radiation from separate sources will interfere at a receiver, but perhaps that is a property of the antenna's mixing of the source?    * Is it possible to perform the same experiment where the measuring system does not provide a chance for the quanta to mix?   * Does reducing the rate of quanta incident at the detector eventually stop interference, or reduce it, or cause it to break down? * * * Further links:$_1$Another description of Pfleegor and Mandel's experiment. $_2$On this site: Why can interference from two independent sources be observed? $_3$A review of independent photon interference, if you subscribe to APS Physics.


title: Physics of the inverted bottle dispenser

text: ![Look at the red arrow](http://i.stack.imgur.com/aBnic.jpg) When you invert a water-bottle in a container, the water rises and then stops at a particular level --- as soon as it touches the hole of the inverted bottle. This will happen no matter how long your water-bottle is. I understand this happens, because once the water level touches the hole, air from outside cant go inside and therefore there is nothing to displace the water that falls out of the container. Now according to the laws of pressure ---- the pressure at the water level must be same everywhere --- whether it's inside the water bottle or outside. And that must be equal to the atmospheric pressure. Therefore the pressure of the water column + air column inside the inverted bottle must be equal to the atmospheric pressure. What I dont understand is, no matter how long a bottle you take, the water level will always stop at the hole. So that means that no matter how long a bottle you take, the pressure of the water column + air column inside the water bottle will be equal to the atmospheric pressure. How could this be possible? Also I'd like to let you know that, if you pierce the upper part of the bottle with a small pin, then the water level rises and overflows out of the container. I'm assuming air from outside rushes in and pushes the water out.


Wikipedia states: > _Sometimes due to specific combinations of properties extended objects > behave as point-like even in their immediate vicinity. For example, > spherical objects interacting in 3-dimensional space whose interactions are > described by the inverse square law behave in such a way as if all their > matter were concentrated in their centers of mass. In Newtonian gravitation > and classical electromagnetism, for example, the respective fields outside > of a spherical object are identical to those of a point particle of equal > charge/mass located at the center of the sphere._ Can you please explain/elaborate more on this? Are they saying, for example, that a source which has a flux emanating from it can be treated as a point particle, even if it's spherical? How can spatially extended objects behave like point particles?


title: Converting between Galactic and Ecliptic coordinates

text: I was hoping someone would be able to tell me the formula to convert between ecliptic and galactic coordinates. I've been able to convert values using http://lambda.gsfc.nasa.gov/toolbox/tb_coordconv.cfm but have no idea how they were able to calculate that. Thanks in advance :)


title: What is Rabi nutation in NMR?

text: What is the mathematical and physical significance of Rabi Nutation in terms of NMR?


title: NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms

text: From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other hand, we have polynomial time algorithms for calculating eigenvalues for matrices. This confuses me. Should not be solving an Ising model be equivalent to calculating the eigenvalue of the Hamiltonian matrix of that model? In that case why is it NP-complete? Is coming up with such a Hamiltonian matrix at the first hand is NP-complete? If it is so, it makes the whole project (coming up with such Hamiltonian and solving it) NP-complete.


title: Prerequisites to start the study of noncommutative geometry in physics

text: What are prerequisites (in mathematics and physics), that one should know about for getting into use of ideas from noncommutative geometry in physics?


title: Single electron pumps, superconducting energy gap

text: I am seeking to find out how many electrons per gate cycle are pumped at different temperatures. The Single Electron Pump is made out of Al, with a $T_c=1.2\,\mathrm{K}$. This single electron pump is two-gate electron pumpthat is based on a single-electron transistor (SET). Diagram of discussed SET: ![enter image description here](http://i.stack.imgur.com/Fvmfh.png) I know that: $\Delta=e^2/2c\implies I=ef$ $\Delta>e^2/2c\implies I=2ef$ $\Delta<e^2/2c\implies I=ef$ However, how do I calculate the superconducting energy gap $\Delta$ with only the transition temperature of Al, $T_c=1.2\,\mathrm{K}$?


Why is Heisenberg's uncertainty principle not an experimental error since it is the error created by photons striking on elementary particles?


title: Caldeira-Leggett dissipation model: Call for Help!

text: I am sorry to be so troublesome and possibly infringing some rules, but could I once more invoke the help of other participants in order to solve my query on the Caldeira-Legget model, namely the introduction of a phase shift, better detailed in Caldeira-Legget Dissipation: cannot get it? Thank you very much. I am so keen to understand this point but got into a "loop" and cannot make sense of it any more...


title: Proving that the hermitian conjugate of the product of two operators is the product of the two hermitian congugate operators in opposite order

text: I have reach a step in a problem of my quantum mechanics textbook that requires me to prove the following. $$\hat{A}=(\hat{Q}\hat{R})^{\dagger} = \hat{R}^{\dagger}\hat{Q}^{\dagger}$$ I tried to prove this by substituting $\hat{A}$ into $<\Psi_1|\hat{A} \Psi_2>$ $$\Rightarrow \int_{-\infty}^\infty \mathrm{\Psi_1}(\hat{Q}\hat{R})^{\dagger}\Psi_2\,\mathrm{d}x$$ Do I integrate by parts next? Any advice would be much appreciated


title: What's the point in shooting down space debris?

text: From this news item on a planned North Korean satellite launch comes the quote: > The Japanese government has pledged to shoot down any debris that falls over > its territory. To me, that sounds strange. The only way I can see in which this would help is if they manage to break the debris into many pieces _before_ it slows down into the atmosphere, so that parts may be small enough to burn up completely. If they manage to shoot it down later, they can at best replace a few large pieces of launcher debris to a large number of small pieces of launcher debris and missile debris. My questions:   * What causes more damage:      * A small number of large pieces of launcher debris?     * A large number of small pieces of launcher debris + debris from the missile used to destroy this debris?   * Is there any historical precedent of shooting down debris from a satellite launch?


title: In bubble-multiverse theories, are the parameters values describing a given universe members of a countably infinite set?

text: I recently had the chance to read Brian Greene's "Hidden Reality" with all its talk of multiverses. He makes a case that there may be an infinite number of identical copies of our world "out there". He motivates this arugment by reference to the fundamental quantum discreteness of our world, so that in an infinite multiverse each world must repeat infinitely often, which is intuitive enough. However, my concern regards two things:   1. How do we count worlds?   2. What constraints are there on their properties? If we constrain ourselves to a multiverse composed of an infinite number of discrete worlds (i.e., "bubbles"), each with its own set of prameters and closed geometry, then the question of uniqueness seems to hinge on the cardinality of the set of values for the parameters for each universe. The cardinality of the set of worlds is $\aleph_0$, so if the parameters for a given universe are $\aleph_0$ as well, then we will have infinite repeition of each world and no uniqueness. If (at least one of) the parameters takes values from a set of cardinality $\mathfrak{c}$, then it seems that uniqueness is preserved, as there will never be enough worlds to exhaust the parameter space. Has anyone else thought about this and come up with an answer? To the extent theory can speak to this, could the set of values for at least one of the fundamental "constants" have cardinality $\mathfrak{c}$?


title: What is Z3 exciton?

text: I am searching and studying excitons and I confronted with a term named Z3 exciton. What is it? And what is its difference with, for instance Z1 or Z2 exciton?


title: Questions regarding rubbing catfur on glass

text: I have certain questions regarding the electrostatic charge developed by this rubbing effects: 1-what is the threshold for electrostatic charges by this process? 2-can we define potential energy associated to the system, post rubbing? 3-If we can define Potential, then whats the efficiency of this process of changing Kinetic energy to potential energy?


title: Mechanical waves edge between material and vacuum

text: I have been thinking about the propagation of EM waves vs. mechanical waves and some of their odd cases. One such case that I haven't been able to puzzle out is what happens when a mechanical wave reaches the end of a medium (such as the outer layer of the atmosphere) and the beginning of a vacuum - outer space itself. Edit: for clarity, I created a simple image demonstrating what I thought should happen: ![Last particle in a vacuum is launched into space](http://i.stack.imgur.com/mU0C5.gif) Basically, my theory was that the "edge" particles (or particles with a large amount of space between them) would continue into space with perpetual motion as the force applied to them is not counteracted by particles in front of them. My question is, after the wave has affected the last molecules on the edge of a vacuum, what happens to it, and, do the molecules on the very edge continue to move indefinitely (or do they return to equilibrium)?


title: Matsubara Frequencies

text: I have to evaluate the following Matsubara sum: $$\frac1\beta \sum \left(\omega^2 +a^2\right)^{-1}$$ for Bosonic-Matsubara frequencies. I know contour integration it the way to go. Therefore, I wrote: $$\frac1\beta \sum \left[(\omega-ia)(\omega +ia)\right]^{-1} = \frac{1}{2i\pi} \times \int dz\, \left[(z-a)(z +a)\right]^{-1}g(z)$$ where $$g(z)=-\frac12\tanh(\beta z /2).$$ My final result is therefore $$\frac1{2a} \coth(\beta a/2).$$ Can someone tell me if this is correct or if I am wrong? And moreover, are there rules/limitations for choosing $g(z)$?


title: Can we 'see' the Kolmogorv microscale

text: My understanding of Kolmogorov Microscale is that in a turbulent fluid, smaller eddies tend to be similiar to larger eddies - until you arrive at the microscale. My understanding (that may be wrong) is that this roughly the smallest eddy in a turbulent flow. If I'm not wrong, then we should expect to see these 'smallest eddies' in some way. In another question by me about the subject, I was given some numbers for an estimation:   * 100 $\mu m$ for a person running in air   * a range for energy dissipation in ceons that translates 0,1mm to 1cm for the length scale While small, I imagine one can design an experiment to observe flow at these scales or create a setup (less speed, more viscosity) with larger microscales. Has this been done and can I see pictures?


title: Integration of Forchheimer equation for unsteady-state permeability determination

text: I would like to know the steps for the integration of Equation 17 in Section 6.2.1.2 of American Petroleum Institute Recommended Practices 40 for Core Analysis to yield: $$\frac{(P_1-P_2)(P_m+f_{Fo}b)z_r}{P_rq_rz_m} = A_1 + A_2 \left(\frac{P_rq_r(P_m+b)}{z_rP_m}\right)$$ This is equation B-1 found in Section 6, page 49 (pg. 165/236 in pdf document). Relationships for the variables are defined on the same page. link to pdf for API RP 40: http://w3.energistics.org/RP40/rp40.pdf This equation is for axial gas flow, used with the pressure falloff method proposed by Stanley C. Jones in his paper "A Rapid Accurate Unsteady-State Klinkenberg Permeameter" - https://www.onepetro.org/journal-paper/SPE-3535-PA The derivation begins on page 123/236 (Section 6.2.1) using the one- dimensional differential form of Forchheimer's equation (5): $$\frac{-dP}{ds} = \frac{C_2 \mu q_s}{C_1 A k_g} + \frac{C_3 \beta \rho_s q^2_s}{C_1 A^2}$$ The density of a real gas is (7): $$\rho = \frac{MP}{zRT}$$ The Klinkenberg relationship is (9): $$k_g = k_\infty (1+\frac{b}{P})$$ The mass of gas flowing out from a reference point equals that flowing through the sample: $$\rho_s q_s = \rho_r q_r$$ At steady-state, isothermal conditions, where the reference temperature is the same as the flowing gas temperature, the previous relationship can be reduced as (16): $$\frac{P_s q_s}{z_s}=\frac{P_r q_r}{z_r}$$ since M, R, and T are constant. *Substitute equations 7, 9, and 16 into equation 5: multiply both sides by $C_1$ $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu q_s}{A k_g} + \frac{C_3 \beta \rho_s q^2_s}{A^2}$$ substitute Klinkenberg relation (eqn 9) $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu q_s}{A k_\infty (1+\frac{b}{P})} + \frac{C_3 \beta \rho_s q^2_s}{A^2}$$ substitute eqn. 16 for $q_s$ in 1st right-hand side term $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu P_r q_r z_s}{A k_\infty z_r (1+\frac{b}{P}) P_s} + \frac{C_3 \beta \rho_s q^2_s}{A^2}$$ $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu P_r q_r z_s}{A k_\infty z_r (P_s+ b)} + \frac{C_3 \beta \rho_s q^2_s}{A^2}$$ substitute eqn. 7 for $\rho_s$ in 2nd right-hand side term $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu P_r q_r z_s}{A k_\infty z_r (P_s+ b)} + C_3 \beta \frac{MP_s}{z_sRT} \frac{q^2_s}{A^2}$$ substitute eqn. 16 for $q_s$ in 2nd right-hand side term $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu P_r q_r z_s}{A k_\infty z_r (P_s+ b)} + \frac{C_3}{A^2} \beta \frac{MP_s}{z_sRT} \left(\frac{P_r q_r z_s}{z_r P_s}\right)^2$$ Canceling terms gives you equation 17 (page 130/236): $$C_1 \left(\frac{-dP}{ds}\right) = \frac{C_2 \mu P_r q_r z_s}{A k_\infty z_r (P_s+ b)} + \frac{C_3 \beta M P^2_r q^2_r z_s}{A^2 RT z^2_r P_s}$$ from this equation the pressure and other pressure dependent variables were integrated with respect to the length of the core to yield equation B-1.


title: Conceptual questions about chaotic signals

text: 1. Are chaotic signals stationary or non stationary signals? They are ergodic and ergodic signals are a kind of stationary signals. But, from statistics point of view, are chaotic signals classified as stationary or non-stationary?   2. I have not found any resource which mentions about the probability distribution function of chaotic signals. Are they assumed to be Gaussian? Any other information will be appreciated. Thank you in advance!


title: Heuristic Motivation for Lagrangian Formalism

text: Does anyone know a good heuristic motivation for the Lagrangian Formalism? I think most physicist just accept at one point that it works and thats that. I think I understand the historic origin. Lagrange was fascinated by the Fermat Principle and searched for a similar principle for massive objects. By guessing he ended up with the classical Lagrangian that reproduces the Newtons second law. I would be interested in a modern viewpoint. Can in be maybe understood, in some way, from a quantum field theoretic or relativistic point of view? I stumbled upon the book chapter by stackexchange member Hans de Vries: http://physics-quest.org/Book_Chapter_Lagrangian.pdf , but he never elobarted on this. For example in What is the physical meaning of the action in Lagrangian mechanics? he just repeats his idea of "least action = least proper time", but does not give any further explanations for this claim. Does anyone know some elaboration on this or some other illuminating idea for why the Lagrangian formalism works (not in classical mechanics, but the more fundamental case of Quantum Field Theory)? PS: I dont think this is a dublicate of What is the physical meaning of the action in Lagrangian mechanics? because the question there is specifically about the physical meaning of the action


title: What is the definition of colour (the quantum state)?

text: I heard somewhere that quarks have a property called 'colour' - what does this mean?


title: Paradox?: What is the form of radiation experienced by a harmonically accelerated observer?

text: Theory predicts that uniform acceleration leads to experiencing thermal radiation (so called Fulling Davies Unruh radiation), associated with the appearance of an event horizon. For non uniform but unidirectional acceleration the shape of the experienced radiation changes from thermal to other spectral densities, but also is predicted to exist. But suppose the acceleration is periodic and oscillatory, i.e. no permanent horizon persists? In particular, what about the case of harmonic motion, for a full cycle, half a cycle, etc.? Here is an even simpler related problem that makes the apparent paradox easier to see. Suppose at proper time t=0, I accelerate at constant acceleration k in the x direction for t0 seconds, presumably experiencing Unruh radiation. Then I accelerate with acceleration -k, (in the -x direction,) for 2*t0 seconds, seeing more Unruh radiation coming from the opposite direction, and then I finish with with acceleration +k for the final t0 seconds. At the end of the 4*t0 proper seconds, I'm back where I started, at rest, without any event horizon. Was the Unruh radiation I felt when reversing acceleration secretly correlated or entangled with the radiation I initially and finally saw? Otherwise, from a more macro scale, I didn't actually necessarily move much, and the acceleration event horizon was instantaneous, evanescent and fleeting, so whence arose the Unruh radiation?


title: Is almost all entropy in our universe entanglement entropy?

text: Our observable universe, or a subregion of our universe many times larger than the observable universe, originated from inflating from a very tiny inflationary patch. Being so small, the initial entropy has to be bounded by a very small value. But everything in our universe originated from that tiny patch after inflation, reheating, and further expansion. Does that mean almost all entropy in the observable universe is entanglement entropy if we assume the wavefunction of the universe never collapses and is linear? Because of the peculiarities of quantum mechanics, a subsystem can have far greater entropy than the system as a whole. The many worlds interpretation explains why we subjectively observe a far greater entropy, from the vantage of an Everett branch.


title: Is there a better choice of coordinates for a bead on a rotating helical wire?

text: A bead of mass $m$ is threaded around a smooth spiral wire and slides downwards without friction due to gravity. The $z$-axis points upwards vertically. Suppose the spiral wire is rotated about the $z$-axis with a fixed angular velocity $\Omega$. Determine the Lagrangian and the equation of motion. This is related to a previous problem in which the wire shape is given as $$ z = k\psi, \hspace{3mm} x = a\cos\psi, \hspace{3mm} y = a\sin\psi $$ where $a$ and $k$ are both positive. My attempt at a solution: We still have $z = k\psi$, but now $x = a\cos(\psi + \Omega t)$ and $y = a\sin(\psi + \Omega t)$. This gives $\dot{z} = k\dot{\psi}$, $\dot{x} = -a(\dot{\psi} + \Omega)\sin(\psi + \Omega t)$ and $\dot{y} = a(\dot{\psi} + \Omega)\cos(\psi + \Omega t)$. Then the kinetic energy is $$ T = \frac{1}{2}m\left(\dot{x}^2 + \dot{y}^2 + \dot{z}^2\right) = \frac{1}{2}m\left(a^2\left(\dot{\psi}^2 + 2\dot{\psi}\Omega + \Omega^2\right) + k^2\dot{\psi}^2\right) $$ and the potential energy is $V = mgz = mgk\psi$. Then the Lagrangian becomes: $$ L = T - V = \frac{1}{2}m\left(a^2\left(\dot{\psi}^2 + 2\dot{\psi}\Omega + \Omega^2\right) + k^2\dot{\psi}^2\right) - mgk\psi. $$ This gives $$ \frac{\partial L}{\partial \psi} = -mgk, \hspace{3mm} \frac{\partial L}{\partial \dot{\psi}} = m\left(a^2\dot{\psi} + a^2\Omega + k^2\dot{\psi}\right), \hspace{3mm} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\psi}}\right) = m\left(a^2 + k^2\right)\ddot{\psi} $$ so plugging into Lagrange's Equation gives $-mgk = m\left(a^2 + k^2\right)\ddot{\psi}$ or $\ddot{\psi} = -\frac{gk}{a^2 + k^2}$, which is the exact same equation of motion as in the case with the coil not rotating. Obviously this isn't correct. I think the problem might be related to my choice of coordinates (in particular, having $\psi$ rotating), but I'm not sure what a better choice of coordinates would be.


title: 1D topological insulator

text: This question is inspired by another one about the simplest model of topological insulator, where 4tnemele showed a nice two band model in the answer. I read that and am wondering if we and push that to one dimension. For example, by analogy to the graphene case, if we have a Hamiltonian in 1D (say x) as $H(k_x)=(k_x-k_0)+m$ for $k_x>0$. When $k_x=k_0$, one has $m>0$. $H(k_x)=(k_x+k_0)+m$ for $k_x<0$. When $k_x=-k_0$, one has $m<0$. A smooth connection in between, we will have a conductive edge (two ends in the 1D structure), right? If I want to make a intuitive picture like below, is it correct? ![enter image description here](http://i.stack.imgur.com/n1qkG.png) Any suggestion for real materials show this behavior?


title: What does the Many-Body Problem say about mathematical physics?

text: I was interested to read the answers to the other Many-Body Problem questions on this site and was left with one nagging question of my own. What does the Many-Body Problem reveal about reality and our attempt to understand it? To explain: if the solutions we find to this problem are in the form of lengthy iterative expansions may this not signify in some way that nature operates along discrete iterative paths? What does this say about our theories if these problems are so difficult/time-consuming for them but so quickly solved by the real world around us? Is this problematic? Will more complete theories-ie possibly string theory- address this or do they even need to?


title: Collision of multiple fluid flows at a point

text: I would like to do an analysis when two or more fluid streams flowing from different direction with different velocity, meet at a point. What happen after the collision? Are there any related studies or papers?


title: Is there a standard reference where entropy is set equal to zero in property tables?

text: For practical considerations, it seems that entropy is only meaningful as a difference between states, like $\Delta s$ going from state A to state B. For an ideal gas, for instance, standard formulas give some infinities for entropy at absolute zero. But we still tabulate values. Looking at these tables, I see the s=0 point to be around -10 to -5 degrees C for air. This doesn't seem to be consistent with other substances (notable steam tables). Is there a commonly accepted rule about where entropy to zero. While that point is arbitrary physically, different steam tables would need to use the same offset to be compatible among each other. I'm interested to know if there's a standard rule for this. Particularly for ideal gases.


title: Any cheap (less than 100 euro) telescope for observing Jupiter?

text: The moon is not challenging. I would like to observe the big planets and their rings.


title: Algebra, commutators and test functions

text: I am trying to make sense out of the algebra of the generators of the conformal group and I am running into some issues regarding how to calculate commutators. For instance, for translations of a "test operator" $\phi(x)$, we know that $ \phi(x+a) = \phi(x) + a^\mu \partial_\mu \phi(x) + \cdots$ if we simply Taylor expand. Moreover, in the Heisenberg picture, we can also write $$ \exp(i a^\mu P_\mu) \; \phi(x) \; \exp(-i a^\mu P_\mu) = \phi(x) + i a^\mu [ P_\mu, \phi(x)] + \cdots $$ from which we conclude that $[P_\mu, \phi(x)] = -i \partial_\mu \phi(x)$. Why then do we say that $P_\mu = -i \partial_\mu$ is the generator of translation? From the above definition, the equality does not hold directly since we used a commutator to define $P_\mu$'s action on $\phi(x)$, but introductory quantum mechanics teaches us that $P_\mu = -i \partial_\mu$. My problem seems to be that I can't reconcile this with rigorous group theoretical concepts. What made this confusion arise was the calculation of commutators of algebra generators for the conformal group. For example, taking $D$ to be the generator of dilatations, to prove that $[D, P_\mu] = i P_\mu$, one needs to apply the commutator $[D, P_\mu]$ onto a test function $\phi(x)$ outside of the commutator after replacing the $D$ and $P_\mu$ by their differential definition. So how do we go from $[P_\mu, \phi(x)] = -i \partial_\mu \phi(x)$ to $P_\mu = - i\partial_\mu$ where the equality makes sense?


title: Why does my gravity simulation do this?

text: For a school project i created a simple 2D gravity sim in Matlab using the simplest possible method. There are 2 nested loops so that the total force and acceleration of every object can be calculated. I found that conservation of momentum in x and y works perfectly. I also use a damping so if two object are very close, their acceleration doesn't explode. Then i place two objects of equal mass on coördinates (-100,0) and (100,0) with initial speeds 0. If i start the sim, the two objects will move towards eachother, pass though eachother and move away again. However, they move away much farther than where they originally started. Would that mean that kynetic energy + pot energy is violated, or is this normal? In the real world this obviously never happens because objects can't pass through eachother. It is not very important to solve the problem, the project is from a few months ago. I just wonder what went wrong. Here is my code. The circle.m must be in the same folder as the gravity.m Some names of variables are in Dutch, but i translated the comments. http://speedy.sh/4wVqm/Matlab-gravity2.rar


title: Effect of humidity on static electricity

text: I am looking for a reference to a scientific paper or textbook that explains the effect of humidity on static electricity (triboelectric effect). Any help will be highly appreciated.


title: Naive visualization of space-time curvature

text: With only a limited knowledge of general relativity, I usually explain space- time curvature (to myself and others) thus: _"If you throw a ball, it will move along a parabola. Initially its vertical speed will be high, then it will slow down, and then speed up again as it approaches the ground._ _"In reality, the ball in moving in a straight line at constant velocity, but the space-time curvature created by the Earth's gravitation makes it appear as if the ball is moving in a curved line at varying velocity. Thus the curvature of space-time is very much visible."_ Is this an accurate description, or is it complete nonsense?


title: How to prove that zero Weyl tensor predicts no deflection of light?

text: There is Nordstrom theory, which can be given as $$ C_{\mu \nu \alpha \beta} = 0. $$ The solution of Einstein equations for this case is conformally flat metric: $$ g^{\mu \nu} = e^{\epsilon \varphi (x)}\eta^{\mu \nu}. $$ How to show that this theory predicts no deflection of light?


title: Convert $\text{mm}^2$ to $\text{m}^2$

text: Stuck on a very very basic concept. I have $1750\text{ mm}^2$ and need to get into $\text{m}^2$. I figured $1750\text{ mm}^2 = (1750\text{ mm}) \cdot (1750\text{ mm})$, but I know this isnt right.


title: How can steel shavings get magnetized while hole tapping?

text: I was tapping a hole yesterday with my uncle into a piece of soft steel. (A tap cuts threads on the inside of a hole, to accept a machine screw or bolt.) The tap was NOT magnetic! As I finished boring through the soft steel, I noticed that the steel shavings (or chips) were magnetized. I didn’t take a picture of it but one can see the steel shavings in the image below. ![enter image description here](http://i.stack.imgur.com/pL631.jpg) I could clearly see that steel shavings were standing perpendicular to the tap itself, not attached by any other means. My question is how does hole tapping (friction) align the domains of the chips? I could conceivably imagine that the soft steel is being heated up that loosens the “weakly bound domains” of the soft steel and that maybe, just maybe, they are aligned by the earth’s magnetic field. However, I would guess that the earth’s magnetic field is too weak over such a short time period. None of this makes any sense to me and therefore, I am asking for help on my question.


title: Could there be a small-scale observation of dark energy?

text: The amount of dark energy in the universe is pinned down fairly precisely by fluctuations in the cosmic microwave background, if I understand correctly. So in that sense there's very good evidence of its existence. What I'm wondering is, could there ever be a laboratory-sized experiment that demonstrates its effects? For comparison, Pound and Rebka demonstrated the existence of gravitational redshift over the height of a five-storey building. What would be the smallest equivalent for showing the existence of dark energy? For instance, suppose I have two super-light ping pong balls floating in space. (On the ISS, say, or on board Voyager 1.) Would dark energy make them drift apart?


title: Differences between 3 apples and 3 kg of apples according to base units

text: 7 base units are defined in physics and other units are derived from these units. When we say "3 kg of apples", we mention its mass and mass is a base unit. But for "3 apples", what is its unit? Is this what unitless is?


Andromeda is made of antimatter. Am I wrong? Why? Of course I do not know that Andromeda is made of antimatter.   **_ _but_ ____** I do not know that Andromeda is made of matter. Does anybody know what is the correct sentence? Why? EDIT   I used Andromeda as a substitute for any non-local galaxy.   Lets put some geometry (due to the first two comments):   Antimatter galaxies (AMGs) have in general AMGs in the neighborough. MGs only MGs.   For instance suppose the hurricanes in the northern hemisphere rotate X way and in the southern rotate anti-X way, and they dont meet ;).   That is to say, it is not mandatory that AMGs colide at all with MGs. I suspect that we have no way to decide if the light we receive is originated in an anti-atom or from an atom.


title: Why is momentum conserved (or rather what makes an object carry on moving infinitely)?

text: I know this is an incredibly simple question, but I am trying to find a very simple explanation to this other than the simple logic that energy is conserved when two items impact and bounce off each other. The question is this: how do particles, atoms, objects just carry on moving in space in their free will (in a "vacuum")? When I am throwing a ball and let go of it (imagining that myself and the ball are in a vacuum of which I am impervious) why does it keep pushing itself forward? I know there's a logic to it that says "what's there to stop it" but really, what is keeping it going, infinitely through a real vacuum? So there's kinetic energy which is being used, and potential energy inside it. Is the kinetic energy flaring out the back of it like a ball shaped rocket? Is it really just the simple logic that "nothing is there to stop it"? Or could there be other methods such as propagation? With dark energy? I know all the Newtonian stuff just to be straight, and I know about friction etc etc and the maths behind those. It's just the acceptance that things just carry on moving that I'm finding difficult. Edit: just realised, okay, energy is not being used. So why is it just cruising? Edit: I can't vote up yet by the way, else I would on all these great answers! Thanks everyone.


title: Can computers survive bubble nucleations?

text: According to string landscape theory, our vacua with a cosmological constant of $10^{-123}$ is a metastable vacua which can decay to a supersymmetric vacua with either a zero or negative cosmological constant. However, such a decay is destined to happen sooner or later, but if we wait for it, it will happen at an unpredictable random location and time. Suppose we have some computer memories and computations that we wish to preserve through a bubble nucleation. Is it possible to engineer a controlled bubble nucleation to a supersymmetric vacua with an exactly zero cosmological constant, and construct computers with data storage such that the memories contained therein can survive the bubble transition to an entirely new law of physics unscathed, and the computers can also continue their computations?


title: How is the logarithmic correction to the entropy of a non extremal black hole derived?

text: I`ve just read, that for non extremal black holes, there exists a logarithmic (and other) correction(s) to the well known term proportional to the area of the horizon such that $S = \frac{A}{4G} + K \ln \left(\frac{A}{4G}\right)$ where K is a constant. How is this logarithmic (and other) correction term(s) derived generally? Or how can I see that there has to be such a logarithmic correction? I`m wondering if there is some kind of a general makroscopic thermodynamic or semiclassical argument (in analogy to some derivations of the first term) that motivates the appearance of the second logarithmic term and does not depend on how the microstates are quantum gravitationally implemented.


title: What is the difference between an accelerometer and an inclinometer?

text: When looking through papers for a controls project I found one that seemed to be referencing the fusing of sensor data from an inclinometer and an accelerometer. I always had thought that it was impossible to separate acceleration due to gravity and acceleration due to other forces. So wouldn't an inclinometer and accelerometer basically be the same thing? Or does an inclinometer typically measure tilt in some other fashion that does not involve gravitational forces?


title: General relativity and the conservation of momentum

text: I'm trying to understand the conservation of momentum in general relativity. Due to the curvature of space-time by matters and energy, the path of a linear motion appears to be distorted. Therefore from the frame of reference of an observer who is at a not-so-curved region of space-time, it appears that the velocity vector of an object that is moving along a straight line in a very-curved space-time is changing? Does that imply that from the observer's frame of reference (in a not-so- curved space-time) momentum of that object (in the very-curved space-time) is not conserved? In this sense is momentum not always conversed in general relativity or did I misinterpret something?


title: Spherical magnets shooting like bullets and demagnetizing

text: I was playing around with a 2x chain of bucky balls and winding it around itself. Suddenly several of the little balls went shooting like bullets across the room in 3 directions and now all of the balls seem substantially less magnetized. What happened?


title: What is the magnetic susceptibility of NdFeB?

text: NdFeB is one of the most popular materials used for making permanent magnets. Yet I could find no link or reference reporting a decent value for the magnetic susceptibility of NdFeB. Magnetic susceptibilities of some common materials has been listed at the following link: http://en.wikipedia.org/wiki/Magnetic_susceptibility#Examples A scientific paper or application notes of a good company involved in magnetism would be really helpful.


title: Harvesting electricity in the air from power lines:

text: Recently I came across this quote on the internet, about a person who apparently set up devices to extract electricity from the "air", on his own property: > I read an article a couple of years ago about this very thing, Someone had > lined the edge of their property with devices to capture bleed-off from the > transmission lines and was using it to power the house. They were able to > cut power bill to less than 10% of normal and the power company sued them > and won. It is illegal to "harvest" unused power from transmission lines. I > don't know if the same would apply to radar/radio/etc... I would do some > checking before I spent a lot of money on equipment. I'm not interested in the legal side of things, but I'm wondering if one set up a device to extract energy in this way, will this method simply be scavenging energy that would otherwise be wasted, or is he somehow stealing the energy that would otherwise be used in the transmission lines? Please explain the physics behind your response.


title: Relativistic Computation?

text: Is it possible to employ relativity to develop computational technology? Here is a really basic example: Build a Computer and Feed it the Problem (say the problem is projected to take 10 years to solve) Start the solution of the problem using algorithm. Now go on to your convenient relativistic jet... accelerate very near the speed of light (utilizing time dilation to your advantage) and make a round trip back to the planet, which takes a reasonable amount of time for the observer within the relativistic jet to complete (say a couple weeks) and allows them to arrive back to their computer with the problem solved. This sort of system could solve very difficult problems "relatively" quickly in the sense that the user would be able to find the solution to their problem in much less time ,relative to themselves, then normal. A more user convenient system can be found if research is being conducted in a heavy gravitational field (time speeds up). For example: If you have a lab close to the surface of a black hole, and you send your computer far away (much less gravitational attraction) a mere couple of minutes could pass in your lab's frame of reference while the computer had thousands of years to work, in it's frame of reference, so then the computer can be fetched back to the lab with the solution found. Unfortunately Earth's gravitational time dilation is not nearly enough to take advantage of this sort of system (which is convenient in that it allows the user to remain stationary). Are there any better ideas? Just for the sake of though experiment and logic.


title: Qubit projections

text: Given the qubit: $$\frac{|0\rangle+i|1\rangle}{\sqrt{2}}$$ What is the corresponding point on the extended complex plane and Bloch sphere? How to perform calculations and get the point representing the state of qubit?


title: Does Charge conjugation change the spin momentum?

text: Charge conjugation of a particle of spin momentum up is an anti-particle of spin momentum up or down?


title: Longitudinal EMAG wave?

text: I'm reading about optical waveguide analysis, and often come across the terms "transverse electric mode" vs. "transverse magnetic mode". As I unerstand, it means that the electric/magnetic field has only transversal component. But how could it have longitudinal component? Aren't the electric and magnetic fields always perpendicular to the direction of propagation?


title: Drift speed of electrons when the object is grounded

text: So I know that the drift speed of electrons is usually pretty slow. Let's say I have a charged sphere and I would ground it over a wire. How fast would the electrons leave the sphere? Would that drift speed be a lot faster than usual? I just want to get the general idea of the speed.


title: Apparent depth and the viewing angle

text: Does the viewing angle of a person observing a coin placed in a bucket of water affect the apparent depth in any way?


title: Does the uncertainity principle violate the law of conservation of energy?

text: What is the scientific view of the beginning of universe? Quantum fluctuation seems to contradict with the law of conservation of energy. Uncertainity Principle does seem to violate the Law of conservation of energy, I read it on **this Wikipedia page** and got confused.


title: Spectra of the Type II String theories

text: The spectrum of the Type II string theory (both IIA and IIB) is given by: \begin{array}{*{20}{c}} \hline & {{\text{Sector}}}& & {{\text{Spectrum}}}& & {{\text{Massless Fields}}} & \\\ \hline & {{\text{R}} - \operatorname{R} }& & {{{\mathbf{8}}_s} \otimes {{\mathbf{8}}_s}}& & {{C_0},{C_1},{C_2},{C_3}{C_4},...} & \\\ \hline & {{\text{NS}} - {\text{NS}}}& & {{{\mathbf{8}}_v} \otimes {{\mathbf{8}}_v}}& & {{g_{\mu \nu }},{F_{\mu \nu }},\Phi ,...} & \\\ \hline & {{\text{R}} - {\text{NS}}}& & {{{\mathbf{8}}_s} \otimes {{\mathbf{8}}_v}}& & {{{\Psi '}_\mu },\lambda ',...} & \\\ \hline & {{\text{NS}} - {\text{R}}}& & {{{\mathbf{8}}_v} \otimes {{\mathbf{8}}_s}}& & {{\Psi _\mu },\lambda ,...} & \hline \end{array} I know that for the Ramond-Ramond fields, the even ones belong to the Type IIB string theory and the odd ones belong to the Type IIA string theory. But what about the rest? Are they there in both Type II string theories? I think it should be the case, because the choice of the GSO projection is only for the R-R sector.


title: What causes a soccer ball to follow a curved path?

text: Soccer players kick the ball in a linear kick, though you find it to turn sideways, not even in one direction. Just mid air it changes that curve's direction. Any physical explanation? Maybe this video will better emphasize this phenomenon: http://www.youtube.com/watch?feature=player_detailpage&v=ZEv7QEFNVq0#t=265s


title: Second quantization

text: In second quantization we use Hamiltonian in form: $$H=\int d^3x [ \psi^{\dagger}(x) h \psi(x)],$$ where $h$ is Hamiltonian density. The field operators have following form: $$\psi = \sum\limits _{i} \phi(x) a(t),$$ $$\psi^{\dagger}=\sum\limits_{i}\phi^{*}(x)a^{\dagger}(t),$$ where $a$ and $a^{\dagger}$ are creation and annihilation operators. I want to try this formalism for Hamiltonian: $$H=\sum\limits_{i}{H}_{i}+\sum\limits_{i,j}V(\vec{r}_{i}-\vec{r}{_j}).$$ In the Mahan book, there is following expression: $$H=\sum\limits_{lm}a^{\dagger}_{l}a_{m}H_{lm}+\sum\limits_{lmij}a^{\dagger}_{j}a^{\dagger}_{m}a_{l}a_{i} \cdot \int d^3 r_{1}d^{3}r_{2}\phi^{*}_{m}(r_{1})\phi_{l}(r_{2})V(r_{1}-r_{2})\phi^{*}_{j}(r_{2})\phi_{i}(r_2).$$ I don't understand how he obtains this formula, especially why there are two integrals in the second term. Is there a simple way to get this form of Hamiltonian?


title: Heating a black body

text: By definition, Black body has absotivity=emmisstivity=1. This means the black body radiates all energy it accepts. Does this mean the black body cant be heated?


title: The physics behind The Great Flood

text: The book of Genesis floats (pardon the pun) some interesting numbers when discussing the Great Flood. For example, it rained for 40 days and 40 nights, and at the end of that time, the entire planet was covered in water. I think we can deduce how much water that would have had to be, estimating that the highest peaks in the Himalayas were covered with water. (8,848 meters above sea level) My questions are, how fast would the rain have had to come to raise the ocean level that high in 40 days and nights, how much would the mass of the earth have changed for this event, and would that significantly alter the strength of gravity on earth?


title: Expectation value of time-dependent Hamiltonian

text: I'm trying to solve a problem in QM with a forced quantum oscillator. In this problem I have a quantum oscillator, which is in the ground state initially. At $t=0$, the force $F(t)=F_0 \sin(\Omega t)$ is switched on and after time $T$ turned off again. I need to find $\langle \hat{H} \rangle$ at time $T$. I started out with this Hamiltonian: $$\hat H=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2-\hat{x}F_0\sin(\Omega t)$$ And I want to solve this problem in the Heisenberg picture. Then $$\langle \hat{H}\rangle=\langle\psi(T)|\hat{H}_S|\psi(T)\rangle=\langle\psi(0)|\hat{H}_H|\psi(0)\rangle\quad,$$ where $\hat{H}_H$ is the Hamiltonian in the Heisenberg picture and $|\psi(0)\rangle$ is the ground state of the harmonic oscillator. Since $\hat{H}_H=U^{\dagger}(T)\hat{H}_SU(T)$, I need to find the time evolution operator $U$. I previously asked a question regarding this operator, but don't see how to apply it to this problem.


title: Why is moment dependent on the distance from the point of rotation to the force?

text: The formula for moment is: $$M = Fd$$ Where F is the force applied on the object and d is the perpendicular distance from the point of rotation to the line of action of the force. Why? Intuitively, it makes sense that moment is dependent on force since the force "increases the intensity". But why distance? Why does the distance from the line of action of the force to the point of intensity affect the moment? I am NOT looking for a derivation of the above formula from the cross product formula, I am looking for intuition. I understand how when I am turning a wrench, if the wrench is shorter its harder to turn it but I don't understand WHY. Thanks.


title: What's the deal with the gyroscope?

text: In this article ("The problem with physics", Tony Rothman, ABC science) the author says in the 5$^\textrm{th}$ paragraph: > For example, one needs only first-semester equations to describe reasonably > well the behaviour of a gyroscope; engineers can then go off and build > gyrocompasses that guide aircraft or missiles to their destinations. But if > you merely ask: "At what, exactly, is the gyroscope pointed?" you are > plunged headlong into one of physics' deepest questions, one that led > Einstein to develop his general theory of relativity — and that, even today, > has no definitive answer. I know of no undergraduate textbook that > acknowledges the question." What is the issue he is referring to?


title: How much of theoretical physics research involves contemplation and reflection?

text: Something that I do when I'm learning physics is to contemplate and put into perspective the stuff that I've learned. I often find that I stumble upon deeper or more subtle questions and issues when I do this. Besides that, it's a kind of pleasure in itself. I'm thinking of going into a theoretical physics area and I'd like to know if this is par for the course for a physicist. (I'm coming from an engineering background and I found that there was nothing much to think about in engineering: the theory is very thin, at least at the Bachelors level. It's mostly doing stuff as opposed to contemplation). So, do I have the right picture of what being a physicist is like? Just trying to dispel any myths from my head..


title: Beam splitters and Mach-Zender interferometers

text: I have a question (my very first here) related to 50/50 beam splitters as used in the Mach-Zehnder interferometers (see for example http://en.wikipedia.org/wiki/Mach-Zehnder_interferometer). Let's concentrate on the input beam splitter: A continuous light beam (the input) is split 50/50, one in the 90deg direction and one in the forward direction of the input beam. Momentum/energy conservation teaches us that deflected part of the beam must exert a (tiny) force on the beam splitter. The forward part of the beam will not. In the single photon (quantum) case, the beam-splitter should get a slight "kick" if the photon is deflected, while the beam splitter is left in its original state if the photon goes straight through. If I got this right so far, we can now increase the energy (and momentum) of the photon and decrease the mass of the beam splitter (now coupled to a sensitive piezo transducer or something) so that a photon "kick" can be recorded if deflected. Now "which way" information is available and interference should be destroyed as I understand this. Would interference be restored if the transducer is disconnected? If yes, how do the photon "know" whether the transducer is connected or not? Maybe this is related to how much the photon/beam splitter system can become entangled to the environment? If the answer is "No", how weak must the photon kick be in order for the interferometer to work? We already know that it works for ordinary light (very tiny photon kicks). Update: Maybe I can reformulate a bit: Why does not the momentum exchange (or lack thereof) between the photon and the beam-splitter (and the trace this leaves in the environment) destroy the interference? Anyway, I see that no-one is interested in this question. Is the question very poor? How can I improve it? If someone just could point me to relevant some literature were transfer of momentum between between photons and optical elements are discussed.. Regards, Carl


title: Averaging decibels

text: Wikipedia: > The decibel (dB) is a logarithmic unit that indicates the ratio of a > physical quantity (usually power or intensity) relative to a specified or > implied reference level. If I measure some physical quantity in decibels, then what is the preferred way to calculate the mean of the measured values? Is it enough to simply average them, or should I convert them back to linear scale, calculate the average, and convert it back to decibels (example)? When should I use which approach, and why?


title: Gauge invariance and the form of the Rarita-Schwinger action

text: in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field strength, $F_{\mu\nu}$. This is because a massless vector field does not transform properly under Lorentz transformations (Weinberg, Vol. I eq. 5.9.22) and requires additional gauge symmetry/redundancy to 'mod out' unphysical degrees of freedom. One of the questions in Banks' Modern QFT textbook (problem 2.10) asks to repeat this general analysis for spin-3/2 and spin-2 fields, for which we know the main examples are the Rarita-Schwinger field and the graviton. For spin-3/2, however, I am confused as to why the Rarita-Schwinger field is described by an action which looks very similar to the Dirac action. How is it that the massless Rarita-Schwinger field, which carries a vector index as well as an additional spin-1/2 index, is able to avoid the ill-behavior under certain Lorentz transformations (which Weinberg calls $S(\alpha,\beta)$) that is exhibited in the massless vector? Naively I would have wanted to write a spin-3/2 field as a $(1,1/2) = (1/2,1/2)\oplus(1/2,0)$ representation of the Lorentz group so that the vector index can be treated independently of the spinor index. Shouldn't the argument for massless photons carry over and imply that I should have some kind of `field strength' for the spin-3/2 field? I suspect that what I'm missing is something trivial and silly. Thanks!


title: relation between first law of thermodynamics and statistical mechanics definition of entropy

text: From the definition of entropy as $S= - Tr (\rho\, ln \rho)$ one obtains that $S = \frac{\langle E \rangle}{T} + \log Z.$ The first law of thermodynamics has $dS = {dE \over T}$. Why is there no $d (\log Z)$ in there? Intuitively I see that $S$ and $E$ are thermodynamic quantities and $\log Z$ is not but formally I don't see why the change in $log Z$ (which is also a function of energy) does not affect the change in entropy.


title: Exam review question on thermodynamics

text: > A cube has a side length of $20\text{ cm}$. An atom in the gas moves around > the cube as shown. It continually bounces off the four lateral walls of the > cube. The atom has a mass of $6.6\times 10^{-27}\text{ kg}$. Because of the > elastic collisions with the walls, the atom maintains its speed of > $300\text{ m/s}$ as it moves around. The figure below shows the view from > above looking down into the cube at the path of the atom. > > ![Top view of > cube](http://dl.dropbox.com/u/11444220/Screen%20Shot%202012-02-02%20at%207.30.47%20PM.png) > > The repeated hits cause there to be a force applied to each of the four > lateral faces. Find the force (in Newtons) that it applies to Face 1. Find > the force it applies to Face 2. MY ANSWER: $$2 \sin(\theta)m v = 2 \sin(60) (6.6\times 10^{-27}) 300\text{ m/s} = 1.2 \times 10^{-24}$$ Do I have the right idea?


title: Orientation of Magnetic Dipoles

text: Does a magnetic dipole (in a permanent magnet) tend to align with the B-field or with the H-field? The current loop (Ampère) model of the magnetic dipole suggests the former, while the magnetic-monopole (Gilbert) model suggests the latter.


title: How to express answers where inequalities are involved?

text: Part of the floor of a workshop is inclined at 10 degrees to the horizontal. This is to allow the safe storage of steel cylinders. A cylinder of mass 7000 kg is stored as shown in the diagram (c the link below). The magnitudes of the forces exerted on the cylinder by the wall and the floor are Rw and Rf newtons respectively. Find the values of Rw and Rf. If the sloping part of the floor was inclined at alpha degrees (0 < alpha < 45) instead of at 10 degrees, show that a) Rw < 70000< Rf, whatever the value of alpha. b) Rw and Rf both increase with alpha. http://www.flickr.com/photos/8625954...in/photostream i did this: R (II to slope) : Rwcos10 = 70000sin10 ---- Rw = 12342.9 N R (perpendicular to slope) : Rf = 70000cos10 + 12342.9sin10 ----- Rf = 71079.9 N I am able to get Rw and Rf on the cylinder. But I don't know how to approach mathematically expressing the second part of the question (the proof part). Parts (a) and (b) of the second part. Could you please help?


title: How accurate are Mayan astronomical "ephemerides"?

text: Because of the hype surrounding the "end" of the Mayan calendar (along with the usual cultural relativism and Western guilt) it is nearly impossible to find an objective quantitative assessment of the accuracy and sophistication of Mayan astronomy. Specifically, how accurate were there "ephemerides" in their time, and extended into the present? Most popular descriptions suggest that we understand enough of their models to apply them to present times, but could we really use them to, say, predict the time of a sunrise or the right ascension of a planet. If so, with what accuracy?


title: Is the spin state of an atom related to the polarization of the photon it spontaneously emits?

text: From literature I've been reading, I find that scientists are able to "map" atomic states onto photon states. Are they talking about spin states and corresponding photon polarization states? Can somebody explain how these two things are related? I am having a tough time keeping track of the orientation of the measurement basis.


title: Existence and uniqueness of solutions to $\nabla^a T_{ab}$ in general (or special) relativity

text: The equation in the title of this question can be a relativistic analogue of the Navier-Stokes equation (in the sense that, in the low-velocity limit, it reduces to Euler's equation when $T_{ab}$ is the stress-energy tensor of a perfect fluid). I imagine that this equation is difficult to analyze in general relativity where the covariant derivative depends on the stress-energy tensor itself. But in special relativity, are solutions known to exist and be unique?


title: Gravitational collapse of a photon cloud

text: The Schwarzschild solution shows that a spherically symmetric, static fluid will undergo gravitational collapse if too much mass-energy is concentrated together (i.e. if the fluid's radius is less than the Schwarzschild radius for its total mass-energy). If a very large number of photons in a spherical arrangement ar travelling towards a central point, which all arrive at the same time, can gravitational collapse occur, or does the non-staticity due to movement allow one to avoid the gravitational collapse that would otherwise occur according to the Schwarzschild solution? Tl;dr: Can a large number of photons create a black hole?


title: How does a force on electrons produce a force on a metal plate

text: In a paper from Cadwell Magnetic damping: Analysis of an eddy current brake using an airtrack about eddy current brakes the author explains the effect qualitatively as follows: > When a nonferromagnetic conductor passes between the poles of a magnet, an > electric field is induced, and circulating currents called eddy currents are > generated. As a result, a magnetic damping force is induced on the eddy > currents which opposes the motion of the conductor. > > As an aluminium plate enters an external magnetic field [...] a Lorentz > force [...] is exerted on the conduction electrons in the aluminium > plate[...] The velocity of the plate $v$ and the magnetic field $B$ are > orthogonal to one another. An induced current moves along a closed path > [...]. > > A horizontal magnetic force is exerted on the portion of the eddy current > that is within the magnetic field. This force **is transmitted to the > aluminium plate** , and is the retarding force associated with the magnetic > braking. (my emphasis) The part about the moving electrons and the Lorentz-force is pretty clear. But how and why is the force **transmitted** to the aluminium plate? I mean we are speaking of conduction electrons, so I just would expect that the Lorentz force alters the path of the electrons, but why does the Lorentz force on those conduction electrons affect the aluminium plate as a whole? And why is the Lorentz-Force which is due to the motion of the aluminium plate with respect to the magnetic field not transferred to the plate as a whole (resulting in a force orthogonal to the direction of motion on the whole plate)? **Edit** Here is an illustration of the set up from the paper: ![](http://i.stack.imgur.com/hCh50.png)


title: Converting Point Gradients to Rotational Representation

text: I'm a PhD student in an unrelated field. It's been a very long time since I've done physics, and I've run into a problem in my research which I think is actually a physics problem. Basically, I have N 3-dimensional points attached to a rigid body. At each of the points, I have a 3-dimensional gradient of an objective function defined over 3D space. I want to find, given these N point gradients, a small transformation which will move the entire rigid body in the direction of the gradient (the "direction of steepest descent" in optimization terms). I think this problem is the same as finding the net force and torque on an object around its center of mass assuming a uniform mass distribution, but I'm not sure about that. Any thoughts? Edit More clarification: I have a cost function in 3D space. For all points, its cost is defined, as well as the gradient of the cost function. I want to find a local minimum of the cost function defined on the points attached to a rigid body by gradient descent. For example, if I have a cube with 8 points, and a cost function that is just 1 over distance from the origin, then descending the gradient of the cost function should push the cube away from the origin. I should be able to somehow compute the _way_ that the cube moves from the _point gradients_ defined at its corners. Translation is easy. I just add up all the point gradients. But what about rotation? Hope that makes it clearer. Some math: We have some cost function C. $$ \text{objective:}~~C(x) : \mathbf{R}^3 \to \mathbf{R} $$ $$ \text{gradient:}~~ \nabla C(x) : \mathbf{R}^3 \to \mathbf{R}^3 $$ We can additionally define a gradient over a set of N points, which are rigidly attached to one another. $$ \text{point-set objective:}~~C(\\{x_1, \ldots, x_N\\}) = \Sigma_i C(x_i)$$ We want to find: $$ \text{goal: }~~ T^* = \text{arg}~\text{min}_{T} C(\\{T x_1, \ldots, T x_N\\}) $$ Where T is a rigid transformation. We can find a local minimum by descending the gradient with respect to T, which should decompose into a translation part, and a rotation part. $$ \text{translation part: }~~ \nabla_{\text{trans}} C(\\{x_1, \ldots x_N\\}) = \Sigma_i \nabla C(x_i)$$ $$ \text{rotation part: }~~ \nabla_{\text{rot}} C = ??$$ EDIT: I found the answer to my question in section 4.2 of this paper: http://www.geometrie.tuwien.ac.at/ig/papers/tr117.pdf The key is to represent the problem using plucker coordinates, and indeed treat the problem as calculating the net wrench from a bunch of point forces.


title: What does this notation mean in terms of quantic numbers, and how to imagine the electrons in this quantic system? (Helium $2^1$ $P$ and $2^3$ $P$)

text: > Helium atom in the $2^1$ $P$ and $2^3$ $P$ excited states Now I'm guessing that 1 electron should be considered in the 1s state, but what about the other? Should I consider the other as simply being in the $p$ orbital ($l=1$) and having the spin as either antiparallel (for the $2^1$ $P$) or parallel (for the $2^3$ $P$) ..because the 1 and 3 can't possibly refer to the azimuthal quantic number $l$... or could I be mistaken?


title: Expressing an adjoint representation Wilson line in terms of the fundamental representation

text: I'm working out some calculations with Wilson lines, defined as path-ordered exponential integrals of a gauge field: $$U = \mathcal{P}\exp\biggl(ig\int_{-\infty}^{\infty}\mathrm{d}x^\mu T^c A_{c\mu}\biggr)$$ Here $g$ is the vertex factor for strong interactions, $A_{c\mu}$ is the gluon field which carries an $\mathrm{SU}(3)$ index $c\in\\{1,\ldots,8\\}$ and a Lorentz index $\mu$, and $T^c$ is an $\mathrm{SU}(3)$ generator in the fundamental representation. (Repeated indices are summed, as usual.) I can also write the equivalent in the adjoint representation, $$W = \mathcal{P}\exp\biggl(ig\int_{-\infty}^{\infty}\mathrm{d}x^\mu \tilde{T^c} A_{c\mu}\biggr)$$ where now $\tilde{T^c}$ is an $\mathrm{SU}(3)$ generator in the adjoint rep. I'm trying to prove the identity $$\boxed{W^{ab} = 2\operatorname{Tr}\bigl[T^a U T^b U^\dagger\bigr]}$$ but I'm kind of stuck on how to handle the generators in the exponentials. So far I see two possible ways to go about this: I can use the identity $(\tilde{T^c})^{jk} = 2i\operatorname{Tr}\bigl(T^c\bigl[T^j,T^k\bigr]\bigr)$ to write $$W^{ab} = \mathcal{P}\exp\biggl(ig\int_{-\infty}^{\infty}\mathrm{d}x^\mu (2i)\operatorname{Tr}\bigl(T^c\bigl[T^a,T^b\bigr]\bigr) A_{c\mu}\biggr)$$ and then try to separate that into two factors of the form $TU$, but I think I need at least some identity like $$T^a \mathcal{P}\exp\bigl(i T^c X_c\bigr) \overset ?= \mathcal{P}\exp\bigl(i (\text{something}) X_c\bigr)$$ and I can't find one. Alternatively, I can start from the other side and expand to first order, $$\begin{align} 2\operatorname{Tr}\bigl[T^a U T^b U^\dagger\bigr] &\approx 2\operatorname{Tr}\biggl[T^a \biggl(1 + ig\int\mathrm{d}x^\mu T^c A_{c\mu}\biggr)T^b \biggl(1 - ig\int\mathrm{d}x^\mu T^c A_{c\mu}\biggr)\biggr] \\\ &= 2\operatorname{Tr}\biggl[T^a T^b + ig\int\mathrm{d}x^\mu \bigl(T^c \bigl[T^b, T^a\bigr]\bigr) A_{c\mu}\biggr]\quad\text{(cyclic property in trace)} \\\ &= \delta^{ab} + g\int\mathrm{d}x^\mu (\tilde{T^c})^{ab} A_{c\mu} \end{align}$$ which gets close, but this leaves me without a needed factor of $i$, and also it's only approximate, since there's no reason to assume higher order terms will completely vanish (e.g. no $g\to 0$ limit or anything like that). Can anyone suggest another method, or fill in the gaps?


title: what is the farthest-away individual star that is visible with the naked eye?

text: My girlfriend and I were watching _Cosmos_ , and something Carl Sagan said got us wondering what the farthest-away visible star is. Obviously "visible to the naked eye" is a fuzzy enough concept that there may be multiple defensible answers, but hopefully not too many. To make the question a little more interesting, let's restrict to individually distinguishable stars; otherwise the answer is pretty clearly some Local Group galaxy, and there aren't many of them to check. The closest thing to a reasonable answer we came up with was this wikipedia list of stars that are more luminous than any closer star. The farthest-away star on that list with a plausibly visible apparent magnitude is Eta Carinae (7500 ly away, magnitude 4.55). However, there are several reasons why I'm not willing to consider this definitively answered:   * It's a wikipedia article, and a poorly sourced one at that. So I don't entirely trust it.   * It sorts stars by bolometric luminosity rather than visual luminosity, so perhaps there's some farther-away star whose spectrum is better-centered in the visible range.   * The farthest-away visible star isn't actually guaranteed to be on a list of that sort, even assuming the other two points are cleared up. Perhaps the farthest-away visible star is only barely visible, and there's some star both closer and absolutely brighter than it which makes the list. Given all these points, **is it actually the case that Eta Carinae is the farthest-away visible star, or is there some visible star that's farther from us?**


title: Identity as a trivial reducible representation

text: In particle physics, I was taught that a representation of a group is a function $r: group \rightarrow matrices\,(n\times n)$ such that $r(g_1)r(g_2)=r(g_1g_2)$ and $r(e)=I_{n\times n}$. Then, that a representation is reducible when you can find a matrix $A$ such that $Ar(g)A^{-1}$ is in diagonal-block form for every element of the group. Then the professor tried to find in complicated ways reducible representations of $SO(N)$, $SU(N)$ and so on. But the trivial function that assigns $I_{n\times n}$ to every value of $g$ is not already a reducible transformation? I know it must be somehow useless, but what did I lose?


title: Specific electron energy gap values $E_{i+1}-E_i$ vs. photons with arbitrary energy $\hbar \omega$

text: The energy levels of electrons in an atom are quantized $E_i$. A photon of a specific momentum $\vec p$ and energy $$\omega=(E_{i+1}-E_i)/\hbar$$ hits an atom and gets absorbed. Okay now say the energy of the photon is $$\omega'=(E_{i+1}-E_i)/\hbar+\Delta E/\hbar,$$ where the value $\Delta E$ is some energy, which is not even near the value of any electron state gap energy. hence the quantity $\Delta E$ cannot be absorbed by means of excitation of the other particle. What happens in this case? Does this energy- wise non-fitting photon just go through the material? Does the photon "spit" in a way that the part $E_{i+1}-E_i$ of the photon energy gets absorbed and a new photon with energy $\Delta E$ will move away? Microscopically, has this anything to do with soft radiation? Okay well, if we model the atom as only being able to suck up energy by changing it's energy levels, this would violate momentum conservation. I guess the atom must start to move too, or if we view it as part of a bigger solid state system, the photon will make the thing wobble a little. From that perspective, we'd have to consider the kinetic energy distribution of the particles composing the material and this would be able to suck up any real energy or momentum value. If we leave the microscopic description (in terms of which I'd like to see an answer to the first question - after all, I'm speaking of the behaviour of a single photon here) and go on and view this as a "gray body" problem, and in case the answer really is that after the photoelectric effect the rest energy gets termalized, then I'd like to know the relation (for a class of materials) between the absorption per frequency and material characterizing parameters like (electron/atom density? emissivity? ...).


title: Changes in Water Bonding Angle

text: I heard something recently in a casual discussion, but have yet to be able to confirm it: is there any evidence that the bonding angle for a water molecule, currently defined as 104.5, has been either steadily growing or shrinking since it was first observed and measured? Is there any possibility that water samples from different locales would produce different results? I am speaking strictly of water in as pure a form as possible, not water affected by impurities or physical manipulation.


title: What is the theoretical justification for a fluid flow's being irrotational?

text: I am not a fluid dynamicist, and I really just began thinking about this problem as my curiousity drew me into building an answer for the question What really allows airplanes to fly?. It is very clear from the answers to the following questions: When is a flow vortex free? Does a wing in a potential flow have lift? that viscous flows cannot be everywhere irrotational. Moreover, some handwaving justification is given in answer to the first question that low viscosity fluid can be irrotational. Now, the spin angular momentum per unit volume of a fluid is the vector $\rho\,\nabla \wedge \vec{v}$. So the assumption of irrotational flow is the assumption of lack of spin angular momentum. I can accept that it is reasonable in some cases to accept this to be true. But is there a deeper theoretical justification as to _why_ and _when_ a fluid's spin angular momentum should be nought? In electromagnetism, we have specific equations - the Ampère and Faraday laws _i.e._ $\nabla\wedge\vec{H} = \vec{J}+\partial_t\vec{D}$ and $\nabla\wedge\vec{E} = -\partial_t\vec{B}$ - for the "source" of such vorticity in the fields. At an elementary level, we can see that in electrostatics conservation of energy behests that $\oint\vec{E}\cdot \mathrm{d} \vec{r} = 0$, so we immediately grasp irrotationalhood for the electrostatic field. Is there any such analogue in fluid dynamics? The Navier-Stokes equation doesn't seem "split" the velocity field up into curl and divergence terms like the Maxwell equations do, or to put it more pithily, the Maxwell equations say that the source distribution must be an exact form $\mathrm{d} \vec{F} = \mu_0 \vec{J}$ and we can thus "invert" $\mathrm{d}$, to within a constant (which we can set to nought the grounds that it would have infinite energy). Are there any other ways to "split" the Navier-Stokes equation like Maxwell's equations are to shed intuitive light on the nature of an irrotational flow?


title: Hamiltonian of Harmonic Oscillator with Spin Term

text: We have the usual Hamiltonian for the 1D Harmonic Oscillator: $\hat{H_{0}}=\frac{\hat{P^2}}{2m} + \frac{1}{2}m \omega \hat{X^2}$ Now a new term has been added to the Hamiltonian, $\hat{H} = \hat{H_0} + \mu B\hat{S_z}$ The system has a spin degree of freedom with $s=\frac{1}{2}$ What are the new eigenstates of the Hamiltonian and what are the energy eigenvalues? We have been denoting the stationary states as $|n,s_z\rangle$ and the non-spin eigenvalues as usual $E = h \omega (n+1/2)$ Can anyone give me some help with this question? I'm guessing that the eigenstates don't change and then it's straightforward to find the eigenvalues but I'm not sure if this is correct (or why it would be correct).


title: How do we measure the range of distant objects despite relativistic effects?

text: When we observe astronomical objects like distant galaxies there are several complicating factors for estimating the distance:   * Relativistic speed result in length contraction   * Relativistic speed results in Doppler-shifts in the frequency of light   * Simultaneity is skewed due to our differing velocities With these effects, how are we able to state with decent accuracy how far away all of the distant galaxies are from us? Do the above effects have an impact on the measurements? I know that determining far distances is mainly done by measuring the red- shift of light from those sources. A higher z number normally means it is farther away (due to the accelerating expansion of the Universe). My question is really about how relativistic effects are accounted for and whether or not they play a big role in the uncertainty of our calculations. I'm looking for an answer geared towards the layperson, but formulas are always a welcome challenge.


title: Virtual-work problem

text: I have a very specific doubt about the next exercise: I have the crane of the picture: ![enter image description here](http://i.stack.imgur.com/O4GH9.png) With a force $F_a=-K\varphi$ Applied on the point B, perpendicular witch AB, and another force $F_b=-K\psi$ Applied on the point C. They ask me to find an equilibrium configuration with the virtual-job theorem. I define a new basis wich moves with the first arm of the crane (AB): $i'=i \sin(\varphi) + j \cos(\varphi)$ $j'=j \sin(\varphi)- i \cos(\varphi)$ And another base which moves with the second crane, as a function of the $"'"$ base $i''=i' \sin(\varphi) + j' \cos(\varphi)$ $j''=j' \sin(\varphi)- i' \cos(\varphi)$ Also, I define the vector $r_1$, which goes from A to B. To find the virtual displacement, I must do: $\frac{\partial r_1}{\partial \varphi}$. BUT (and this is my question), the arm AB hasn't any relation with the movement of the BC arm. Then, in order to find the virtual dispement I must find $\frac{\partial r_1}{\partial \psi}$ which should be zero. **But** , I can write the vector $r_1$ as a function of the double $ '$basis, then derive it, and have something different from zero. So ¿$\frac{\partial r_1}{\partial \psi}=0$?


title: Boiling Order (Thermodynamics)

text: Three pots made of the same material are placed one inside the other and all pots are filled with water to the same level. If heat is applied to the bottom of the outermost pot what will be the boiling order of the water in each pot and why? PS: Clarified with the person who asked me this question--The pots are suspended inside one another and so there is no metal to metal contact with the inner pots.


title: Wave function decomposition

text: **Problem:** Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$,   1. find the eigenvalues and eigenfunctions of $\hat{H}$ and    2. derive the form of $\Psi(t)$. **Solution:** I apply the $\hat{U}(t)$ time-evolution operator on $\Psi_0$ to get $\Psi(t)$: $\Psi(t)=\hat{U}(t)\Psi_0=\exp\left(-\frac{i}{\hbar} H t\right) \Psi_0$. In order to calculate the effect of $\hat{H}$ on $\Psi_0$, I need to write $\Psi_0$ as a linear combination of the spherical harmonics $Y_{\ell m}$. I have a list of the first few (up to $\ell=2$) spherical harmonics in my textbook, but I am not so sure on how to proceed. The best guess I can think of is: $Y_{20}=\sqrt{\frac{5}{16\pi}}(3\cos^2(\theta)-1)$ if I consider that $\cos^2(\theta)=1-\sin^2(\theta)$. Then $\Psi_0 = \frac{2}{3}A - \left(\frac{A}{3}\sqrt{\frac{16\pi}{5}} \right)Y_{20}$. **My questions are these (edited more than once)** :   1. How do I tackle the constant term ? Do I use $Y_{00}=\frac{1}{2}\sqrt{\frac{1}{\pi}}$ for that?   2. Is it physically meaningful to have $Y_{\ell m}$ terms with different $\ell$ in the same wave function, like e.g. $Y_{00} + Y_{20}$ ? I am asking because until now I was encountering only terms with the same $\ell$ but different $m$. I presume it is.   3. Is there anything fundamentally wrong with my strategy?   4. How do I calculate constant A? I used the normalization condition $|\Psi_0|^2=1$, and since $\hat{U}(t)$ is unitary, $\Psi(t)$ will also be normalized. Finally, $\Psi_0 = \sqrt{\frac{5}{6}}Y_{00}-\sqrt{\frac{1}{6}}Y_{20}$. At this point I have the following question:   5. How do I calculate $\exp\left(-\frac{i}{\hbar}Ht\right)$ say on $Y_{20}$? I recollect a theorem which states that if operator $\hat{A}$ has eigenvalues $\alpha$, then operator $f(\hat{A})$ has the same eigenfunctions with eigenvalues $f(\alpha)$ ($f$ needs to fullfil some criteria that escape me). Anyway, the "problem" is that in this case $\hat{H}$ is a function of both $L^2$ _and_ $L_z$. Given that $Y_{\ell m}$ constitute a common set of eigenfunctions for both $L^2$ and $L_z$, could I claim that $Y_{20}$ is also an eigenfunction of $H(L,L_z)$ with eigenvalues $\frac{\hbar^2 \ell (\ell+1)}{2I} + gB(m\hbar), m \in Z$ ? **EDIT** : I use this Mathematica snippet to verify my results (up until now): $\text{In:=} \text{FullSimplify}\left[\frac{4}{3} \sqrt{\pi } A Y_{00}-\frac{A}{3} \sqrt{\frac{16 \pi }{5}} Y_{20}\text{ /.}\,\\\ \left\\{Y_{20}\to \sqrt{\frac{5}{16 \pi }} \left(3 \text{Cos}^2(t)-1\right),Y_{00}\to \frac{1}{2}\sqrt{\frac{1}{\pi }}\right\\}\right]\\\ \text{Out=}A-A \text{Cos}^2[t]$


title: Help understanding a Magnetic Levitation "Physics Toy"

text: I was shown a toy, yesterday, which I would like help understanding qualitatively. ![](http://d1jqecz1iy566e.cloudfront.net/large/lt004.jpg) A fellow engineer showed me a kit which included three main parts: 1.) A base (black box), approximately 4 inches square and 2 inches high, weighing about 2 pounds (quite heavy for its size). No batteries. 2.) A rigid plexiglass window, also 4 inches square, which rested nicely on top of the base. 3.) A top. This object was about 1.5 inches tall. a flat metal washer (ID = about 0.25", OD = about 1.0") was mounted on the top-side of the top, and a plastic washer slid down over the top post of the top, to keep the metal washer securely in place. The owner said that the system was designed like that so that multiple metal washers could be used. (We only used one.) OPERATION: 1.) Place the top in the very center of the plexiglass and give it a good spin. 2.) With the top spinning in the very center of the plexiglass, gently and slowly lift the plexiglas about 1 inch off of the base. The top will levitate up off the plexiglass about 4 inches, and hang suspended for about 1/2 a second before it falls off to a side (as if it were sliding off of an invisible 4-inch-tall and 4-inch radius hemisphere. EDIT: We must have been doing something wrong. I found a video of this top staying levitated for almost 2 minutes: www.fatbraintoys.com/toy_companies/fascinations/levitron_platinum_pro_combo.cfm?display=video 3.) I noticed that on one of my trial runs, the washers flew off the top AFTER it had levitated and BEFORE it had fallen to the desk. QUESTIONS: 1.) (My main question) How does this thing work? 2.) Anyone know where this kit (and/or other like-minded novelties) are sold? (I tried "Think Geek," but was disappointed in their selection of mag-lev products.) EDIT: Found a very similar toy at: www.fatbraintoys.com/toy_companies/fascinations/levitron_platinum_pro_combo.cfm


title: Minimal temperature achievable by vanilla Peltier element?

text: I wonder, are there any fundamental issues leading to reduced performance of Peltier elements at cryogenic temperatures (-100C and lower)? What is theoretical/practical minimum temperature achievable by a cascade of Peltier elements, provided that each next element have about 3 times less power than previous in cascade, so that they are not overwhelmed by selfheating? Let's say first element is water-cooled down to 20C when dissipating 150W. **Update** : After extensive tests, I've found out that in any setup I cannot get below -19C using any number or combination of Chinese (r) Peltier elements (I've tried alot of different ones in different combinations).


title: Laptops in space

text: I was reading an article regarding the Shuttle's GPCs and how they stack up against commercially-made hardware on http://www.nasa.gov/mission_pages/shuttle/flyout/flyfeature_shuttlecomputers.html and was finding it quite interesting. One thing that caught my attention was that they have a lot of issues caused by radiation with their off-the-shelf laptops (IBM/Lenovo Thinkpads), with the memory in each machine typically getting fried 2-3 times per mission (and a lot more on Hubble missions), with a high probability that it will happen while crossing the South Atlantic Anomaly. That got me thinking. The Thinkpads, like most off the shelf computers, are plastic. Sure, they have RF shielding, but that exists mostly to appease regulators regarding RF emissions and would offer little to no radiation protection. However, there are laptops on the market with metal cases (most notably the Apple laptops, made from a milled block of aluminium). Would an off-the-shelf machine in a metal casing hold up better in space compared to an otherwise identical machine in a plastic case? I understand some plastics can be quite effective radiation blockers, and metals can actually make the situation worse (though I'm by no means an expert on nuclear physics, I'm just interested). Are there any cases on record of a matal-clad laptop going into space, and if so how did it hold up compared to the plastic- clad thinkpads? Apologies if this is the wrong SE site to be asking this on, but while it is to do with computers I think the answer to this question lies in physics so it seems to me that this is the correct place to be asking a question like this.


title: Detecting radiation from tritium keychain

text: I am measuring radiation from 6 tritium glow keychains. Usual gamma detector does not detect anything (this is expected), but on beta detector with thin window I am getting about 5 counts per second. Count rate drops by mere 10% when using 1mm plastic shield. My understanding is that I am not actually detecting betas, but rather weak secondary X-Ray radiation generated when electrons travel through keychain's plastic. Is that correct? ![enter image description here](http://i.stack.imgur.com/DufVf.jpg)


title: Electric field outside a capacitor

text: I know that the electric field outside of a capacitor is 0 and I know it is easy to calculate using Gauss's law. We create cylindrical envelope that holds the same amount of charges (of opposite signs) in each plate. My question is why can't I pick an envelope which include only part of one of the plate? Guess's law states, specifically, that I can pick any envelope I want. Note: I have encountered this question a couple of years a go and I got an answer which I was not completely happy with.


title: Container divided in two parts with ideal gas

text: I wanted to ask your help with the following problem: Consider a closed container, that is, with insulated walls impervious, and rigid. The container is divided into two parts by a wall having the same properties. Each part contains a portion of the same ideal gas. initially on the left has a volume $V_{1}$, $N_{1}$ atoms and a temperature $T_{1}$. On the right is $V_{2}$, $N_{2}$ and $T_{2}$. The wall that separates the gas loses its rigidity property, but still is insulated and waterproof. Suppose that $T_{1}\neq T_{2}$, $V_{1}\neq V_{2}$ and $N_1 \neq N_2$, initially. Find the final thermodynamic state. I understand that by thermodynamic end state refers to temperatures, volumes, entropies, number of atoms, change energy, work and heat. However, I could not get the volumes as a function of baseline variables and neither obtaining the entropies.


title: Reverse bias P-N junction

text: I am not able to understand why the PN junction does not conduct when reverse biased. Can't electrons travel from the N side through the power supply to the P side where they can jump from one hole to other to traverse the P side and then cross the depletion layer back in to the N region ?


title: Proving that the free energy is extensive

text: If I have two system of an Ideal gas $A$ and $B$ each of these system has a partition function: $Z_{A,B} = \left ( \frac{V_{A,B}}{\lambda_T} \right )^{N_{A,B}}$ Where: $\lambda_T = \left ( \frac{m}{2\pi\beta \hbar } \right )^{\frac{1}{2}} $ The free energy is: $F_{A,B} = -kT \ln \left ( Z_{A,B} \right ) = -kT N_{A,B}\ln \left (\frac{V_{A,B}}{\lambda_T}\right)$ For the free energy to be extensive the following must be true: $F_{A} + F_B = F_{A+B} \Rightarrow Z_A \cdot Z_B = Z_{A+B}$ However: $Z_A \cdot Z_B = \left (\frac{V_A}{\lambda_T} \right )^{N_A} \left ( \frac{V_B}{\lambda_T} \right)^{N_B}$ and: $Z_{A+B} = \left ( \frac{V_{A+B}}{\lambda_T} \right )^{N_A + N_B}$ So, for $Z_A + Z_B = Z_{A+B}$ to be true $V_A^{N_A} V_{B}^{N_B} = \left(V_A + V_B \right)^{N_A + N_B}$ must be true as well but this isn't true for any system. Since we cannot create energy by mixing two containers of Argon in the same pressure and temperature, something in my understanding is wrong. Where is my fault?


title: How do particles, such as electrons become visible?

text: Quantum mechanics says that atoms are invisible - they do not have some specified location, only a probability distribution. So, how can we see them? If there is to be particle-antiparticle annihilation (or other interactions), the particles must have a fixed location, right? So, is this process just random? Is it impossible to know whether a pair will annihilate, but only know the probability?


title: DC Motor Torque Constant

text: I am very new to DC motors and to stackexchange. Please correct me if anything I said does not make sense. For DC motors, the equation looks like this: $P = \tau\dot{\theta}$ where $P$ is power, $\tau$ is torque and $\dot{\theta}$ is angular velocity. The power can be expressed as $P=VI$ (voltage times current) and $\tau=K_\tau I$ ($K_\tau$ is the torque constant). Thus, $VI = K_\tau I \dot{\theta}$ It seems from the above equation that the angular velocity is directly proportional to the voltage applied to the motor and the current flowing through the motor is irrelevant. For a PWM controlled motor, the effective voltage is $V_{in} \times \textrm{duty}\%$. For different loads applied to the motor, the angular velocity should be the same if the PWM duty is fixed. However, in reality, when the load increases, the RPM goes down for a given PWM duty. So, the question is:   * Is $K_\tau$ constant across different loads for a motor (in this case, a DC shunt motor)?   * What is the 'effective voltage' that should be used in the equation? If $K_\tau$ is constant with different loads for a given motor, then the 'effective voltage' must **not** be a simple $V_{in} \times \textrm{duty}%$. When the load increases with a fixed PWM duty, there must be a decrease in the 'effective voltage' since the RPM goes down. If the 'effective voltage' is the voltage difference between Motor+ and Motor-, I did measure it and the voltage difference drops when the load increases for a given PWM duty. Is it that the back EMF causing the reduction in the voltage difference? Thank you in advance.


title: Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

text: Operators can be cyclically interchanged inside a trace:   $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero:   $${\rm Tr} ([A,B])~=~0.$$ But what about the commutator of the position and momentum operators for a quantum particle? On the one hand: $${\rm Tr}([x,p])~=~0,$$   while on the other hand: $$[x,p]~=~i\hbar.$$ How does this work out?


title: Impulse from absorbing a photon? Is there an increase in rest mass?

text: I'm going through A P French's special relativity. In one chapter (6) the following is set up: > Suppose that a stationary particle of mass $M_0$ is struck by a photon of > energy $Q$, which is completely absorbed. The combined system will have mass > $M'$ and will recoil with a velocity $v=\beta c$. It then goes on to say that conservation of energy implies: $$E=M_0 c^2+Q=M' c^2 $$ (1) and conservation of momentum implies: $$p=Q/c=M' \beta c$$ (2) Using the first equation to solve for $M'$, and plugging $M'$ into the second equation to solve for $\beta$, we wind up with: $$\beta=\frac{Q}{M_0 c^2+Q}$$ However, if I use what was developed earlier in the chapter, $M'=\gamma M_0$, and assume equation (1) [ignoring (2)], I get a completely different answer. Likewise, if I assume equation (2), I get yet a different answer, for $\beta$. (So, the first solution would be solving $M_0 c^2+Q=\gamma M_0 c^2$ for $\beta$, with $\gamma=(1-\beta^2)^{-1/2}$. The second solution would be solving $Q/c=\gamma M_0 \beta c$ for $\beta$. I didn't include the solutions I got because they're [from what I gather] incorrect, and because the equations for them aren't too interesting/enlightening/short to write) I figure the only way to reconcile these problems is if $M'$ doesn't only depend on $\beta$, but if instead the rest mass actually increases from before the collision, so that I have $M_0 c^2+Q=\gamma M_0' c^2$ and $Q/c=\gamma M_0 ' \beta c$. Then, I'd have two equations and two unknowns, so the result in the book would hold, and I can calculate the new rest mass. My question is: Is this a real effect? If so, is there a name for it? It just seems a bit remarkable, and it's something that if I had ignored momentum and only looked at energy (or vice versa), I would have missed completely.


title: Quantum Ripples?

text: Would someone please explain what quantum ripples are? I've heard of gravitational waves, are they the same thing? I overheard someone saying that it could allow for possible faster than light speed travel? Is this true or complete nonsense.


title: What stochastic process is associated with spontaneous electromagnetic radiation?

text: Imagine one have an ideal sensor, which can convert the emission to some kinds of signal (typically, voltage, and suppose no noise at all), then what process can describe the measure data? Is it related to Stochastic electrodynamics?


title: Given a state function of a particle, can we determine its mass?

text: The quantum state of a system is supposed to contain all the information that can be obtained about the system such as its energy, momentum...etc. So I have 2 questions: 1-If someone gave us a quantum state of a single particle, can we tell of what mass it is? 2-Another question is that, given a quantum state can one tell what the Hamiltonian is?


title: Calculating lifetime of a pi meson via Heisenberg uncertainty relationship?

text: This is a problem from my textbook: "A proton or neutron sometimes 'violates' conservation of energy by emitting and then reabsorbing a pi meson, which has a mass 135MeV/$c^2$. This is possible as long as the pi meson is reabsorbed within a short enough period of time $\Delta t$ consistent with the uncertainty principle. Consider $p \to p + \pi$. By what amount $\Delta E$ is energy conservation violated? (ignore any kinetic energies)" For this, I thought it appropriate to take the "violation" of energy conservation's only constituent to be the rest energy of the pion, $E_0 = mc^2 = 135MeV$. "For how long a time $\Delta t$ can the pi meson exist?" Using the energy-time uncertainty relation, I do the following calculations: $$\Delta E \Delta t \approx \hbar $$ $$\Delta t \approx \frac{\hbar}{\Delta E}$$ Taking $\Delta E $ to be the rest energy of the pion calculated above, the resulting $\Delta t$ comes out to be $4.88 \times 10^{-24}s$. But checking the Wikipedia article on pi mesons, the lifetime is said to be $2.6 \times 10^{-8}s$. Is there something that I'm missing in this process? Was my choice of the rest energy for $\Delta E$ perhaps incorrect? Why am I coming out with the wrong value for the lifetime of the pion?


title: Find largest angle

text: The two homogeneous bars AB and BC are connected with a pin at B and placed between rough vertical walls. If the coefficient of static friction between each bar and the wall is 0.4, determine the largest angle θ for which the assembly will remain at rest. ![](http://screensnapr.com/e/vU9M3X.jpg) Attempt at solution, so i use the fbd of the whole body, and take summation of forces on x axis, and i got Na = Nc (normal at point a is equal to point c). then, i take summation of forces on y axis, so thats Fa - 6N -8N +Fb = 0 (where fa and fb are friction forces on a and b), and since fa = fb (their normals are equal), i use f = uN and yielded Na = 17.5 N. I then take summation of moments at point B( bar of A to B) and heres eqn (6)(300)(cos(x)) - 17.5 (600)(cos(x)) + 7(600)(sin(x)) = 0 and the value of angle is 64 degress. the book says 10 degress. Where did I go wrong?


title: Finding symmetry of a part of an equation, given the group transformation property of another part

text: I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super- symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to the Bogolomony equation for bps monopoles. He finds out that when the simple root used for embedding the SU(2) monopole is simple then the bosonic zero modes which I believe is the solution for the higgs field transform as $1 \oplus 0$, and when it is not a simple root it transforms as two doublets. Hope this is not required for my question. He founds it in the following way. The fermionic zero modes are the solution to $\psi$ given by $$\psi = \chi \otimes (1\,\,\, 0) - i\sigma_2 \xi^{\star}\otimes(0\,\,\,1) $$ The zero modes are eigenvectors of $J=j+s$ the sum of the spatial rotation, and global gauge transformation. $\psi$ transforms as a $J \otimes \frac {1}{2}$ representation. My question is if the zero modes of $\psi$ is a triplet of SU(2) how, do I conclude that zero modes of $\chi$ transforms as $1 \oplus 0$ wrt to $J$, as stated in the paper? Also, if $\psi$ is a doublet, how do the zero modes of $\chi$ transform,


title: What is a good way to reason in physics?

text: I have recently made the decision to study Physics seriously. However, in the past, I've had some difficulty with the subject because of my primarily mathematical background. I find that sometimes even if my reasoning is axiomatically correct, it doesn't lead to the right answer for simple reasons like I made a wrong assumption, or I hadn't assumed something I should have. So, I have three questions: 1) Is there some system or method I can use to determine what assumptions I should be making? This is important to me because it seems the problems I've encountered traditionally seem to be open-ended in this regard when they're not simple plug-and-chug questions. 2) There's also the question of definitions. I am uncomfortable with the fact that most definitions are not rigorous enough; this usually leaves some room for interpretation and consequently, it's unclear how they may be applied to certain situations. How do I know if the interpretation I've made of the definition is accurate? 3) Thirdly, when I'm reasoning with a problem, I'm not sure if at each step, what I've concluded is accurate or would actually happen in reality. This applies especially to problems where I'm asked to make simplifying assumptions (such as, "assume the surface is friction-less" or "assume the rod supporting the beam is mass-less") which simply wouldn't be applicable in the real world. So, I'm not really sure what to model my reasoning after in these cases; how can I get around this? Also, is solving such problems really helpful as far as learning actual Physics goes? I can only speak for classical Physics, as I haven't progressed much further. Is Modern Physics the same in this regard, or is it more axiomatic? Finally, if there really isn't any systematic approach to this, then how can we really call the process Science? Isn't the scientific method all about finding agreement? Thanks for all the help in advance!


title: Phasor representation of voltage in frequency domain

text: In a text on application of electromagnetism in transmission line, there introduces a phasor for the voltage (in frequency domain) $$\tilde{V}(x) = V^+e^{-i\beta x} + V^-e^{i\beta x.}$$ Here $V^+$ and $V^-$ are the amplitude of the incoming wave and reflected wave. My question is the exponential $e^{-i\beta x}$ and $e^{i\beta x}$ are already complex, so should $V^+$ and $V^-$ be real or complex? and why?


title: Are probability-preserving variations of QT with respect to the Born rule mathematically possible?

text: Is it possible to create (m)any theoretically workable framework(s) - that do(es) produce probabilities - by taking QM and replacing the Born(-like) rule(s) with something that is not equivalent to it (them)? And, only if so, would that still be called a quantum theory (not necessarily physical such as QM) or does one call that something else?


title: Approximating Rolling/Sliding in 2D Shape

text: I'm trying to find more information on how a 2D shape (could be defined by a function, such as ellipse, or by a polygon) will roll across a surface. The shape could be nearly circular or quite eccentric which will obviously have a huge impact on how well it rolls. There is a wealth of information on a rolling circle/sphere/cylinder that is fairly simple but I can't find anything beyond that. Sitting down trying to figure out the dynamics of how a shape will move becomes incredibly complex very quickly even in 2D, let alone in 3D. I'm just looking for some reading on the topic and potentially a method for approximating rolling or sliding on a 2D shape. I haven't found much other than a sliding rectangle or a rolling circle though. Any recommended reading or direction on where to look?