Wire forces in an inductor

For an inductor, the electromotive force comes from Faraday's law, as I understand it. The governing equation includes the number of turns, the magnetic flux (field and area), and a time differential. $$ \text{emf} = n \frac{ d \phi }{ dt } = n \frac{ d ( B A) }{ dt } $$ The idea is that $B$ changes, because that comes directly from the current value. So does $A$ want to change too? Since increasing area would decrease the flux, I picture a natural tendency to close or open the area in the same way as the inductor does for the current, because they're contained in the same derivative. Does that imply that the wire experiences an outward or inward force? Inductors you can buy tend to be fairly well secured, but if they weren't, is there are combination of $I$ and $dI/dt$ that would make the loop balloon out, and also contract in? I ask because I'm curious if this is used as artificial muscle fibers. Make a connected series of loops like a chain, and then alter the current to contract and expand it. Other examples of such artificial muscle fibers I've seen use less direct material properties. But I've not seen the idea mentioned before, and in fact, I've never seen it mentioned that inductors experience wire forces. I'm not really sure how to formalize that in the first place.


Is thermal noise "quantum random"?

Is the randomness that can be extracted from thermal noise "as random" (that is, even theoretically inaccessible to measurement according to our knowledge of quantum mechanics, and not just random for practical purposes) as the one that comes from "true" quantum phenomena, like radioactive decay of nuclei? Due to the uncertainty principle, we can't predict both a particles position and speed infinitely accurate, and therefore we shouldn't be able to predict thermal noise, so it would seem that there is at least some level of quantum mechanics involved, but is that the same randomness that dictates the decay of nuclei, tunneling and other quantum phenomena? So, is there a true distinction between thermal and "quantic" noise on a theoretical level, or are they basically the same? In other words, is the thermal movement of atoms and molecules fundamentally deterministic or indeterministic?


Degeneracy in quater-wave stack

Consider a 1D photonic crystal, the quarter-wave stack, and its band structure. A famous conclusion is that there's no gap at Brillouin zone's center. In other words, successive bands are degenerate at k = 0. It's this degeneracy I try to understand. My current understanding is that, at k = 0, the quarter-wave stack supports standing waves and since each layer is exactly a quarter-wavelength in thickness, the energy densities of a certain pair of successive bands correspond to the same energy level. Is there any alternative explanation of this fact? Can one predict this degeneracy from group theory?


What is the physical meaning of the norm of the electromagnetic four-potential?

In SI units this would be $\frac{1}{c^2}\phi^2 - A_x^2 - A_y^2 - A_z^2$. Is this just not a physically meaningful quantity at all because it's not gauge invariant?


Does space curvature automatically imply extra dimensions?

Total newbie with basically no physics knowledge here :) I would welcome any correction to the steps of my reasoning that lead to my question, which could easily turn out to be invalid :) My current understanding is that General Relativity is definitely accepted by the vast majority of scientists, and that according to General Relativity spacetime is curved by the presence of mass. I also seem to understand that not as many scientists (although probably still the majority) agree that extra dimensions of space exist. But if we agree that spacetime can "curve", aren't we automatically saying that extra dimensions of space exist? I mean, if I grab a stick and I bend it, it becomes curved with respect to our good old 3D space. The stick needs to be in a 3D space, with respect to which it can be straight or curved. So, if the thing that gets curved is not an object in space but space itself, doesn't space need to be in "another" space in order to be curved? I apologize in advance if I used inaccurate terms here, which I most likely did :)


What happens when a charged (negative or positive) object touches an insulator?

I know that insulators do not conduct electricity because they do not allow the free movement of electrons. Let's assume the object that's going to touch the insulator is negatively charged. Does some charge get transferred to the the insulator or does no charge get transferred at all? I want to understand this more conceptually. I don't really get why insulators are used in electrostatic experiments.


Trying to link Eulerian and Lagrangian perturbations

I'm trying to make sense of the link between a (linear) Eulerian (i.e., at a given point) and Lagrangian (following a fluid element) perturbation. I will here express not only where I'm stuck, but also my current understanding of the topic, so I beg that if you find any misconception, please point it out! **>> Background** So far, I (think I) know that in the **Eulerian** perturbation, if $h(\boldsymbol{r},t)$ is the perturbed quantity, where $\boldsymbol{r}=\boldsymbol{r}_0+\boldsymbol{\delta r}$ is the new position vector and $\boldsymbol{\delta r}$ is a small displacement around $\boldsymbol{r}_0$, then $$h(\boldsymbol{r},t)=h_0(\boldsymbol{r})+h'(\boldsymbol{r},t),\ \ \ \ \ \ \ \ (1)$$ where $h_0(\boldsymbol{r})$ is the quantity at an equilibrium state evaluated at $\boldsymbol{r}$ and $h'(\boldsymbol{r},t)$ is a small perturbation. On the other hand, the **Lagrangian** perturbation $\delta h(\boldsymbol{r})$ can be written as $$\delta h(\boldsymbol{r})=h(\boldsymbol{r}_0+\boldsymbol{\delta r})-h_0(\boldsymbol{r}_0)=\left[h(\boldsymbol{r}_0)+\boldsymbol{\delta r}\cdot \nabla h_0(\boldsymbol{r}_0)\right]-h_0(\boldsymbol{r})$$ or, by using the relation found for the Eulerian perturbation, eq. $(1)$, evaluated at $\boldsymbol{r}_0$, $$\delta h(\boldsymbol{r})=h'(\boldsymbol{r}_0)+\boldsymbol{\delta r}\cdot \nabla h_0(\boldsymbol{r}_0).$$ **>> The question** What I have trouble relating is $\delta h(\boldsymbol{r})$ and $h'(\boldsymbol{r},t)$. Both are small perturbations but, in this case, there is a explicit dependance on $\boldsymbol{r}_0$ by $\delta h(\boldsymbol{r})$. However, if I fix $\boldsymbol{r}_0$, both represent a perturbation about that point so...why are they different? My intuition (probably messed up by know; thanks QM) tells me that they are equal, but the notation is different in almost every book I've read on the topic. Can anyone shine some light here?


Is there any physics that cannot be expressed in terms of Lagrange equations?

A lot of physics, such as classical mechanics, General Relativity, Quantum Mechanics etc can be expressed in terms of Lagrangian Mechanics and Hamiltonian Principles. But sometimes I just can't help wonder whether is it ever _possible_ (in the future, maybe) to discover a physical law that can't be expressed in terms of Lagrangian Equations? Or to put it in other words, can we list down for all the physical laws that can be expressed in terms of Lagrangian equations, what are the mathematical characteristics of them( such as it must not contain derivatives higher than 2, all the solutions must be linear etc)?


Why can't massless particle exceed speed of light?

Why massless particle can't exceed speed of light?


What does an electron's wavevector mean inside of a crystal?

With a plane wave, I always took the direction of the wavevector, $k$, as the direction of propogation (magnitude proportional to the inverse wavelength). Alternatively, it could represent the momentum (minus a factor $\hbar$) of a particle. However inside a crystal, the electron wavevector and the electron velocity are not necessarily in the same direction. I'm thinking here of a 2D material with a cylindrical Fermi surface where the momentum may have a z component, but the Fermi velocity does not. In everyday cases you would expect momentum and velocity to be in the same direction, moreover I considered the propogation of the wave to be in the same direction as its particle analogue. I realise that inside a crystal the electrons are no longer simple plane waves, but what then does the $k$ vector mean?


Derivation of cylindrical line heat source problem?

I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is $$\theta(\eta)=C_{1}\int_0^\eta \\! \frac{e^{-u}}{u} \, \mathrm{d}u + C_{2}$$ and my boundary conditions are $-4\pi kh\frac{\partial\theta}{\partial \eta}|_{\eta=0}=q_{0}$ and $\theta\to0$ as $\eta\to\infty$. When I apply the first boundary condition using the derivative I get $1/0$ which is indeterminate so what do I do there? Also, the second boundary condition is supposed to give the exponential integral function, but I am not sure where that comes from. I guess I am stuck on applying the boundary conditions to this. The derivation says the final answer should be $$\theta(\eta)=\frac{-q_{0}}{4\pi h k}\int_\eta^\infty \\! \frac{e^{-u}}{u} \, \mathrm{d}u$$


Calculating the velocity needed to make a distance in the horizontal with a given height and no upward motion

In 1780, in what is now referred to as "Brady's Leap," Captain Sam Brady of the U.S. Continental Army escaped certain death from his enemies by running over the edge of the cliff above Ohio's Cuyahoga River, which is confined at that spot to a gorge. He landed safely on the far side of the river. It was reported that he leapt 22ft across while falling 20ft ![enter image description here](http://i.stack.imgur.com/7MH00.jpg) Now how do I calculate the minimum speed needed to make the "jump"?


How much more efficient is a road bike than a mountain bike?

What would be a good estimate on the difference of efficiency between a road bike and a mountain bike? A number of links cite all the usual reasons: thinner tires, better aerodynamics... But I'm not able to find one where they quantify this efficiency


Simultaneous conservation of linear and angular momentum

Suppose there is a ring in the space where there is no gravity. The width of the ring is $r$ which is negligible compared to its inner radius $R$. The ring is in horizontal position. Now imagine a human starts to move on this ring. Before the start of the motion angular and linear momentum are zero and system is isolated. When the human walks on the ring linear and angular momentum must remain zero. $$m_{human}*R^2*\omega_{human} = -M_{ring}*R^2*\omega_{ring} \rightarrow \omega_{human}=-\frac{M_{ring}}{m_{human}}*\omega_{ring}$$ There is no problem with this part. Now we consider the linear momentum. The linear momentum of the ring after the beginning of motion is zero because for every element of mass there is another element of mass moving in the opposite direction. However, the linear momentum of the human which can be considered a point mass is not zero. The magnitude of the linear momentum of the human is $ m_{human} \cdot \omega_{human} \times R$. So the total linear momentum is not zero despite the fact that the initial linear momentum was zero and the system is isolated. If there is not anything wrong I have to conclude that $\omega_{human} = 0$. What am I missing here?


Dual Pairs in Four Dimensions

Following the conversation here, I am wondering if anyone knows of an example of dual pair with 4-dimensional N=1 SUSY which relates a non-Abelian gauge theory on one side to a theory with a Lagrangian description but no non- trivial gauge group. Cannot think of one off the top of my head, which doesn't mean it does not exist or even is well-known...


How to calculate the kinetic energy of supernova ejecta?

I believed I could use $0.5\times M_{ej} V_{ej}^2$, with $M_{ej}$ being the ejected mass and $V_{ej}$ being the velocity of the ejected mass. But I noticed in this and this that the the mean velocity is somehow calculated as following : $V_{mean}=(3/5)^{0.5}\times V_{ej}$. And so the kinetic energy was calculated as : $(3/5)(0.5)\times M_{ej} V_{mean}^2 =(3/10)\times M_{ej} V_{mean}^2$. So is there any explanation for how this factor of $0.3$ or the mean velocity was calculated ?


Does GR provide a maximum electric field limit?

Does GR provide a limit to the maximum electric field? I've gotten conflicting information regarding this, and am quite confused. I will try to quote exactly when possible so as not to confuse things more with my paraphrasing. The author of the Motion Mountain physics textbook claims in his book there is a limit, and clarifies on his site that ( http://www.motionmountain.net/wiki/index.php?title=Dislike_Page ): > "electromagnetic fields are limited in magnitude. Now, every electromagnetic > field contains energy, and energy density is limited by general relativity: > if energy density is too high, a black hole appears. The smallest possible > black hole then leads to a field limit. If you deny an upper field limit, > you deny general relativity. However, general relativity has been confirmed > in every experiment so far." This sounds very obvious and intuitive to me. However one of my physics TA's got very upset when I used this in a thought experiment when discussing some limits in physics. When I told him the textbook I got that from, he looked it up, and commented on the Motion Mountain wiki website his argument: > "I'd like to add something here. Suppose for a moment the energy density > limit is correct, then if an object is sped up until it length contracts > enough such that the energy density is greater than this limit, does it turn > into a black hole? No, quoting John Baez "The answer is that a black hole > does not form"* > http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html > *The claim that "if energy density is too high, a black hole appears" is an > incorrect oversimplification. The issue is that in GR, gravity depends on > more than just the energy density (component T^00 of the stress energy > tensor). So we can't just look at the "relativistic mass" (E/c^2) to judge > whether a black hole forms. It actually isn't even enough to look at the > "invariant mass", for while the trace T of the stress energy tensor for a > particle is just its invariant mass, for an electromagnetic field it is > identically zero even though electromagnetic fields curve spacetime in GR. > So none of these concepts of mass are sufficient when discussing gravity > using GR (especially when considering electromagnetic fields), because > gravity couples to the entire stress energy tensor. I hope this was > helpful." This makes much less sense to me, and I don't understand how energy density tending to infinity could EVER avoid being a black hole. No offense to my TA, but I'm skeptical as he's disagreeing with a textbook author. Plus, the author's response was that my TA is another Einstein denier, and not worth responding to. So I'd like a third party's answer on this. Does GR provide a theoretical limit to the strength of an electric field? Is it best to just ignore my TA on this one?


Does Heisenberg's uncertainty under time evolution always grow?

Recently there have been some interesting questions on standard QM and especially on uncertainty principle and I enjoyed reviewing these basic concepts. And I came to realize I have an interesting question of my own. I guess the answer should be known but I wasn't able to resolve the problem myself so I hope it's not entirely trivial. So, what do we know about the error of simultaneous measurement under time evolution? More precisely, is it always true that for $t \geq 0$ $$\left<x(t)^2\right>\left<p(t)^2\right> \geq \left<x(0)^2\right>\left<p(0)^2\right>$$ (here argument $(t)$ denotes expectation in evolved state $\psi(t)$, or equivalently for operator in Heisenberg picture). I tried to get general bounds from Schrodinger equation and decomposition into energy eigenstates, etc. but I don't see any way of proving this. I know this statement is true for a free Gaussian wave packet. In this case we obtain equality, in fact (because the packet stays Gaussian and because it minimizes HUP). I believe this is in fact the best we can get and for other distributions we would obtain strict inequality. So, to summarize the questions >   1. Is the statement true? >   2. If so, how does one prove it? And is there an intuitive way to see it > is true? >


Who of these cyclist travel takes less time?

There are two cyclist. They start off with equal velocity $v_0$. The first one bikes a straight path, while the other bikes through a valley, something like this: ![Like this ______/](http://i.stack.imgur.com/pQ8pp.jpg) We assume friction doesn't affect either cyclist, so only gravitational potential energy plays a role here. Who should arrive first? I think its about to find the angles of the second way where the time its a minimun. Someone said I should use the conservation of mechanical energy, but I don't understand how could add the two different velocities (one simply contant, while the other is related to the gravitational potential).


Why are carbon nanotubes are so difficult to create?

I have heard that the future of the techonolgy are the carbon nanotubes due to their resistance and conductivity, but why are they so difficult to create?


Light dispersion in water

Simple question: Water is clearly not disspersive if we have it, for example, in a glass of water. We don't see the rainbow through a glass of water. However drops of water disperse light in the atmosphere and allow us to see the rainbow, am I right? Why is this? In which conditions is water disspersive and in which conditions it's not?.


Deformation due to isostatic compression

When loading a solid material in the shape of a cylinder in isostatic compression, what is the relationship for the change in length versus change in diameter? For example. If I know that when I apply an isostatic load of 6000 psi to a cylindrical core of rock that is initially at a volume of 11,351 cc the deformation will be ~10.8 cc. Knowing the initial length and diameter of my cylinder, how do I predict the new length and diameter?


How exactly to show that s-matrix elements diverges because time-ordering is not well determined?

Let's have s-matrix: $$ S_{\alpha \beta} = \langle \alpha | \hat {S} | \beta \rangle , $$ $$\hat{S} = \hat{T}e^{-i\int \hat{L}(x)d^{4}x}, \quad \hat{T} \left( \hat{\Psi}(t) \hat{\Psi}(t') \right) = \theta (t - t')\hat{\Psi}(t)\hat{\Psi}(t') \pm \theta (t' - t)\hat{\Psi}(t')\hat{\Psi}(t). $$ How to show that matrix elements diverges because time ordering is not well-defined operation when $t=t'$?


What are some approaches to discrete space-time used in modern physics?

This thought gave rise to some new questions in my mind. What are the consequences for:   1. How would it affect duality i.e. particle, wave property of photons?   2. How does this statement affect the information theoretical aspect (entropy) of the universe? **Update:** Given a volume V of space, is the entropy (maximum information that can be store) in this volume changed when this statement is applied?   3. How is a black hole affect by this statement? **Update** How is entropy changed inside the black hole?   4. Could one consequence be that the universe is hologram, since the construction isn’t continues?   5. Would the smallest quantified space be planck's constant? Is there an equivalent constant for time? I hope to get some of your feedbacks regarding this statement.


Does mutual gravitational attraction imply a negative pressure?

Imagine a small ball of gravitating dust particles initially at rest at the center of a large volume $V$. Following John Baez, Einstein's field equations say (in units of $c=8 \pi G=1$): $$\frac{\ddot{V}}{V} = - \frac{1}{2} (\rho + 3 p)$$ Now traditionally dust particles have a pressure $p=0$. However the dust particles have a mutual gravitational attraction for each other. Could this mutual attraction be described within the framework of GR by assigning a negative pressure $p$ to the ball of dust particles? Another way of looking at it is to say that the ball of dust particles has a certain amount of negative gravitational self-energy which counteracts some of the gravitational effects of its positive rest mass energy. Apparently the concept of gravitational energy in GR is complex so maybe negative pressure is a better way to think about it.


Time reversal invariance and statistics

To what extend does the behaviour of time reversal invariance depend on the statistics of the particle under consideration? More explicitly: To what extend does the action of the time reversal operator depend on the commutation relations of say fermions, bosons, etc.? Given a certain Hilbert space (say $\mathbb{C}^N$) with $N>2$ (so we are not talking Spin 1/2). What is the general way to construct a Time reversal symmetry operator? I am looking forward to your response.


the physical properties of leak testing.

Description: To test for leaks a pressure bubble test is performed. This is adding a specified pressure and submerging the test object under water to see if air leaks emerge. The requirement is that no leaks shall be detected when a certain pressure 1 is applied during a predefined time 1. I’m currently looking for a way to speed up this process. When decreasing the time, pressure should logically increase to draw the same conclusions. However what I would like to know is if there is more know about this relationship or/and if it is possible to superimpose this if this relationships is too complicated.


Ski Jumper's vertical velocity after $246.5m$ record?

What would be the vertical velocity of this ski jumper (ski flyer), after he first touches down, after he breaks the record with a $246.5m$ jump? What g force would he experience as he slows down? Video (the last video) I think he gets at least 6 seconds of airtime from $26$ to $32$ second mark. The ramp gradient at take off is listed at $11$ degrees and the take off speed is about $105km/h$. Some statistics] are here. [1] [2]


Covariant Quantisation and the Time Operator in String Theory

Covariant quantisation in string theory is accomplished by giving the commutator relations $[X^\mu(\sigma,\tau),P^\nu(\sigma',\tau)] = i \eta^{\mu\nu} \delta(\sigma - \sigma')$. Although string theory is not a field theory this approach is identical to the quantisation of fields $X^\mu$ in 1+1 dimensions. From this point of view it is also clear how to proceed from here (e.g. getting rid of the negative norm states due to the "wrong" sign of the zeroth component). However, there are two issues that I do not quite understand. Physical states are supposed to have positive energy, i.e. $P^0 \geq 0$. Especially when supersymmetry comes into the game, the algebra so to say dictates positive energies. But when the energy is bounded from below it is not possible to have a time operator obeying the commutator relation $[x^0,p^0] = -i$ where $x$ and $p$ are the above Operators $X$ and $P$ integrated over $\sigma$. For the case of quantum mechanics this has been proved by Pauli in 1933. The prove goes roughly like this: For an operator $H$ which is bounded from below let there be a Hermitian time operator $T$ with $[T,H] = i$ It is possible to shift the spectrum of H because of $e^{-i\epsilon T} H e^{i\epsilon T} = H - \epsilon$ for arbitrary $\epsilon$. That means for an eigenstate $|\psi\rangle$ with $H|\psi\rangle = E|\psi\rangle$ there is also an eigenstate $|\phi\rangle = e^{i\epsilon T}|\psi\rangle$ with $H|\phi\rangle = (E-\epsilon)|\phi\rangle$ For $\epsilon > E$ one has to either conclude that H can not be bounded from below or that T is not a Hermitian operator because it would generate states that lie outside of the Hilbert space. One possible solution could be to allow negative $p^0$ and to apply some kind of Feynman-Stueckelberg approach by reversing the sign of the world-sheet coordinate $\tau$. In this sense the decay of the vacuum into a positive and negative energy state is simply the propagator of a free state. However, this would not be possible in the supersymmetric case where the states have $p^0>0$. The other issue related to defining the zeroth component of the commutator is the construction of the Hilbert space of the physical states. In his book String Theory Polchinski starts with a "vacuum" state $|0;k\rangle$ with the norm $ \langle 0;k|0;k'\rangle \sim \delta^{26}(k-k')$ However, when integrating over $k$ one ends up with a superficial delta function because of the additional mass shell constraint. Polchinski solves the issue by introducing a reduced product $ \langle 0;k||0;k'\rangle \sim \delta^{25}(k-k')$ This seems to be a logical step because selecting the physical states by $L_0|\psi\rangle = \alpha'(p^2+m^2)|\psi\rangle = 0$ yields a Hilbert space whose functions depend on 25 momentum components. Still, the full Hilbert space is not normalisable (this differs from what is usually done in QFT during gauge fixing, where the Hilbert space is constructed via an equivalence relation for the gauge and not by selecting a subspace). The question is whether the Hilbert space constructed in such a way does exist at all. Also it seems as if the number of spacetime operators is reduced by one, thus rendering the steps taken for the covariant quantisation needless. (REMARK: I edited my question to make the point a bit clearer. This is not meant to be a criticism against string theory. It is just that during studying string theory I was confronted by someone with this issue. I just want to understand this detail of the basics because I haven't yet been able to resolve it. I hope somebody can help me.)


Dyson Series Second Order Term Importance vs. Time

Given a time ordered Dyson series expansion of $$H_I=e^{-\frac{i}{\hbar}\sigma_3t}V_0\sin(\omega t)\sigma_1e^{\frac{i}{\hbar}\sigma_3t}$$ $${\cal T}\exp\left[-\frac{i}{\hbar}\int_0^tH_I(t')dt'\right]=I-\frac{i}{\hbar}\int_0^t dt' V_0\sin(\omega t')\sigma_1e^{\frac{2i}{\hbar}\Delta\sigma_3t'}+\ldots.$$ where $\sigma_i$ are the Pauli matrices, what is the condition for the second or higher order terms to become non-negligible? I know it scales like $V_0>V_0^2$ for the first to second order, but what is the role of the time? at very long times, the second term must become more and more important, what is that time scale? (based on this question: Rigorous justification for rotating wave approximation )


3D Quantum harmonic oscillator

For an _isotropic_ 3D QHO in a potential $V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2)$. I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger equation would be of the form $\psi(x,y,z)=f(x)g(y)h(z)$. Explicitly, what would it be? Is it $\psi(x,y,z) = cH_{n_x}H_{n_y}H_{n_z}e^{-{m\omega\over2\hbar}(x^2+y^2+z^2)}$, where $H_{n_i}$ are the $ith$ Hermite polynomial? (A side query, surely since the potential is radial, there is a polar coordinate form of solution which might be better? But this is not asked for in the question. Also, does _isotropic_ just mean that the potential is spherically symmetric?) How many linearly independent states have energy $E=\hbar\omega({3\over2}+N)$? Am I supposed to be counting the number of combinations of $n_x,n_y,n_z$ s.t. $n_x+n_y+n_z = N$? I vaguely remember some notion $(n,l)$ mentioned once, but I can't remember what it is nor find the bit of notes on this. Thanks in advance.


Berry curvature of Landau levels

If we consider an electron on a two dimensional surface with a magnetic field normal to the surface, we know the states the electron can occupy are Landau levels. If we additionally impose periodic boundary conditions, i.e., that these states are eigenstates of the magnetic translation operators, then our surface becomes a torus, and the eigenstates are generalized Bloch states $\psi_{\mathbf{k}}(x,y) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(x,y)$, with $k_i$ taking values in $\frac{2\pi}{L_i}n, 1 < n < N_{\Phi}$, where $L_i$ are the system dimensions, and $N_{\Phi}$ is the number of flux quanta passing through the area $L_x L_y$. For this boundary condition to be meaningful, we further require that the magnetic translation operators in either direction commute, so that $L_x L_y = 2\pi N_{\Phi} \ell^2$ ($\ell$ is the magnetic length). We can write down wavefunctions $\psi_{\mathbf{k}}(x,y)$ that statisfy these boundary conditions, as done, for example, by Yoshioka (http://prb.aps.org/pdf/PRB/v29/i12/p6833_1) and Haldane and Rezayi (http://prb.aps.org/pdf/PRB/v31/i4/p2529_1). To calculate the Berry curvature, one could simply calculate the Berry connection in some gauge, $$ \mathbf{A} = \int d^2x \,\, u_{\mathbf{k}}^*(\mathbf{r}) \nabla_k u_{\mathbf{k}}^*(\mathbf{r}) $$ and then the curvature $$ F_{xy} = \partial_{k_x}A_y - \partial{k_y}Ax $$ We know the Berry curvature for the Landau levels should be constant in $\mathbf{k}$. My question is, how does one do this in practice? It seems that the calculation of the Berry connection (in the Landau gauge $A = -Bx\hat{k_y}$) reduces to calculating the expectation value of the x position $\left<x\right>$. However, while this calculation would be trivial in an infinite plane, with periodic boundary conditions the wavefunctions become much more complicated.


Why do balls in a spinning ellipsoid move to the minor axis plane?

There is a question concerning the Physics of a small child's tall that has been bothering me for some time now. I have investigated this to a small degree, but I have not been able to find a satisfactory answer and I believe the resulting answer must be of an elementary nature. My newborn son has a small plastic toy that is an ellipsoid filled with small, plastic balls (approximately 10-15 balls). The ellipsoid can be revolved around an axis of revolution (the axis of revolution is the major axis of the planar ellipse) and when one spins the ellipsoid, the small, plastic balls find there way (almost immediately) to the plane that is perpendicular to the axis of revolution and contains the minor axis of the planar ellipse where they continue to spin in said plane as long as their speed is sufficiently large. ![Spinning ellipsoid](http://i.stack.imgur.com/popDF.png) In the image, the red curve is a metal rod (elephant's arms) that keep the ellipsoid in suspension, the green line is the axis of revolution, and the blue curve is the curve that the small, plastic, balls move to (almost immediately) and rotate along as long as the speed is sufficiently fast enough. Is there an elementary explanation (elementary = 2 undergraduate courses in Physics) for what I am observing? I am also interested in any explanation of what I am observing that would lead me to learning something new.


What is the relationship between Force and Kinetic Energy?

I am an Middle-School (Grade 8) student, who is self-learning classical mechanics. I saw this equation from Classical Mechanics: $\vec F=m \vec a$, the following equation is pretty straight forward Force equals to mass x acceleration but I as a newbie to this subject am left very confused. As you may see the $a$ in the formula is an vector and so is the $F$. That in mind, it must mean force & acceleration is an direction we apply to an object. That seems extremely familiar to Kinetic Energy as Kinetic energy tends to be applied to an object in an direction, yet its equation is $E = \frac{1} {2}mv^2 $ is not too similar except from it pertaining to do with mass and movement. That in mind, my question is what is the relationship between Kinetic Energy & Force?


What is the energy distribution of light if it has an infinite length? ![Electromagnetic wave](http://i.stack.imgur.com/wgZiU.png) I have read in one of the answers here on phys.SE that light has actually an infinite length. But then what is the energy distribution of that electromagnetic wave?


Does Feynman path integral include discontinuous trajectories?

While reading this derivation of relation of Schrödinger equation to Feynman path integral, I noticed that $q_i$ can differ form $q_{i+1}$ very much, and when the limit of $N\to\infty$ is taken, there remain lots of paths, which are discontinuous (almost) everywhere — i.e. paths consisting of disconnected points. Do I understand this wrongly? How do such discontinuous paths disappear on taking the limit? Or maybe they have zero contribution to integral?


Statics with ladder and wall

> _A uniform ladder of mass m and length L rests against the wall. The > coefficients of static friction between the floor and the ladder and between > the wall and the ladder are equal to each other (μ). What is the maximum > value of angle θ that the ladder can make with the wall without sliding._ My way to solve this: since ladder is not moving Torque on the ladder $$ -\frac{mgL}{2}\sin\theta-\mu N_{f}L\cos\theta+N_{f}L\sin\theta=0$$ Horizontal forces $$ N_{w}-f_{f}=0$$ Vertical forces $$ N_{f}+f_{w}-mg=0$$ where $N_{f}$ and $N_{w}$ is normal forcé due to floor and Wall, $f_{w}$ and $f_{f}$ friction due to Wall and floor. Now I solve for $N_{f}$ the first equation, to get: $$N_{f}=\frac{mg\tan\theta}{2(\tan\theta-\mu)}$$ Sincé the angle is the situation where the ladder is about to start sliding, friction forces will have their máximum value, so: $$f_{f}=\mu N_{f}$$ from 2º equation: $$N_{w}=f_{f}$$ and for same reason like befor $$f_{w}=\mu N_{w}$$ substituting all this values in the 3º equation I get: $$\frac{\tan\theta}{2(\tan\theta-\mu)}(1+\mu^{2})=1$$ and solving this for $\tan\theta$: $$\tan\theta=\frac{2\mu}{1+\mu^{2}}$$ but aparently this is wrong. Could someone check this and point where is the mistake. Ty


Probability of spontaneous Boltzmann brain formation

I was reading through: http://www.scottaaronson.com/papers/giqtm3.pdf But I can't make sense of page 61 discussing Boltzmann brains. Specifically the fact that it says: > But the problem is worse. Since in an eternal universe, you would have > inﬁnitely many Boltzmann-brain doppelgängers, any observer with your > memories and experiences seems inﬁnitely more likely to be a Boltzmann > brain, than to have arisen via the “normal” processes of Darwinian evolution > and so on starting from a Big Bang! I don't understand what this last statement follows from and it seems very unintuitive. A Boltzmann-doppelgänger brain arising by chance seems very improbable since that would require the formation of a brain with a full brain state, i.e. essentially all the individual atoms/molecules appearing in the exact (or at least close enough) configuration. In other words, a very high level of organization. Whereas the Big Bang scenario merely assumes a hydrogen/helium cloud and that the basic laws of nature are in place. This seems to be a much simpler object as a starting point and hence much more probable. This starting point, once it has occurred (maybe due to chance) then evolves (through the relatively simple laws of nature) into the state we have now, and gives rise to processes like natural selection etc. which eventually bring about intelligent life and the very high level of organization in the brain. Evolution through natural selection, while fundamentally also depending on stochastic events, seems much more probable since natural selection guides the process, allowing it to "tune in" on the target state, rather than there being a requirement of the target state just emerging in one step as with the Boltzmann-doppelgänger. Another argument is that the human brain, while seemingly a very complex structure, is probably the result (as part of fetus/child development) from a much simpler process, described in the human DNA. Like how a seemingly complex fractal can emerge from a simple equation. Hence, it seems a much easier way of producing a brain by chance is to produce this simpler mechanism by chance "and let it run" until it forms a brain, rather than demanding that a full brain itself appears by chance as in the Boltzmann case. But given Scott Aaronson is very smart there must be something about this I'm missing. Because I'm reading the statement as saying that the Boltzmann- doppelgänger are more probable than the big bang, but to me it seems the converse is true.


Question on Conformal Field Theory

Since every question has to be asked in a seperate topic, I'm asking a question refering to the following topic: Beginners questions concerning Conformal Field Theory In particula I'm refering to the subsection "Implementing a symmetry on operators" of the answer given by Lubos Motl. It is clear to me that we obtain $\delta\phi=i\epsilon [L_{m},\phi]=i \epsilon z^{m+1} \partial_{z} \phi$ and that $z^{m+1} \partial_{z}$ are the generators of conformal symmetry. What confuses me though is that these generators fulfill the Witt-Algebra and not the Virasoro Algebra. Since we're in the quantum case and we want to show that the theory is invariant on a quantum level, shouldn't we get generators that fulfill the Virasoro Algebra? Thanks in advance for the responses. Best regards.


Give me examples of crackpots who were right after all

I am interested in examples of crackpots coming up with correct results in physics. Why do mainstream physicists look down so much upon "crackpots"?


Fuel and heated iron?

If I have a pile full of fuel (gasoline) - and I also have heated iron ( very very hot - color orange) and I put the Iron in the pile , Would the gasoline will burn ? I mean - does gasoline need heat or actual fire in order to burn ? nb - I can't test it since I don't have the means to do it.


Why doesn't sound infinitely loop?

Let say we have a microphone and a speaker. The microphone will receive the surrounding sound, and the speaker will amplify it. Assume we have unlimited power for the microphone and speaker, so they can work forever. When we speak to the microphone, the sound should be repeated by the speaker. While the speaker generates sound, the sound should input to the microphone, and the microphone should repeat to the speaker. So, the sound should never stop, in a loop. But in reality, we don't have this issue. What's going on? Do my assumptions have something missing? Thanks.


Electric Field Lines between two non parallel plates

**A potential difference is applied between two metal plates that are not parallel. Which diagram shows the electric field between the plates?** ![enter image description here](http://i.stack.imgur.com/lFmHv.png) A is the answer BUT WHY explanation needed !


Constructing Ward identity associated with conserved currents

Consider constructing the Ward identity associated with Lorentz invariance. It is possible to find a 3rd rank tensor $B^{\rho \mu \nu}$ antisymmetric in the first two indices, then the stress-energy tensor can be made symmetric. Once done, the conserved current coming from the classical analysis is of the form $$j^{\mu \nu \rho} = T_B^{\mu \nu}x^{\rho} - T_B^{\mu \rho}x^{\nu}$$ This ensures the symmetry of the conserved current which can be seen most easily be invoking the conservation law $$\partial_{\mu}j^{\mu \nu \rho} = 0 $$ and $$\partial_{\mu}T_B^{\mu \nu} =\partial_{\mu} (T^{\mu \nu}_C + \partial_{\rho}B^{\rho \mu \nu}) = 0.$$ Let $X$ denote a set of $n$ fields. The Ward identity associated with Lorentz invariance is then $$\partial_{\mu} \langle (T^{\mu}x^{\rho} - T^{\mu \rho}x^{\nu})X\rangle = \sum_i \delta(x-x_i)\left[ x^{\nu}_i \partial^{\rho}_i - x^{\rho}_i\partial^{\nu}_i\langle X \rangle - iS^{\nu \rho}_i \langle X \rangle\right].\tag{1}$$ This is then equal to $$\langle (T^{\rho \nu} - T^{\nu \rho})X \rangle = -i\sum_i \delta (x-x_i)S^{\nu \rho}_i\langle X \rangle,$$ which states that the stress tensor is symmetric within correlation functions, except at the position of the other fields of the correlator. My question is: how is this last equation and statement derived? I think the Ward identity associated with translation invariance is used after perhaps splitting (1) up like so: $$\sum_i^n x^{\nu}_i \sum_i^n \delta(x-x_i)\partial^{\rho}_i \langle X \rangle - \sum_i^n x^{\rho}_i \sum_i^n \delta(x-x_i)\partial^{\nu}_i \langle X \rangle - i\sum_i^n\delta(x-x_i)S^{\nu \rho}_i\langle X \rangle $$ and then replacing $$\partial_{\mu}\langle T^{\mu}_{\,\,\,\rho}X \rangle = -\sum_i \delta (x-x_i)\frac{\partial}{\partial x^{\rho}_i} \langle X \rangle$$ for example. The result I am getting is that $$\langle ((\partial_{\mu}T^{\mu \nu})x^{\rho} - (\partial_{\mu}T^{\mu \rho})x^{\nu} + T^{\rho \nu} - T^{\nu \rho})X \rangle = \sum_i x^{\nu}_i \partial_{\mu}\langle T^{\mu \rho}X \rangle + \sum_i x^{\rho}_i \partial_{\mu} \langle T^{\mu \nu} X \rangle - i\sum_i\delta(x-x_i)S^{\nu \rho}_i\langle X \rangle$$ To obtain the required result, this means that e.g$$ \sum_i x^{\nu}_i \partial_{\mu} \langle T^{\mu \rho}X \rangle = \langle(\partial_{\mu}T^{\mu \rho})x^{\nu} X \rangle,$$ but why is this the case? Regarding the statement at the end, do they mean that when the position in space $x$ happens to coincide with one of the points where the field $\Phi_i \in X$ takes on the value $x_i$ (so $x = x_i$) then the r.h.s tends to infinity and the equation is then nonsensical?


Is this theory about Universe information true?

I recently saw this video about information and randomness. At some point, it states that a completely predictable universe would infringe the second law of thermodynamics, because it would imply that the overall information of the Universe would be conserved throughout time, thus leading to a costant entropy Universe. Also, it states that the additionally information that is increasing entropy is coming from QM randomness. I recognize it's a really fascinating theory, but is it true? Is the equivalence between information and entropy sensible?


Action of the Poincare Group on a Scalar Function

Let $F(x^\mu)$ is a scalar function; i.e. $F(x^\mu): \mathbb{R}^{1,3} \rightarrow \mathbb{R}$. How the Poincare Group $P(1,3)$ will act on it; i.e., by which formula I can calculate it for a specific function $F(x^\mu)$? Edit 1: We know that a subgroup of Poincare group $P(1,3)$ leaves the hypersurface $\Sigma: x^+ = 0$ invariant where $x^+ \equiv x^0 + x^3$. To show this a formula is needed.


Feynman diagrams in effective theories

I've been seeing many feynman diagrams lately that I can't quite interpret yet. I've heard a basic Quantum Field Theory lecture and so to me, a Feynman diagram is simply a mnemonic picture to quickly write down (and remember) all of the possible terms in the perturbation series of the matrix element in scattering. But they seem to be much more than that. Consider for example this diagram of proton-neutron scattering via pion exchange. The picture seems to have an intuitive meaning, but how can it be a valid Feynman diagram? Wouldn't that imply that there is an unterlying theory with a proper Lagrangian from which Feynman rules can be derived that would assign a number to this diagram? **Question: When describing nucleon-nucleon interactions in this way, do people actually write down a Lagrangian and derive those rules?** (I am confused because I've seen diagrams like that in several books and they usually write down the cross sections simply from analogy arguments to other theories.) Any clarification will be greatly appreciated, David


Has somebody ever tried to measure the gravity oscillations of nearby rotating masses?

I'd like to extend this question: Are Newton's gravity waves detectable by a laser interferometer? but with some changes. Has somebody ever tried to measure the gravity oscillations nearby (< some meters) of such rotating masses (as in Carl Brannens question)? Say some kgs or grams rotating on a beam with the highest rpm technically possible? (In a vacuum chamber of course!) I dont know what will "radiate" stronger , some grams at 100 000 rpm or some kg at say, 5000 rpm (and some cm distance), therefore, has anybody ever tried something like that in lab? I am aware, that such a experiment is plagued by mechanical (elastic) coupling through air, supporting structures etc, but I think that one can make use of mechanical resonance in the detecting device, helping a lot for selectivity. The fact that the oscillations come from a quadrupole(frequency doubling) , whereas mechanical coupling is not at double frequency, might help to "see" something. As often, its a question of signal/noise ratio :=(


What is the intrinsic colour of the star?

Hey I have this question and I am wrestling with it all day and I am completely lost. The V magnitude of a star is 15.1, B-V =1.6, and absolute magnitude Mv= 1.3 The extinction in the direction of the star in the visual band is av = 1 mag/kpc. What is the intrinsic colour? Can somebody help me? Edit: Ok, apparently I have to calculate first the distance and after that the excess colour and then I can calculate the intristic colour. I guess I need this formula for the distance: V = Mv + 5log(r/10pc) + Av But if i fill this in and get the R I get a different r and I dont know what the next step if to go to the excess colour.


From spectrum/dispersion relation to the partition function

I know the spectrum/dispersion relation for a bosonic system. $$E \left( \mathbf{k} \right) = \cdots$$ Is there a general method for writing down the partition function when the spectrum of the system is known? Thanks in advance!


Putting a spoon in boiling tea brings the froth down. Why?

Two questions really: 1\. Why does putting a spoon in boiling tea bring the froth down? 2\. Also, the tip of spoon (outside tea) never heats up. I cannot imagine that normal air can have so much of a cooling effect. The heat conduction from spoon inside tea should be far more than cooling effect of air. Isnt it?


Spring Force in a Dynamic Equation

I am working on dynamic simulation of a bipedal robot, I have dynamic equations of a preliminary structure. But Now I have to add some springs to the dynamics. I am having a problem of how to account for the forces that will be produced by spring. I have solved the ODE by calculating the force vectors and it runs good, but now I want to solve using Lagrangian equations. The problem is I am able to compute the Kinetic Energy using the Jacobians and generalised variables but I am having the problem accounting the spring Forces in the Potential Energy. ![enter image description here](http://i.stack.imgur.com/elaSl.png) As You can see in the image, My question, How to account for the Force $F$ on $M_1$, $M_2$ and $M_3$, in a Dynamic equation using Lagrange's method.


Measuring radon level

In theory what are the techniques used to measure the levels of radon in air? On which principle/law do they rely?


Why is initial position negative?

> A student on ground throws a ball straight up, the ball leaves students hand > with speed of 15m/s when hand is 2.0mm above ground How long is the ball in > the air before it hits the ground? I was able to figure out the answer to this question using $V_0 = 15\text{ m/s}$, $t_0 = 0\text{ s}$, $Y_0 = 0\text{ m}$, $Y_1 = -2\text{ m}$, $a = -9.8\ \mathrm{m/s^2}$ and using $S_f = S_i + V_i t + \frac{1}{2} a t^2$ I could find $t_1$. My question is, why is $y_1$ negative? I also remember learning that $a$ is always negative, at least in these kind of problems.


Can all laws of physics derived by a single or lists of more general laws?

(This is actually my first question - please understand my lack of knowledge and bad English..) I have always been curious about deriving thousands of laws from more general ones.. Can general laws or models of physics explain minor concepts completely? For instance, can Standard Model explain, well, F=ma?


Is this expression of electric field correct?

I know that the electric field component of electromagnetic wave should be written as the first equation shown, but some times it can be expressed in terms of complex amplitude of electric field as in second equation. I later found integration in the exponent term and I didn't know why this term appeared. Which one is correct, or more general? $$ \mathbf{E}(\mathbf{r},t)=\mathrm{Re}\left[\hat{\mathbf e}F(x,y) A(z,t)\exp(i\beta_0-i\omega_0t)\right], $$ $$ \mathbf{E}=\exp\left(-j\int_0^z\beta_\text{eq}\mathrm dz\right)\mathbf{A}. $$


Why does sound need air in order to travel?

I know that in space, because of vacuum that we can't hear voices or sound to be specific, why is that? Why does sound need a medium to go through?


If 1 screw can support 120 lbs, how much weight can 25 screws support?

While this is mainly about my personal home improvement project, I figure this would be a good opportunity for some of you to apply your physics skills to a real-world situation. My situation is this: I am hanging 500 lbs of drywall (two layers weighing about 250 lbs each) on 5 rows of metal furring channels (hat channels) which are mounted onto RSIC (sound insulation clips) which are screwed into studs with drywall screws. There are 25 total clip assemblies across all 5 rows of channels. This wall is suspended (ie, mounted solely on channels and none of the edges touch the walls, floor, or ceiling). Here's a diagram of a single clip assembly: ![digaram](http://i.stack.imgur.com/7AywV.jpg) **Edit 11/1/2012 4:56AM EDT** Below is a diagram of my actual clip assembly array. The wall is 13' 10" long and 8' tall. The vertical black lines represent the studs and are spaced apart at about 16" (it's not exact because I had to add a couple studs and the frame joins up with another frame at one point in the wall), the red dots are the clip assemblies, and the horizontal grey lines are the hat channels that are mounted onto the clips. Ignore the green dots and yellow arrows. ![clip array](http://i.stack.imgur.com/d8Uwc.jpg) The problem: Since drywall screws are not as strong as wood screws which I should have used, I am worried that the 25 screw/clip assemblies may not be strong enough to bear the weight of 500 lbs. To test the strength of a single screw, I mounted a single clip into a dummy stud and ended up being able to hang 120 lbs of dumbells on it. This mock assembly has been up for over a month and shows no signs of being on the verge of snapping. I could probably hang another 30 lbs on it before it gave away. So, using simple math, it seems I would be able to say, "Well, if one screw can support 120 lbs then 25 screws can support 3,000 lbs!" Of course, I am sure the math is not so simple. I am sure there is a curve there somewhere where adding more screws certainly allows for greater weight capacity, but the weight capacity deteriorates the more screws and more weight you add even if the number of screws and amount of weight was proportionate. So is it really as easy as using simple math to solve this problem or does it require something more advanced? BTW, I just took stabs at tags here. I am hopinq this is an actual physics question and that it will not get deleted. I've already posted a similar question as this on SE DIY site, but I think it's not so much related to handiwork as it isnecessarily


What evidence exists for string theory viability?

I know that string theory is still under heavy development, and as such it still can't make predictions (or not that many predictions anyways). On the other hand, it is clear from the number of years this theory has been under development and from the large number of theoretical physicists studying it, that it is considered a good and viable candidate as a quantum gravity theory. > So, what is the evidence that this is true? Why is it considered such a good > candidate as the correct quantum gravity theory? Without wanting to sound inflammatory in the least, it has been under heavy development for a very long time and it's still not able to make predictions, for example, or still makes outlandish statements (like extra dimensions) that would require a high amount of experimental evidence to be accepted. So - if so many people believe it is the way to go, there have to be good reasons, right? What are they?


Showing that the probability density of a linear harmonic oscillator is periodic

The complete question I am trying to answer is the following: Show that the probability density of a linear harmonic oscillator in an arbitrary superposition state is periodic with period equal to the period of the oscillator. I have: $$\Psi(x,t)=\sum\limits_n c_n \psi_n(x) \exp\left(-\frac{iE_nt}{\hbar}\right)$$ Therefore, the probability density is: $$P(x,t)=\Psi^*(x,t)\Psi(x,t)=\sum\limits_n c_n^2 \psi_n^2(x) $$ (The exponential term should cancel out because of the conjugation). So I get the probability density as being time-independent, which seems wrong, according to how the question is stated. In the solution in the book, we have:$$P(x,t)=\Psi^*(x,t)\Psi(x,t)=\sum\limits_m \sum\limits_n c_m^* c_n\psi_m^*(x)\psi_n(x) \exp\left(\frac{i(E_m-E_n)t}{\hbar}\right)$$ I don't understand why, in the solution, we have the conjugate of the wavefunction have the subscript $m$, while the wavefunction itself has the subscript $n$. It changes the whole thing, when we introduces different subscripts. Moreover, the formula for probability density is $$P(x,t)=\Psi^*(x,t)\Psi(x,t)$$not $$P(x,t)=\Psi_m^*(x,t)\Psi_n(x,t)$$ What is going on here?


What is "surface fluid adhesion energy"?

This is related to my previous question. Pardon me for asking so many questions recently. My physics knowledge is not that good, and some answers are hard to find. In the question in the link, I asked what was the correct interpretation for $\beta_i$ in the energy formulation: $$\mathcal{F}(S_1,S_2,S_3)= \sum_{1\leq i<j\leq 3}\sigma_{ij}P_\Omega(S_i \cap S_j)+\sum_{i=1}^3 \beta_i P_{\partial \Omega}(S_i)+\sum_{i=1}^3 g \rho_i\int_{S_i} z dV$$ I got an answer which said that the correct interpretation for $\beta_i$ is **surface fluid adhesion energy**. I tried googling the term, but I did not find anything clear. My question is > What is **surface fluid adhesion energy** and how is this related to the > fluid-solid interfacial tension? What references are there on this subject? * * * Here is the context: So, in my hypothesis the container $\Omega$ is partitioned in $S_1,S_2,S_3$, the three fluids, with prescribed volumes $v_i$ and prescribed densities $\rho_i$. I took into account in the formulation of the energy the interfacial tensions, the gravity, and the contact of the fluids with the container $\Omega$ (these are not my ideas; they are taken from other similar mathematical articles). I will denote $P_\Omega(S)$ the perimeter of $S$ situated in the interior of $\Omega$ and $P_{\partial \Omega}(S)$ the perimeter of $S$ which is situated on the boundary of $\Omega$. I will not be very rigorous in what I'm about to write: I will write, for example $P_\Omega(S_i\cap S_j)$ the perimeter of the intersection $S_i\cap S_j$ even if as set theory intersection, this is void. Still, I think that the idea will be clear. * * *


Unconstrained action in Yang-Mills theory

Is there any gauge in which Yang-Mills theory (4d, non-SUSY) can be written as a local action containing only the propagating modes?


In turbulence theory, what happens if i take space average of fluctuating part?

According to Reynolds decomposition, velocity field is split into two time average and fluctuating parts: $$u_{({\bf x},t)}=\overline u_{(\bf x)}+u'_{({\bf x},t)}$$ we know that time average of fluctuating part is zero $\overline u'=0$, but what if i take space average of fluctuating part?


Casimir effect when the zeta function has a pole

let us suppose the Casimir force/ Vaccumm energy of a certain system $$ E_{casimir}= \sum _{n} \omega _{n} $$ which is formally equal to $ E_{casimir}= \zeta _{spec}(-1) $ with $ \zeta (s)_{spectral}=\sum_{n}\omega _{n} ^{-s}$ however what would happen if the spectral function has a pole at $ s=1 $ what would be then the value of $ E_{casimir} $ can this pole problem be overcome by regularization ?? perhaps in this case we could take the finite part $ FP \zeta (-1)_{spec}= \sum_{n}\omega_{n}-\int_{0}^{\infty}\omega (x)dn(x)$ the secon integral would be the SEMICLASSICAL approximation ot the frecuencies energies $ E_{n}= \hbar \omega _{n} $


What is the rationale behind representing a state function by a complex valued function in QM?

What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a real valued which directly gives the probability. My question is why should we use complex valued state functions? There must be some strong non-mathematical reason which I am not able to find in books I have referred?


Error propagation estimations for sine and cosine

My lab manual gives this: $B$ is a function of $A$, Greek are uncertainties... $$B + \beta = \sin(A + \alpha) = \sin(A)\cdot\cos(\alpha) + \sin(\alpha)\cdot\cos(A)$$ \--> because $\alpha$ is taken to be (at least relatively) small, $\cos(\alpha) \to 1$, and $\sin(\alpha) \to \alpha$, measured in radians. I see that $\cos(\alpha) \to 1$, but I would have expected that $\sin(\alpha)$ for a small would, by the same logic, go to 0. Has it got something to do with the rate of change of either function near zero? Why is $\cos(\alpha)$ of small $\alpha$ not also proportional or written by relation to $\alpha$?


Cassini 's and Pioneers RTG power output datafile

I'd like to have access to the datafile of the electrical output of the RTG (Radioisotope Thermoelectric Generator) of the Pioneer's and Cassini missions, as can be seen in the figures 1 of the papers of Anderson,2002 and Peter S. Cooper, publ 2009.   I know that I can grab the aprox values directly from the published graphs with the help of software, but accessing the original data is better. I g-searched but I did not found them. Can someone give advice, pls? The question is specific but the answers can say more about the difficulties and solutions of accessing scientific data.


Stability of square of masses on strings under rotation

Imagine we have a square of masses, $m$, connected by light inextensible strings, length $l$, rotating around it's centre at angular speed, $\omega$. It's easy enough to show that there must be a tension in these strings, $T=\frac{ml\omega^2}{2}$. However if we then increase the mass of any two opposite corners resolving forces seems to show that there is no solution for continued circular motion except when the two heavier masses are a distance $l$ from the c.o.m and the lighter ones, a distance 0. I.e. the problem reduces to a line. Is this true, or has my logic just broken down at some point? Also, the same seems to happen if you keep all the masses equal and change any angle just a little. Does this mean that a rotating square (or any other regular shape) of equal masses such as this is the only equilibrium position for the family of systems of shapes and masses such as this? Here's a diagram if anyone's having trouble picturing it, just a random one from the net that had most of the right features (ignore the labels), it is colourful though... ![](http://i.stack.imgur.com/20dec.png)


Ideal gas system with diathermic piston externally manipulated

I wanted to ask about the following problem: An ideal monatomic gas is separated into two volumes $V_{1}$ and $V_{2}$ through a diathermic piston, such that each volume containing $N$ atoms and the two sides are at the same temperature $T_{0}$. The entire system is isolated from the outside by means of insulating walls. The plunger is reversibly externally manipulated until the two gases are in thermodynamic equilibrium with each other. After the problem has different questions, however, I have not been able to answer the following: How is called the type of process? Ie is isothermal, adiabatic, etc. Show that $\Delta S_{1}$ = $-\Delta S_{2}$, where $\Delta S_{1}$ and $\Delta S_{2}$ are the changes in entropy of the two gases.


Electricity from lightning

1. According to the internet, a lightning strike contains about 5 billion joules or 5 GJ. How was this calculated?    2. Another thing: Once the lightning strike a metal grounded rod, a current will flow. What type of that current and how much current? I think it will be a 325 MV current which is the voltage of the lightning. Is that correct? Edit: after i read a little bit about lightning , i found it was such a big scale , so how about a small electrostatic charge , say from a 1 MV van de graaff generator, i have a spark source ( my CRT TV :D ) and i will buy a multimeter and conduct some experiments , but i need to know what to expect


Wave functions for 2D potential with spin interactions

So consider a 2D system with a circular potential and a spin-orbit interaction: $V(r) = V_0 \theta(r_0 - r) + c r_0 V_0 L_z S_z \delta(r-r_0)$ where $\theta$ is step function. So the operators $L_z$ and $S_z$ commute with the Hamiltonian are are therefore conserved quantities. For the same reasons we can write the wave function as a product of radial and orbital parts (and spin parts too): $R(r) e^{i l \theta}$ where $\theta$ here is the polar angle and $l$ is the orbital quantum number. A spinor can be affixed to the wave function but seems unnecessary as no transitions can occur for the spin. My question regards adding another spin interaction to $V(r)$ of the type $b_z S_z\theta(r_0-r)$ that only acts within the circular potential. Will the form of the wave functions change as a result of this addition? My thought is that the wave functions remain the same since once again spin should be preserved so the spinors will not see any further structure. The only purpose of this new interaction will be to alter the effective potential of the circular well - the potential will be shifted either up or down depending on the direction of the spin ($m_s$ = up or down). So is my reasoning correct? I understand that this problem becomes much more difficult in 3D when the full spin-orbit interaction is used since then you will have a lack of commutation.


What happens to proteins under x ray/gamma ray radiation?

A simple question to which I am yet to find an answer. I have not found any papers which would go into detail about it, there is some stuff on protein damage in x-ray crystallography but they are looking at the crystallized proteins, I am interested in "natural" proteins...


Higher dimensions

1) How we determine whether the higher dimensions are Unstable or Unpredictable? Or on the basis of what assumption we make this prediction? ![Dimensions](http://i.stack.imgur.com/0jX6u.gif) (Source of Image: Max Tegmark. See also this Wikipedia article.)


If two ends were a certain "length" apart were they therefore at rest (or at least rigid) to each other?

Considering the definition of the SI unit of "length" [1] and [2 (" method a.")] I'm missing any requirements about the two "ends" of the required " _path travelled by light_ " being "at rest to each other", or at least "rigid to each other". Are such requirements perhaps presumed to be understood implicitly? Accordingly, if some particular pair of "ends", $A$ and $B$, in some particular trial, were characterized as having been a certain "lenght of $x \, \text{m}$ apart from each other", where "$x$" is some particular positive real number and "$\text{m}$" denotes the SI base unit "metre", is it then understood:   * that ends $A$ and $B$ had observed exchanging signal pings between each other; and not only once for each trial, but for any of their signal indications throughout a sufficiently extended trial?,   * that for any two (distinct) signal indications $A_J$ and $A_K$ of end $A$ during this trial the corresponding ping durations of end $A$ were equal to each other:   $\tau_A[ \,_J, \,_{\circledR}^{B \circledR AJ} ] = \tau_A[ \,_K, \,_{\circledR}^{B \circledR AK} ]$ ?,   * that for any two (distinct) signal indications $B_P$ and $B_Q$ of end $B$ during this trial the corresponding ping durations of end $B$ were equal to each other:   $\tau_B[ \,_P, \,_{\circledR}^{A \circledR BP} ] = \tau_B[ \,_Q, \,_{\circledR}^{A \circledR BQ} ]$ ?, and   * that for any signal indications $A_J$ of end $A$ and any signal indication $B_P$ of end $B$ during this trial the corresponding ping durations were equal to each other:   $\tau_A[ \,_J, \,_{\circledR}^{B \circledR AJ} ] = \tau_B[ \,_P, \,_{\circledR}^{A \circledR BP} ] = \frac{2 x}{c} \text{m} $ ?. References:   [1] SI brochure (8th edition, 2006), Section 2.1.1.1; http://www.bipm.org/en/si/base_units/metre.html (" _The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second._ "). [2] "the mise en pratique of the definition of the metre"; http://www.bipm.org/en/publications/mep.html


Lawrence Krauss' Theory of Origin

I don't understand how a vacuum, the absence of matter, can hold energy. How can it hold energy when Einstein proved that matter is energy? And a second related question; how does the energy in a vacuum allow the Universe to produce itself? And how does that account for why the Universe is expanding? I saw Krauss' lecture on YouTube called Universe from Nothing and only understood some of it, even though he was fantastic at putting things in layman's terms and making these concepts easy to visualize. However, he had a bunch of "asides", so it was a bit confusing. He mentioned that the curvature of the Universe was flat, but then also went on to say how that was the worst possible scenario. Why is a flat Universe such a bad thing?


Rotational Constant and Moment of Inertia of Fluorine gas

I have come across some homework question on thermodynamics which needs me to calculate $B$ of $F_2$ My attempt: $B= \frac{h}{8\pi^2cI}$ where $I=\mu r^2=\frac{m_1m_2}{m_1+m_2} r^2$ Atomic mass of fluorine atom is $18.998$ $g$ ${mol}^{-1}$ and Bond length is $141.8\times10^{-12}$ $B=\frac{6.626\times 10^{-34}}{8 \pi^2 (3\times10^{10})\frac{\frac{18.998^2}{18.998\times2}}{6.02\times 10^{23} \times 1000}(141.8\times10^{-12})^2}\approx0.882$ which I think it is wrong because when I continue to calculate the rotational temperature $\Theta_R$, the result is way too low. $\Theta_R=\frac{hcB}{k_B}=\frac{(6.63\times10^-34)(3\times10^{10})(0.882)}{1.38\times10^{-23}}=1.27 K$, $K$ for kelvin. Please help me find the mistakes in the calculations...


Electromagnetic wave propagation through two lossless dielectrics

In Elements of Electromagnetics (Sadiku, 3rd edition, Section 10.8), the author says to consider two lossless dielectric materials joined at an interface $z=0$. Here two lossless dielectric materials can be translated to mean that the conductivity of medium 1 (located to the left of $z=0$, i.e. for $z<0$) is equal to the conductivity of medium 2 (located to the right of $z=0$, i.e. for $z>0$) which is equal to 0 Siemens/meter. Recall that intrinsic impedance is given by $$\eta = \sqrt{\frac{j\omega\mu}{\sigma+j\omega\varepsilon}}$$So if $\sigma=0$, then $$\eta = \sqrt{\frac{j\omega\mu}{j\omega\varepsilon}} = \sqrt{\frac{\mu}{\varepsilon}}$$This means that the intrinsic impedances $\eta_1$ and $\eta_2$ are real numbers (with units) since $\sigma_1=\sigma_2=0$ S/m and $\mu$ and $\varepsilon$ are taken to be just real constants (with units), i.e. they do not change with space or time, except at $z=0$ where there is a transition from one lossless dielectric to another of course. Also note that the reflection coefficient $\Gamma$ is given by $$\Gamma = \frac{E_{r0}}{E_{i0}} = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$$ where $E_{r0}$ is the amplitude of the reflected wave at $z=0$ (the interface) and $E_{i0}$ is the amplitude of the reflected wave at $z=0$ (the interface). The author considers the case where $\eta_2 > \eta_1$ (meaning $\Gamma > 0$). A normally incident electromagnetic wave (normal to the interface) reaches the interface and sends some EM energy through to medium 2 in the form of a transmitted electromagnetic wave and reflects some EM energy in the form of a reflected electromagnetic wave. The incident and reflected waves have different amplitudes. The electric field in medium 1 is the sum of the incident and the reflected waves (described mathematically below) $$ \vec{E}_{1} = \text{Re}\left[\underbrace{\left[\vec{E}_{i0}e^{-\gamma_1z} + \vec{E}_{r0}e^{\gamma_1 z}\right]}_{\text{Spatial variation}}\underbrace{\left[e^{j\omega t}\right]}_{\text{Temporal variation}}\right]$$ Here, $\gamma_1$, is the propagation constant (most generally a complex number) in medium 1 given by $$ \gamma = \underbrace{\alpha}_{\omega\sqrt{\frac{\mu\varepsilon}{2}\left[\sqrt{1+\left[\frac{\sigma}{\omega\varepsilon}\right]^2}-1\right]}} + j\underbrace{\beta}_{\omega\sqrt{\frac{\mu\varepsilon}{2}\left[\sqrt{1+\left[\frac{\sigma}{\omega\varepsilon}\right]^2}+1\right]}} $$ But since $\sigma_1=0$ S/m, $\gamma_1=j\beta=j\left[\omega\sqrt{\mu\varepsilon}\right]$. Since $e^{j\theta}=\cos\theta+j\sin\theta$, taking the real part of a complex exponential is a compact way of writing a sinusoidal wave, which is very relevant here as there are three sinusoidal waves involved here -- the incident wave representing the incident electromagnetic wave's electric field, the reflected wave representing the reflected electromagnetic wave's electric field, and the transmitted wave representing the transmitted electromagnetic wave's electric field. With this background, here is what I do not understand. **Problem** : The author claims, just before Eqs. 10.88 and 10.89, that the amplitude of $\vec{E}_{1}$, $\left|\vec{E}_{1}\right|$ is maximum for when $$-\beta_1 z_{\text{max}} = n\pi$$Further, the author claims the amplitude of $\vec{E}_{1}$ is minimum for when $$-\beta_1 z_{\text{min}} = \left(2n+1\right)\frac{\pi}{2}$$I cannot seem to replicate this result. Here is my attempt at a proof. **Partial proof** : Assuming the electric field oscillates in the $\mathbf{a}_x$ direction (alternately called $\mathbf{e}_x$ or $\hat{x}$ direction) and $E_{r0}=\left|\vec{E}_{r0}\right|$, $E_{i0}=\left|\vec{E}_{i0}\right|$ ![Proof up to phasor envelope representation](http://i.stack.imgur.com/5aUGr.png) To determine the magnitude of this, it may be helpful to consider the geometry of this situation by using a phasor diagram. The phasor diagram would plot two vectors (phasors) – one of magnitude $E_{i0}$ and the other of magnitude $\Gamma E_{i0}$ (remember $\Gamma>0$ in this case but $0\le\left|\Gamma\right|\le 1$ holds in general). If the reference phasor is $\cos\left(\beta_1 z\right)$, then $\cos\left(\beta_1 z-\omega t\right)$ will be located a certain distance in the clockwise direction away from the reference and $\cos\left(\beta_1 z\right)$ will be located the same distance in the counterclockwise direction. The resulting vector (or phasor) sum of the two will have a magnitude of $\sqrt{E_{i0}^2 + \left(\Gamma E_{i0}\right)^2}$. I am trying to figure out the phase angle for this resulting vector. I am currently stuck here, hoping someone can help me out. Thanks in advance! EDIT: Here's the picture I drew.![Phasor diagram](http://i.stack.imgur.com/PN8Ox.png) It seems like this all simplifies to finding $\phi$ using the law of sines / law of cosines. The goal is to see for what values of $\phi$ will $\left|\vec{E}_1\right|$ be maximum. Hopefully, it'll end up validating what the book says.


Ultrarelativistic limit of helicity values

Let's have helicity operator: $$ \hat {h} = \frac{(\hat {\mathbf S} \cdot \hat {\mathbf p})}{|\mathbf p|}. $$ For massive case spin $s$ representation it has $2s + 1$ eigenvalues $s, s - 1,..., -s$, for massless one it has one or two eigenvalues $\pm s$. How to show that ultrarelativistic limit of massive case gives massless one? I mean that effect like relativistic aberration leads us to the result that in ultrarelativistic case we will see only two possible values of projection of spin on motion direction.


Schroedinger equation. Mass. Charge

The mass of a particle appears in Schroedinger equation but it does not appear its charge, although both terms have their effect on movement.Why?


What is a geometrical object?

From the Wikipedia link for Geometry: > Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is > a branch of mathematics concerned with questions of shape, size, relative > position of figures, and the properties of space. Up to the work of Riemann and Gauss, this definition would have made clear to me examples of geometrical objects: a line, square, cube, hypercube; each of these possessing geometrical properties such as the number of sides, angles between faces, the dimension of space that contains them etc. Hence a geometrical object was a set of measurements associated with an object using a ruler. After the work of Einstein and Minkowski who showed that time and space were a part of one another, would it be correct to say:   1. a geometrical object is a set of measurements associated with an object of distance _and time_ using a ruler and a clock ?   2. Geometrical objects includes an interval of time, the invariant space-time interval?


Energy of an EM wave compared to energy of a photon

Several posts on this forum ask the question about the role of amplitude in calculating the energy of an Em wave. This struck me as odd since I learned that E=hv. There is no amplitude in the Planck relation. But I see why this might arise as a question since every other description of waves includes the amplitude in energy calculation. This posts on this forum highlight the question: The energy of an electromagnetic wave Energy in an EM wave should depend on frequency Where is the amplitude of electromagnetic waves in the equation of energy of e/m waves? In one post I read that 'The macroscopic electric field of a wave consisting of photons does have an amplitude which is statistically built up from the individual photons.' My interpretation of the answers in these posts is that the photon is a fundamental unit with a fixed amplitude. In the same way that the speed of light is a fixed velocity, it appears that the amplitude of a photon is also fixed. So this leads to the question of what is the amplitude of a photon Amplitude of an electromagnetic wave containing a single photon I cant seem to determine whether the question was actually answered in that post. But why is it not valid to just use the fact that in a wave, energy is proportional to amplitude squared Why is energy in a wave proportional to amplitude squared E=hv E~A^2 so the amplitude of a photon should be a~(hv)^1/2


Black magic "Hartree" approximation

The question is about an unusual looking version of the Hartree or mean field approximation. The context is several papers I've been reading recently about the out of equilibrium dynamics of phase transitions in the early universe [1-3]. The procedure is to shift a (scalar) field $\Psi \to \psi + \langle \Psi \rangle$, where $\psi$ are fluctuations and $\langle \Psi \rangle$ is a background field. This is perfectly fine. Then you simplify the dynamics of the fluctuations by replacing cubic and quartic terms by quadratic terms like so: $$ \psi^4 \to 6 \langle \psi^2 \rangle \psi^2 - 8 \langle \psi \rangle^3 \psi + 6 \langle \psi \rangle^4 - 3 \langle \psi^2 \rangle^2, $$ $$ \psi^3 \to 3 \langle \psi^2 \rangle \psi - 2 \langle \psi \rangle^3. $$ See footnote 11 of [3] for these full formulas; the other refs simplify these by using $\langle \psi \rangle = 0$. Also note this is supposed to work for _bosonic_ fields. In the papers these substitutions just come out of the blue. I get that the idea is to make the problem solvable by reducing the order of the interaction terms (and then eventually determining a self-consistent $\langle\Psi\rangle$), but I'm very confused about the numerical coefficients. Where do they come from? Is there a systematic way to derive them, or consistency condition they must satisfy? Why the alternating signs? What is the relation of this "Hartree" approximation to _the_ Hartree approximation which has to do with minimizing the energy for separable wavefunctions or summing up a particular class of Feynman diagrams? I've tried the seemingly likely stories:   * It looks at first like maybe something to do with Wick contractions, but the signs are a problem even if all the numbers came out the right size - which they don't.   * Repeatedly applying the mean field theory rule $AB \to A\langle B\rangle + \langle A\rangle B - \langle A\rangle\langle B\rangle$ doesn't seem to work out either.   * Neither does writing $\psi^4 = (\psi - \langle\psi\rangle)^4 + \cdots$ and crossing off the $(\psi - \langle\psi\rangle)^4$ term, or same thing for $\psi^4 = (\psi^2 - \langle\psi^2\rangle)^2 + \cdots$   * It looks similar to the cumulant expansion but I can't get it out of that either.   * The online encyclopedia of integer sequences was completely useless! A literature pointer would be acceptable if the derivation is long, but please give me an article that _explains_ the result! So far I've only found things which _quote_ it and act as if it's the most obvious thing in the world. Sorry if it is... I feel like I'm missing something very basic. :)   1. Boyanovsky, D., Cormier, D., de Vega, H., & Holman, R. (1997). Out of equilibrium dynamics of an inflationary phase transition. Physical Review D, 55(6), 3373–3388. doi:10.1103/PhysRevD.55.3373   2. Boyanovsky, D., & Holman, R. (1994). Nonequilibrium evolution of scalar fields in FRW cosmologies. Physical Review D, 49(6), 2769–2785. doi:10.1103/PhysRevD.49.2769   3. Chang, S.-J. (1975). Quantum fluctuations in a φ^{4} field theory. I. Stability of the vacuum. Physical Review D, 12(4), 1071–1088. doi:10.1103/PhysRevD.12.1071


Infinity of running couplings

A Landau pole \- an infinity occurring in the running of coupling constants in QFT is a known phenomena. How does the Landau pole energy scale behave if we increase the order of our calculation, (more loops) especially in the case of Higgs quadrilinear coupling?


Determing time to complete known distance with constant acceleration

**Answer: $a=v^2/d$ is the formula I needed.** This is a problem in a programming assignment, I haven't taken physics. I have a starting speed (0 m/s), an final speed (208.33m/s) and the distance it took to reach that speed (200km). I need to get the time it will take to travel any arbitrary distance lower than 200km. From what I remember in highschool, I used basic calculus to get $a/2(t^2)=distance$ or for what I have $t=\sqrt(400000(m/s)/a)$. what I don't know is the acceleration, usually to calculate acceleration I'd need time. I'm not sure where to go from here.


Projectile motion of the water surface from water discharging from a pipe

I am looking at pipe flow into a pit. How do I calculate the outside water surface path. I understand how to calculate projectile motion from calculus in terms of a velocity, point object and projectile angle however I do not know how to add thickness and calculate the outside path. We can ignore drag force unless someone can provide suggested values based on field or lab validation. Also I can accept that it will be hard to calculate the expansion of water discharging from the pipe into the pit so unless someone knows suggested values from field or lab validation then this can also be ignored.


If the moon was twice as big but twice as far away, would there be any difference?

I'm just going to go ahead and steal this question question directly of Reddit since I have more trust in the answers I get on this site. So, if the moon was twice as big but also twice as far way, would there be any difference in the effects that the moon now has on earth, like tides and such? Also, would there be any difference in the moons orbital motion apart from the fact that the time it takes for the moon to rotate around earth would be longer? What will increase is the mass and diameter(by a factor of 2) of the moon. The density of the moon will stay the same.


What is the smallest particle exhibiting gravitational properties?

I've long been taught that all matter having mass, possesses attractive forces somewhat akin to gravity. As such, imagine we can 'teleport' a gravitonic detection device that can accurately measure the gravitational forces regardless of strength present at each location, it can detect all gravitational forces ordered by direction of pull within a given radius of up to let say [through a range of settings/switches of 0.001 nanometer to 100,000 km. We teleport these detectors equi-distantly apart in the earth, from troposphere to molten mantle to core's center of mass. measure and plot all the forces detected. Supposedly, there is no gravity at earths center of mass...that means gravity can be concentrated and nullified. Logic tends to indicate 'no gravity=no weight. Therefore why does all reference material say massive objects have tremendously large core pressures and squeeze atoms to extremely dense masses. If gravity is nullified, what is source of this crushing, without usurping all gravitational common sense?


Lamination on a conductor?

How can lamination help reduce eddy currents? I understood that it reduces them, by making their circulation incomplete, or something like that. Can anyone explain this please?


Helium plasma in space and its properties

It is said, that, "Most of the Helium in the Universe, is in a plasma state". Plasma's are now talked of, as the forth state of matter, but this does not seem to be a majority opinion. Plasma's are also normally regarded, to be at a high temperature, I believe because they are formed from elements that would normally be solids at Earth temperatures. Since the mass of Helium in the Universe is > 20% of the total mass, and the average temperature (CMB), of the Universe, is 2.7K, there must be a lot of Helium Plasma in the Universe at, or below this temperature. My questions are:   1. 2.7K is very close to the Lambada point of atomic Helium. Does Helium plasma, still have the same Lambada properties as atomic Helium? If so, will Helium plasma, below it Lambada point, be both superfluid and supercondutive?   2. Is there a point, that Helium plasma reverts to atomic Helium and what causes it to revert? I assume there is a point, since some of the small amount of Helium we still have on Earth, trapped in pockets, and which dates from the Earth's birth, would be of the plasma verity, because it origins, are in the Universe.   3. The majority of the Helium in the Universe, is said to be Helium-4. I presume, this is Helium-4 plasma. Apart from the properties mentioned in question 1, are there any other major differences between Helium-4 plasma and atomic Helium-4. If the answer to this question is yes, and the answer to questions 1 & 2 is no, then the differences between the two states, seems so great, that maybe, there is a case for the plasma, to have a different classification, and a different name, even though, their atomic structure, is very similar.


Time-reversal symmetry

For a quantum system with time-reversal symmetry, other than the absence of a magnetic field, can we infer anything else about the system?


Proof that the Earth rotates?

What is the proof, without leaving the Earth, and involving only basic physics, that the earth rotates around its axis? By basic physics I mean the physics that the early physicists must've used to deduce that it rotates, not relativity.


RC circuit theory and voltage in the capacitor vs Ohm's Law

Consider a simple series RC circuit at steady state (capacitor is full). I've been told that once the capacitor is full we can literally "cut" the circuit because no current can flow. That argument makes sense, but because the capacitor is full, doesn't that also mean it has a potential difference across it? In other words, the capacitor is a new battery? Then we should apply Kirchhoff Law in the circuit if we want to find the current? (even though I just said it was zero). What is going on? What is this contradiction? There is no current in the circuit yet clearly the potential difference in the capacitor is NOT zero. **EDIT** ![enter image description here](http://i.stack.imgur.com/ICDhG.jpg) For part (c), why do I have to use $I = I_0 e^{-t/ \tau}$? Clearly at steady state, the current of the right circuit is just the potential difference across the capacitor (which now can be treated as a battery) and the resistance. As pointed out before I can only apply Ohm's Law to resistors and batteries, but at steady state the capacitor _can_ be thought of as a battery. So the contradiction lies here. On the left circuit the current is simply $10V = I_1(50k \Omega)$, and on the right it should be $10V = I_2(100k \Omega)$ **only because this is steady state** **EDIT2** Let me clarify my question again. I am also confused why $I = I_0 e^{-t/ \tau}$ is used? Why does the capacitor discharge? Yes it's considered an isolated circuit once the switch is shut, but does that mean even though part of the circuit is still hooked up to the battery, that the current doesn't continue to build to prevent discharge?


Could the Schrödinger equation be nonlinear?

Is there any specific reasons why so few consider the possibility that there might be something underlying the Schrödinger equation which is nonlinear? For instance, can't quantum gravity (QG) be nonlinear like general relativity (GR)?


Mars Curiosity Power System

Curiosity's power source comes from a radioisotope thermoelectric generator (RTG) rather than from solar panels like the two Mars Rovers. 2,000 watts of thermal power from alpha particle emission in 4.8 kg of Pu-238 (half-life 87.8 years) yields about 120 watts of electric power via the Seebeck effect (thermocouples). This source slowly declines in power output (~1%/year), and is expected to provide a reliable power source for years. > It appears that the efficiency of Curiosity's RTG is only 120/2000 = 6%. Why > is the RTG efficiency so small? Is there a problem maintaining greater > hot/cold junction delta temperatures or are there other issues?


Mass vs. weight on a spring-loaded bathroom scale

You stand on a spring-loaded bathroom scale in a bathroom. The scale "reads" your mass. What is the scale actually _measuring_? Similarly, you stand on a spring-loaded bathroom scale in an elevator that is accelerating upward at 2.0 m/s^2. The scale "reads" your mass. What is the scale _measuring_? A) Your mass B) Your weight C) The force of the scale pushing up on your feet D) The force of your feet pushing down on the scale To me, C seems correct for both of them, but B also seems correct for the first part, as weight is equal to the force of the scale on your feet for that part. Can anyone corroborate or deny my assumptions?


Opening the fridge door to cool a room

I'm well aware that the default answer to this textbook default question is "it doesn't work", but still, I believe it does. To cool the insides of the fridge, the compressor must do work, and since the efficiency isn't 100% you are constantly warming the whole room to cool it's insides, the winning move here is simply turning the fridge off. However, let's suppose the fridge must stay on, wouldn't it be better to open the door? In other words: Isn't opened fridge turned on better than closed fridge turned on for the whole room temperature?


Question about units in Lagrangian dynamics (inertia matrix)

I have a 3 degree of freedom system and my equation of motion is like this: $$M(q)q_{dd} + C(q,q_d)q_d+G(q)~=~0$$ $M(q)$: inertia matrix $C(q,q_d)$: Coriolis-centrifugal matrix $G(q)$: potential matrix. where $q_d$ is the first derivative of q etc and q is a vector of my variables. $q=[θ,γ,a]^T$ "θ" and "γ" are angles in (rad) and "a" is length in (m). Now my question is this: I have 3 equations with 3 unknowns but 2 variables are in terms of angles and 1 in terms of distance, so the elements of my inertia matrix don't have the same units. Is that wrong? I mean each equation is consistent in units. The first 2 have units of kg(m/s)^2 while the 3rd has units of kg(m/s^2). I can make the whole thing dimensionless but I read somewhere that the inertia matrix should be symmetric(which it is symmetric at the moment). If I make my equation of motion dimensionless then inertia matrix won't be symmetric anymore. What shall I do? Any suggestions are welcome


Effect of CO2 on the environment, besides greenhouse effect and ocean acidification

What effects does the raising level of CO2 in the atmosphere have, besides being a greenhouse gas and responsible for the increased acidity of the oceans? What other environmental damages does it cause?


The Gluon - Does It Exhibit Wave Properties?

Do Gluons have frequencies and wavelengths? I assume that they do, but have been unable to find anything on point in SE or Wikipedia. Just beginning to study university-level physics here.


Stress-energy tensor explicitly in terms of the metric tensor

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $R_{\mu\nu}$ and scalar curvature $R$ are replaced with an explicit expression involving the metric tensor $g_{\mu\nu}$. Using the equations $R=g^{\mu\nu}R_{\mu\nu}$ (relating the scalar curvature to the trace of the Ricci curvature tensor) and $R_{\mu\nu}=R^\lambda_{\mu\lambda\nu}$ (relating the Ricci curvature tensor to the trace of the Riemann curvature tensor), would anyone be willing to give recommendations on how to proceed, or already know the equation?


Maxwell's Theory Of Wave Propagation

What is Maxwell's theory of Wave propagation and what is its physical interpretation?


Why water from upper level not goes into lower level in this drainage system

In the following typical water drainage system, I'd like to know why waste water from upper floor (let say, from 2nd floor's toilet) does not come to the lower floor's equipment via the common pipes (the 1st floor's sink), according to the equal level principle. In what case would this happen? If the water tends to go through the main pipes but not the branches, is there chance for it to go to the sink? ![drainage system](http://i.stack.imgur.com/QqKcF.jpg)


Unbalanced wheatstone bridge

Suppose I have an unbalanced Wheatstone bridge (resistances $R_1, R_2, R_3, R_4, R_g$), to which I apply some voltage $V$. I need to find the voltage across the the galvanometer. Solving the system of equations obtained by Kirchoff's Law would yield the answer, but it will definitely be enormous. I recall seeing somewhere it could be written as the determinant of a somewhat elegant matrix, but I can't see how I would find it (Cramer's rule applied to the system is way too ugly, and solving first, then backtracking to finding the determinant is unpractical). Does anybody know how to approach this problem?


How does one solve the Schroedinger equation for a 2D, time-dependent harmonic potential?

This is the Schroedinger equation with a particular 2D harmonic potential: $$\begin{multline}i\hbar\frac{\partial}{\partial t}\Psi(x_1,x_2,t) = \\\ \Biggl[-\frac{\hbar^2}{2m}\nabla^2 + \frac{1}{2} m\omega^2\Biggl(\biggl(x_1 - \frac{x_0(t)}{2}\biggr)^2 + \biggl(x_2 + \frac{x_0(t)}{2}\biggr)^2\Biggr)\Biggr]\Psi(x_1,x_2,t)\end{multline}$$ Can anyone please tell me what the upside down triangle means? I know its the second derivative, but since the problem I have is of variables $x_1$, $x_2$, and $t$, do I take the second derivative of time? Or does that upside down triangle only does it for $x_1$ and $x_2$? $x_0$ is the separation distance between $x_1$ and $x_2$ and its a function of time because the two double HO potential wells are moving apart from each other. Also, how would I go about doing separation of variables here? Would I move all the time pieces to one side and all the x pieces to the other side? Won't I have to do separation of variables twice? Where the second time is when I have to separate $x_1$ and $x_2$?


The Aharonov-Bohm effect is purely classical, right?

Every discussion I've ever seen of the Aharonov-Bohm effect makes a big deal of its being a quantum effect with no classical analogue. But as far as I can tell it is present already at the classical level in QED. It also seems to have a close analogue in GR: the Riemann curvature outside an infinite straight cosmic string is identically zero, but an interferometer encircling it will see a phase shift that depends on its mass density. Is there something I'm missing? If not, can someone point me to a reasonably trustworthy textbook or paper that makes the point that it's classical, especially one that also mentions the GR analogy? (edit: By "present at the classical level" I mean that if you take the QED Lagrangian and derive classical equations of motion from it, you get a classical theory of Maxwell's electromagnetism coupled to a charged wave in which the A-B effect apparently exists just as in QED. This theory was never investigated before the quantum era, but it could have been, and the A-B effect could have been found then, as far as I can tell. I'm hoping for a published paper by a well-known author that says the above explicitly, in part because I'd like to add it to Wikipedia. I'm not interested in attempts to explain away the effect as being due to an external electric field, at least not for the purposes of this question. This is a theory question and there's no electric field in theory.)


Low frequency electromagnetic waves in General Relativity

I am becoming familiar with the Geometric Optics approximation in General Relativity which (to summarise) says that EM waves follow null geodesics under the geometric optics approximation. In the context of General Relativity the Geometric Optics approximation amounts to the assumption that: $\lambda << R$ where $\lambda$ is the wavelength and R is some measure of curvature (~ 100 meters near Black Holes). So under this approximation EM waves follow null geodesics. Additionally the "photon" model can make sense too, and we can say "photons follow null geodesics". But what about the "general case"? In the general case the order of curvature is the same as (or less) than that of the wavelength. So the usual statements do not obviously apply. Where is the full case discussed and what implications are there? To give some extreme examples for those who have not thought about this before:   1. In a Schwarzchild model at R=2M photons can get into circular null orbits. But clearly this provides an IR cut-off as they would be in a Bohr-like orbit with a max wavelength $\lambda_{max}=4\pi M$.   2. The usual "equivalence principle" may fail as the local trivialisation cannot extend far enough into the curvature region (and $\lambda$ is larger still). So the equivalence laboratory can detect such waves, but cannot measure their behaviour and prove that they follow geodesics (local straight lines). I am struggling to find any coherent account of this, just odd facts and claims. **EDIT** In large part this question is answered by the comment from Genneth, and the links related to this Wikipedia article: Maxwell's equations in curved spacetime.


Relationship between amplitude and frequency of a wave

I wanted to ask what is the relationship between amplitude and frequency of a wave. Some say there isn't , some say that there is , but from their answers the relationship is still unclear.


Why formation of rain is adiabatic?

I recently went through the wikipedia page for Rain. It said that > There are four main mechanisms for cooling the air to its dew point: > adiabatic cooling, conductive cooling,.... It proceeds as > Adiabatic cooling occurs when air rises and expands. Why is this a adiabatic process?


kinematics question about runner

Here's the question: Your goal is to complete your 10,000-m run in less than 30.0 min. Apparently you've started too slowly because at 27.0 min, you still have 1100 m to go! You can accelerate at 0.20 m/s2 . For how many seconds should you accelerate (and after this have constant velocity) in order to complete the run on time? So i can only think of using the equation x=xo+vot+0.5at^2 to solve for vinitial, but after that idk what to do, not sure if my first step is even correct


Free energy of the critical U(N) model

* Can someone help explain how the equations 30, 31 and 34 were obtained in this paper.  At a conceptual level I am wondering looking at equation 34 as to if they mean that $\lambda$ is somehow the anomalous dimension of the single-trace operators in this critical U(N) model - although looking at equation 30 it seemed that $\lambda$ is just the Hubbard-Stratonivich auxiliary field (though its not clear to me as to how is equation 30 derived). That factor of $(l + \frac{1}{2})$ is confusing me everywhere. Also what they call as $1/g$ in equation 30 is what I would think of as the threshold value of the translationaly invariant condensate at which the symmetry breaks. So it is like an order parameter and I can't see that as something that was there in the UV lagrangian as a coupling constant as they seem to imagine it here.   * Although I understand how equation 1 in this paper was derived but I still don;t see how equation 35 somehow follows immediately from it.


How can metal objects become electrically charged, if current flow means that an equal number of electrons enter/leave the object?

I am trying to answer to the question in the title. I am restricting my question to metal objects only. Here is my logical reasoning:   1. Current is the flow of charge over time.   2. In a circuit (simple series circuit, let's say), the flow of current is the same at every point in the circuit.   3. Therefore, the same # of coulombs of charge is flowing at every point in the circuit.   4. Electrons are the "material" of the charge that is flowing.   5. Therefore, equal flow of charge at every point in the circuit must mean equal flow of electrons at every point in the circuit.   6. Therefore, current can never cause a metal object to become positively or negatively charged, because the net number of electrons in the metal object will never change due to current flow. (!) Of course, objects CAN become electrically charged, gaining or losing electrons. So something is wrong with my reasoning or my premises. I just don't know what it is. Where am I going wrong?


How does Ptolemy's table, of angles in air and angles in water, fit perfectly on a parabola?

In "The Feynman Lectures on Physics" Vol.1 Optics Ch.26, Table 26-1 Ptolemy's list of angle in air and angle in water are given, in connection with Snell's law of refraction. I quote below values from that table: The angles are in degrees. Angle in air, Angle in water: (10, 8); (20, 15.5); (30, 22.5); (40, 29); (50, 35); (60, 40.5); (70, 45.5); (80, 50). In table 26-2 the corresponding angles according to Snell's law are given, to show the remarkable agreement with Ptolemy's list. In the text above the tables, a statement says all these values 'fit pefectly on a parabola'. I am unable to see the how these values fit on a parabola. How do you get the parabola from the listed values in the table?