title: Deriving shallow water equations from Euler's equations

text: I would like to derive the one-dimensional shallow water equations from Eulers's equations. This works perfectly for the conservation of mass. Especially the meaning of the longitudinal fluid velocity $\bar u$ in the shallow water equations becomes clear. It can be interpreted as average of the longitudinal velocity in Euler's equations over the height above ground level. But, in Euler's balance of momentum the longitudinal velocity occurs squared and the average of the squared velocity is not necessarily equal to the square of the averaged velocity. I am stuck at this point. Next, there follows what I have so far. Assumptions:   * propagation in the direction of the $x$-axis (unit vector $\vec{e}_x$)   * $y$-axis points upwards (unit vector $\vec{e}_y$)   * everything is constant in $z$-direction (unit vector $\vec{e}_z$; this direction is mostly left out). Volume integrals become area integrals and surface integrals become line integrals. If the path runs with positive circulation the outer surface normal can be calculated via $d\vec{r}\times\vec{e}_z$.   * incompressible medium; The density $\rho(x,y,t)$ is constant.   * The ground at $y=0$ is flat. The water height is $h(x,t)$. Region of water: $0\leq y \leq h(x,t)$, region of air: $h(x,t) < y$.   * The static relative pressure (w.r.t. atmospheric pressure) is $p(x,y,t)=g\rho(h(x,y,t)-y)$ where $g$ is the gravitation constant.   * No friction. We describe the fluid motion through the height above ground $h(x,t)$ and the velocity filed $\vec{v}(x,y,t)$ in the fluid region. At first the "working" case of Euler's mass balance. A fluid motion satisfies Euler's mass balance if for all parts $A$ of the fluid cross-section area in the $(x,y)$-plane there holds the equation $$ \partial_t\int_{A} \rho dA + \int_{\partial A} \rho\, \vec{v}(x,y,t)\cdot d\vec{r}\times\vec{e}_z = 0. $$ Euler's mass balance leads to the mass balance of the shallow water equations if one restricts the choice of areas to stripes $A:=\\{(x,y)\in[x_1,x_2]\times\mathbb{R}\mid 0\leq y \leq h(x,t)\\}$ for $x_1 < x_2$. On the bottom $y=0$ the velocity $v(x,0,t)$ is parallel to the path element $d\vec{r}$, the spat product in the path integral over $\partial A$ is zero and thus the contribution of this section of $\partial A$ to the path integral in Euler's mass balance is zero. Because of the height changing with time there is actually a normal component of the velocity at the top. But, this is already considered in the time- dependent area in the area integral. It is easier to consider the fluid mass between the half planes $A_x:=\\{(x,y,z)\in\mathbb{R}^3\mid y\geq 0\\}$ (note the fixed $x$) at $x=x_1$ and $x=x_2$. The growth of this fluid mass together with the out-flow of the fluid mass through the planes $A_{x_1}$, $A_{x_2}$ must balance to zero. This directly leads directly to the formula $$ \partial_t \int_{x_1}^{x_2}h(x,t)dx + \left[\int_{0}^{h(x,t)}u(x,y,t)dy\right]_{x=x_1}^{x_2} = 0 $$ where $u(x,y,t)=\vec{e}_x\cdot \vec{v}(x,y,t)$ is the $x$-component of the fluid velocity $\vec{v}$. Thereby, we have already divided through the constant density $\rho$. With the averaged longitudinal velocity $$ \bar u(x,t):=\frac1{h(x,t)}\int_0^{h(x,t)}u(x,y,t)dy $$ the last formula gives after differentiation w.r.t. $x_2$ and renaming $x_2\mapsto x$ the mass balance of the shallow water equations: ## $$ \partial_t h(x,t) + \partial_x \bigl(h(x,t) \bar u(x,t)\bigr) =0 $$ Now, the more difficult case of the momentum balance. The momentum balance from Euler's equations is satisfied if for all parts $A$ of the fluid cross- section area in the $(x,y)$-plane the equation $$ \partial_t \int_{A} \rho \vec{v} d A + \int_{\partial A} \rho \vec{v} \vec{v}\cdot d \vec{r}\times \vec{e}_z = \int_{\partial A} p d \vec{r}\times \vec{e}_z = \int_{\partial A} g\rho \left(h(x,t)-y\right) d \vec{r}\times \vec{e}_z $$ is satisfied. We get rid of the constant density and only consider the $x$-component of this formula by multiplying it with $\frac1{\rho}\vec{e}_x$ and we restrict the area to sections $A=\\{(x,y)\in[x_1,x_2]\times\mathbb{R}\mid 0\leq y\leq h(x,t)\\}$ with $x_1<x_2$. For simplification we again integrate over the half planes $A_{x_1}$ and $A_{x_2}$ $$ \partial_t\int_{x_1}^{x_2} \int_0^{h(x,t)} u(x,y,t) d y\, d x \+ \left[ \int_{y=0}^{h(x,t)} u^2(x,y,t)d y \right]_{x=x_1}^{x_2}= \left[ \frac12 gh(x,t)^2 \right]_{x=x_1}^{x_2} $$ Substituting $\int_0^h u dy=h\cdot \bar u$ yields $$ \partial_t \int_{x_1}^{x_2} h(x,t)\bar u(x,t) d x \+ \left[ \int_{y=0}^{h(x,t)} u^2(x,y,t)d y \right]_{x=x_1}^{x_2}= \left[ \frac12 gh(x,t)^2 \right]_{x=x_1}^{x_2}. $$ Here, I am stuck. The substitution $\int_0^h u^2 dy = h {\bar u}^2$ is possible if $u(x,y,t)$ is independent of the height $y$. If this is really the case what is the reasoning for this assumption? How do we come to know the vertical distribution of $u$? If this assumption is true then we arrive at the impulse balance for the shallow water equations through differentiation w.r.t. $x_2$ and renaming $x_2\mapsto x$: $$ \partial_t\bigl(h\bar u\bigr)(x,t) + \partial_x \left(\left((h\bar u)(x,t)\right)^2 - \frac 12 g h(x,t)^2\right)=0 $$ There is a similar derivation at http://www.whoi.edu/fileserver.do?id=136564&pt=10&p=85713. But, there it is just assumed that $u$ does not depend on $y$.


title: What is the intrinsic colour of the star?

text: 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.


title: Rolling resistance and static friction

text: I am a bit confused about the relation between rolling resistance and static friction. I have often heard that it is the static friction that lets the wheel roll. Consider the following two cases: a) A tire of a car of mass $m$ moving with positive acceleration $a$ on a street   b) A wheel of a moving wagon (of mass $m$) which is pulled by a string (with force $F_\mathrm{string}$).   1. How do the free body diagrams look like? (including static friction, rolling friction and friction forces at the axis)   2. What is the resulting force accelerating the center of mass of the wheel in case b) (in dependence of the frictional coefficients and $m$)


title: From spectrum/dispersion relation to the partition function

text: 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!


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

text: 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?


title: Spring Force in a Dynamic Equation

text: 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.


title: Why do hot objects prefer to emit photons over electrons ? Is there electron-positron annihilation?

text: Why do hot objects prefer to emit photons over electrons ? Is there electron- positron annihilation ? If so , why ? Im confused by this.


title: Measuring radon level

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


title: Interaction energy calculation in QFT

text: Suppose I have a QFT action describing interaction of two objects and I can perturbative write the action as $S=S_0+S_i$ where $S_0$ is the non- interacting, unperturbed action and $S_i$ is the interaction part (to appropriate order in perturbation). Then the interaction energy could be written as $E_i=\langle 1-e^{-S_i}\rangle_{C,0}$ where we are taking connected correlation function in the unperturbed background. How to derive this? I guess in terms of partition function it is $Z=Z_0e^{\langle iS_i-1\rangle_{C,0}}$. Many thanks in advance!


title: What would happen if...? (Scenarios involving a ball in an electric field)

text: * First I shall define two table tennis balls: Ball $A$ is coated with a conducting material and ball $B$ is an insulator.   * Then I'll define two scenarios: Scenario $I$ is a ball held from a nonconducting string between two plates of opposite charges (a pendulum). Scenario $II$ is a ball suspended in vacuum between two plates of opposite charges in zero gravity.   * Now I'll define two starting points: Point $1$ is for the ball to start right in the middle of the charged plates. Point $2$ is for the ball to start at the side where it touches one of the plates. For each ball, in either scenario and either starting condition, what are all the changes that would happen in the system as time passes?


title: Poynting Vector between Capacitor - With electrons in between!

text: Consider a capacitor with voltage $V = V_0 cos(\omega t)$, radius $a$ and separation $d$. Electrons are distributed uniformly with number density $n$. I want to find the poynting vector between the capacitor. This is a step-up from the usual question where there is only free space. In this case, conduction current $J_f \neq 0$. We must bear that in mind when calculating the B-field. Also, the electrons are polarized (can be treated as a dielectric medium), so the E-field is enhanced. **Calculation of E-field** Equation of motion is: $$ m\ddot x = -eE_0 e^{i\omega t} $$ Letting $x = x_0 e^{i\omega t}$, I solve for $x$: $$x = \frac{eE_0}{m\omega^2}e^{i\omega t} = \frac{eV_0}{m\omega^2d} e^{i\omega t} $$ Polarization is given by: $$P =nex = \frac{ne^2V_0}{m\omega^2d} e^{i\omega t}$$ Polarization is related to electric field by: $$ \vec P = \epsilon_0(\epsilon_r -1)\vec E $$ $$ \frac{eV_0}{m\omega^2d} e^{i\omega t} = \epsilon_0(\epsilon_r -1)\vec E $$ Thus, electric field is given by: $$E = \frac{eV_0}{m\omega^2d\epsilon_0 (\epsilon_r -1)} cos (\omega t) $$ **Calculation of B-field** $$\frac{1}{\mu_0}B(2\pi r) = I_c + I_d = (\pi r^2)\left(nev + \epsilon_0\epsilon_r \left|\frac{\partial E}{\partial t}\right|\right) $$ Then differentiate $x$ to get $v$, and differentiate $E$ to get $\frac{\partial E}{\partial t}$ to get B-field. **Calculation of Poynting Vector $S$** Since E-field is along $\hat z$, and B-field is azimuthtal along $\hat \theta$, $|\vec S| = \frac{1}{\mu_0}|(\vec E \times \vec B)| = \frac{1}{\mu_0}|\vec E||\vec B|$ we simply multiply both magnitudes. Is my calculation right? And finally, what happens when $\omega$ approaches plasma frequency $\omega_p$ from above? When $\omega = \omega_p$, $n \propto \sqrt \epsilon_r \rightarrow 0$, so the poynting vector shoots up. But in reality there will always be a damping force (electrostatic attraction in this case), so it doesn't explode to infinity.


title: Why is initial position negative?

text: > 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.


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

text: (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?


title: How much energy would it take to maintain the temperature in my pool?

text: Lets sat that I have a 1000 gallon pool sitting pretty at 82.4 degrees F (28 C). How much energy would it lose on a cold day (50 F or 10 C) and only the top was above ground (25 Sq Feet.) Would this be greatly effected if the day was colder (0 degrees C) or if the sides if the pool were exposed to (100 Sq Feel)?


title: Initial conditions at Lagrange Points

text: I'm simulating the restricted three body problem. But I don't know how to found the initial conditions of the object located at the Lagrange points analytically or computationally. I mean my problem is that the object located at the Lagrange points as initial position have to remain at the same distance relative to the second object. I've already look the derivation of the Lagrange points, but when deriving it, people look at the system as if were stopped in a moment in time and find the saddle points when $\vec{F}_{\Omega}=\vec{0}$. My problem is the following: > _**Problem:** Given three objects with masses, $m_1,m_2,m_3$, such that > $m_3<<m_1,m_2$. Find the initial conditions of the third object, such that > it's initial position is one of the Lagrange points and that it remains > orbiting always in the Lagrange point for a long period of time._ I can use all the initial conditions of the first, second, and third object. I also know the solution to the first,second and third objects position velocity for any time. I just have to know the initial conditions such as the constraint above is satisfied. I managed to find a initial velocity by putting numbers, in following picture(but that's not the idea): Initial Conditions for this picture: $x_1=1.0 $cm ,   $y_1=6.0 \cdot 10^3$ cm ,   $vx_1=0$cm/s   $vy_1=1.0 \cdot 10^6$ cm/s   $m_1=8.0 \cdot 10^{22}$g $x_2=4.0 \cdot 10^3 $cm ,   $y_2=6.0 \cdot 10^3$ cm ,   $vx_2=0$cm/s   $vy_2=2.0 \cdot 10^6$ cm/s   $m_2=2.0 \cdot 10^{21}$g $x_3=3.194 \cdot 10^3 $cm ,$L_1$ $x$ component   $y_3=6.0 \cdot 10^3$ cm , $L_1$ $y$ component   $vx_3=0$cm/s ???   $vy_3=1.080680314 \cdot 10^6$ cm/s ???   $m_3 =0$g ![Problem](http://i.stack.imgur.com/67moS.png)


> This post is a specialization of the post: Can the proportion of dark energy > change? Can the cosmological constant change with time? If so, is there a measurement of this evolution up to now, and a prediction for future?   How measure such a change?


title: Looking for the name of a physical phenomenon in fluids' mechanics

text: I have not even an idea about how I would search for that on Google, that is why I'm trying my chance here. As electrical engineer I have no clue about fluid mechanics. We all now that when water is pumped very fast into firefighters tube, it gets very rigid and tends to be straight. What is this effect called, I'm interested in doing some research about the forces applied by such a tube from its initial folded position to the final position. Thanks


title: Is this expression of electric field correct?

text: 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? $$ \mathbf{E}(\mathbf{r},t)=\mathrm{Re}\left[\hat{\mathbf e}F(x,y) A(z,t)\exp(i\beta_0-i\omega_0t)\right], $$ $$ \mathbf{E}=\exp\left(-j\int_0^z\beta_\text{eq}\mathrm dz\right)\mathbf{A}. $$


title: Approximating electric field of two point charges

text: This is the question:               A point charge of +8 nC is placed on the x-axis at x = 1.     A second point charge of +6 nC is placed on the -x-axis at x = -1.     Find an approximation for the electric field for large values of x on the x-axis.     The result should look like a 14 nC charge at the origin plus a term that decays     with the inverse cube of x. Evaluate the electric field at x = 5 using the correct     expression and your approximation. What is the ratio of your approximation to     the correct value?      I found that $E = \frac {1}{4\pi\epsilon_0}[\frac {6*10^{-9}}{(x+1)^2} + \frac {8*10^{-9}}{(x-1)^2}]$. So the actual value for $E(x=5)=5.9918$ V/m. I am not sure how to get the approximation. When I do it I just see the first part: $\frac {1}{4\pi\epsilon_0}*[\frac {14*10^{-9}}{x^2}]$. I see it as when $x$ goes to $\infty$ that $(x+1)^2\approx x^2$ and $(x-1)^2\approx x^2$ How do I get the $+ \frac {term}{x^3}$ on the approximation.


title: Does an elliptical orbit of a satellite maintain its orientation as the Earth revolves?

text: Imagine a satellite in an elliptical orbit around Earth. As the earth travels around the sun, does the elliptical orbit of the satellite swing around Earth, as pictured in A, or does it maintain a consistent orientation as pictured in B? Are both of these options wrong? ## Option A ![diagram of elliptical orbit changing orientation](http://i.stack.imgur.com/kg40Z.png) ## Option B ![diagram of elliptical orbit not changing orientation](http://i.stack.imgur.com/9bO1B.png)


title: Can a linear momentum generate angular momentum at collision?

text: I'm trying to get the facts straight here. Suppose I'm throwing a ball with no angular momentum. It collides with the ground and Newton's third law tells us that a force opposite to the gravity will be applied to the ball at collision. During collision, the momentum will be converted to potential energy and then become kinetic energy again. Am I right to say that some of the kinetic energy can be converted to angular momentum, thus making the ball rebounce not straight up? (assuming some friction)


title: Why does sound need air in order to travel?

text: 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?


title: What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)

text: I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's equations and their correspondent hamiltonian vector field, while conservation of energy in the hamiltonian formalism requires the Hamiltonian to be explicitly independent of time. I'd go as far as saying it's a deeper principle in this respect. On the other hand, its direct implications for the trajectories and the motion of particles are quite elusive to me. I've seen mostly physical systems where both energy and phase space volume get conserved and I've seen a bunch of dissipative systems, say a damped oscillator, where both of them aren't conserved. I'm looking for examples somewhere in the middle in order to capture their respective uniqueness: phase space volume preservation but no energy conservation, a hypothetical situation where energy is conserved but Liouville's theorem doesn't hold and so on. I'd rather know an example out of classical mechanics than statistical mechanics, which I know it's where Liouville's theorem comes particularly in handy but I know almost nothing about it, but please go ahead if you think it can shed some light on the matter.


title: Why is there no UV catastrophe (divergence) in turbulence?

text: I have just read that as the Reynolds number is increased, the separation of macroscopic and microscopic scales increases and that this also means that there is no UV catastrophy (or equivalently UV devergence?) in turbulence. I do not understand what this means, for example does the increased separation of the macro and microscales just mean that the spectrum is broadening? And what exactly would an UV catastrophe mean in the context of turbulence and how can I see (technically and mathematically) that it does in fact not exist? I have only a rough intuition what it could mean when considering the turbulence problem from a QFT approach, namely that it from this point of view the infinit Reynolds number limit should be renormalizable (?)... Any comments that would help me to understand what the sentence in the first paragraph exactly means would be appreciated.


title: Wall pressure of a fluid flow in a pipe of variable radius

text: Using all cylindrical coordinates, pipe with z-axis vertically upward and radius of $r = G(z)$, flow is incompressible, inviscid and steady, Using appropriate boundary conditions I want to find the wall pressure.Flow is axis symmetric and irrotational, and of the form $(a(r,z), 0, c(r,z))$ in cylindrical coordinates. The boundary conditions I have are the the fluid is stationary at the wall. $a(G(z), z) = c(G(z), z) = 0, $ Using Bernoilli's equation for steady flow $\frac{p}{\rho} + \frac{v^2}{2} + \Omega = G$ $\frac{p}{\rho} + \frac{c^2}{2} = G - \Omega$ $p_w = (G - \Omega - \frac{c^2}{2})\rho$ But I don't think this is correct for some reason.


title: What is the formal definition of a stellar day?

text: I'm having trouble understanding precisely what a _stellar day_ is. Neither the USNO nor the IERS sites provide a definition. And Wikipedia's description as the "rotation period relative to the fixed stars" as "the span of time it takes for the Earth to make one entire rotation with respect to the celestial background or a distant star" is confusing, since, relative to fixed stars, the earth is both rotating and precessing, so that the rotation period relative to fixed stars as such, is not a single value, but will be different for stars with different equatorial coordinates. I assume that the stellar day is simply the earth's rotation period on its axis, but I'm not sure that's right, or how to state it formally (e.g. with respect to inertial frames). * * * I understand why the stellar day is distinct from and longer than the sidereal day, since the coordinate system that defines a sidereal day is rotating slowly against the rotation of the earth.


title: Geometric interpretation of hidden SUSY

text: Hidden supersymmetry, which is the classical(non-super) symmetry in the form of susy, acting on a non-Grassmann space (e.g., Grassmann space is $(t,x,\theta,\bar{\theta})$, corresponding non-Grassmann space is $(t,x)$)...see the review article on hidden supersymmetry arXiv:1004.5489. In the language of Lie geometry, we can understand SUSY as a vector operating on a Grassmann space. Is there an analogous geometric explanation for hidden SUSY? Any comment is highly appreciated.


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

text: 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


title: What evidence exists for string theory viability?

text: 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?


title: EM attenuation in free space at different frequencies

text: Can anyone explain intuitively why electromagnetic radiation (according to the basic path loss equation in communications) experiences more attenuation at higher frequencies? The equation states that the path loss $ L $ is proportional to $ d\times f $, where $ d $ is the distance the wave has propagated and $ f $ is the frequency of the wave.


title: Why do rockets have multiple stages?

text: What is the advantage for rockets to have multiple stages? Wouldn't a single stage with the same amount of fuel weigh less? **Note** I would like a quantitative answer, if possible :-) ![alt text](http://i.stack.imgur.com/lswdV.jpg)


title: Physically induced latency in internet connections

text: As a trade I am a software architect. It does happen that I have to design sites where the servers are located in different parts of the world. Now an essential parameter to consider is latency: that is the time it takes between when a packet is sent from one server, and when it is received on the other side. Of course, the packet might go through other systems etcetera. However, the hallmark of a good connection is low latency. > Question: what is the minimal latency that can be obtained over a coaxial > wire (i.e. what is the maximum speed reachable by an electrical impulse over > a coaxial wire)? > > Question: what is the minimal latency that can be obtained over a fiber > optic connection (if different from distance/c)?


title: Time Reversal in Euclidean Spacetime - unitary or antiunitary?

text: **(pre-request)** We know that **time reversal operator** $T$ is an **anti- unitary operator** in Minkowsi Spacetime. i.e. $$ T z=z^*T $$ where the complex number $z$ becomes its complex conjugate. See, for example, Peskin and Schroeder ``An Introduction To Quantum Field Theory'' p.67 Eq (3.133). * * * **(Question)** Is **Euclidean time reversal operator** $T_E$ an **unitary operator** , i.e. $$ T_E z=z T_E \;\;\;(?) $$ or an **anti-unitary operator** in Euclidean Spacetime? Why is necessarily that or why is it necessarily not that? **Or should $T_E$ be an unitary operator like Parity $P$ instead?** However, here see the attempt of a PRL paper, Euclidean continuation of the Dirac fermion  where time reversal $T_E$ is given by, on page 3: $$ T_E z=z^*T_E \;\;\;(?) $$ $T_E$ is still an **anti-unitary operator**! It seems to me if one consider Euclidean spacetime, the Euclidean signature is the same sign, say $(-,-,-,-,\dots)$, then **Parity $P$** acts on the space is equivalent to the **Euclidean time reversal operator $T_E$** acts on Euclidean time (which is now like one of the spatial dimensions). So **shouldn't $T_E$ be an unitary operator as Parity $P$?** * * * [Other Refs] i. Please, you may read, an earlier Phys.SE question here, probably is poorly formulated, so cannot draw the efforts of people to answer the question. Here let me try an easier way and focus on one issue only. ii. Osterwalder-Schrader (OS) approach, and ``A continuous Wick rotation for spinor fields and supersymmetry in Euclidean space'' by Peter van Nieuwenhuizen, Andrew Waldron.


title: Is there a way to formulate relativistic dynamics in a way that "hides" the finite speed of light?

text: I'm not referring to the dimensional choice that makes $c=1$; rather I'm imagining something more about replacing all references that apparently involve velocities with the appropriate $\gamma$ factors or rapidity. In this description, the kinematic feature that maps to speed would then be defined on $[0,\infty)$, and we wouldn't have to deal with the question of "why is the speed of light finite, and has the value that it has?" $^*$ $^*$ Or, at least we wouldn't have to deal with it as a fundamental question about the universe.


title: Two expressions for expectation value of energy

text: I was looking up expectation value of energy for a free particle on the following webpage: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/expect.html It says that $E=\frac{p^2}{2m}$ and therefore $\langle E\rangle=\frac{\langle p^2\rangle}{2m}$ This leads to $$\langle E\rangle=\int\limits_{-\infty}^{+\infty}\Psi^*\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi\,dx$$ However, it also has on the bottom of the page: "In general, the expectation value for any observable quantity is found by putting the quantum mechanical operator for that observable in the integral of the wavefunction over space". Now, I know that the operator for $E$ is $i\hbar\frac{\partial}{\partial t}$. So shouldn't $\langle E\rangle$ be: $$\langle E\rangle=\int\limits_{-\infty}^{+\infty}\Psi^*(i\hbar)\frac{\partial}{\partial t}\Psi\,dx$$


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

text: 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?


title: Reference request for the terminology and usage of different types of plasmas

text: I'm looking into discharge models, specifically for _combustion prcesses_ , and I'm swamped by the number of different types of plasmas and their respective treatments (thermal plasmas, non-thermal plasmas, different degrees of inoization, excitation levels, diluted systems, etc. etc.). I'm looking for an overview, maybe in form of a dedicated review, which gives a characterization of these different types, probably also in terms of temperatur(es). I'm not sure how standardized the terminology in this field is at this point. In the end I would like to find out which of these types is used in which applications (i.e. in which practical engineering designs). A comment on the respective model approaches would then be very helpful too.


title: What is "surface fluid adhesion energy"?

text: 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. * * *


title: Unconstrained action in Yang-Mills theory

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


title: Do gravitational waves slow down as they pass through matter?

text: I've heard that gravitational waves travel at the speed of light, and have some parallels to electromagnetic waves. EM waves slow down as they pass through matter (speed of light in glass is slower than in vacuum, for instance). Do gravitational waves also slow down as they pass through matter? If so, are there any effects like Cherenkov radiation when matter passes through a medium at a speed greater than the velocity of gravity in that medium?


title: Why do charges enter the capacitor in the first place?

text: I have been studying Capacitors for the past year now and the one thing I don't understand is how a charge is stored on the capacitor. Essentially, a circuit with a capacitor is an incomplete circuit right? So why do the electrons start to gather up on one of the plates. Its like they are tricked into thinking its a complete circuit and get trapped once they realise it isn't! What is this sorcery! What is happening to make the electrons do that and if it simply is electrons moving and then turning back or getting stuck, would a switch have the same effect? (Assuming the switch is open). Hopefully someone can finally fill the major gap in my knowledge because everything else I read doesn't explain this.


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

text: 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?


title: How does the Double Slit Experiment work in detail?

text: What is the best detailed description/visualisation of the experiment available? Describing what is actually measured, how the data is analysed, correlated and interpreted when for example large molecules are send through the apparatus? What are the active components? What role does time and correlation within the measured data play? How for example did the guys in Vienna http://www.livescience.com/19268-quantum-double-slit-experiment-largest- molecules.html actually confirm an interference pattern? did they? What do they detect? How? What is the actual physical magnitude they measure at the "screen" showing the interference?


title: Casimir effect when the zeta function has a pole

text: 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} $


title: Doppler redshift in special relativity

text: I came across this exercise in Elementary General Relativity by Alan MacDonald: A source of light pulses moves with speed v directly away from an observer at rest in an inertial frame. Let $ \Delta t_e $ be the time between the emission of pulses, and $ \Delta t_o $ be the time between their reception at the observer. Show that $ \Delta t_o = \Delta t_e + v\Delta t_e $. Based on my understanding of special relativity, the space-time interval between two events as measured from two inertial frames of reference should be the same. Therefore, $$ \Delta t_e^2 = \Delta t_o^2 - \Delta x^2 $$ $$ \implies \Delta t_e^2 = \Delta t_o^2 - v^2\Delta t_o^2 $$ $$ \implies \Delta t_o = (1 - v^2)^{-1/2}\Delta t_e $$ which is not the same relation. What is wrong with my reasoning?


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

text: 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?


How does a mobile phone vibrate without any external force? By Newton's law, any body can't move without any external force


title: Dual photon in d=3

text: In some papers (such as http://arxiv.org/abs/hep-th/9910184 and http://arxiv.org/find/all/1/all:+AND+kapustin+AND+topological+disorder/0/1/0/all/0/1) I am reading it is always referred at "the dual particle of a photon" in three (2+1) dimension. This particle is a scalar $\sigma$, compared with the photon which is a vector field $A^{\mu} \quad \mu=0,1,2$. What is meant with that? Is it just because the photon in $d=3$ as $1$ degree of freedom, just as the scalar, or there is a deeper reasoning behind that? Would a theory formulated in terms of the "dual photon" $\sigma$ be the same theory as the one formulated in terms of the original photon $A^{\mu}$? How can one express the original photon in terms of its dual?


title: Why do people say the phase oscillates in time and the amplitude stays the same but the intensity of a traveling beam does oscillate with time?

text: I'm confused why people say the phase oscillates in time and the amplitude stays the same (the reason for having complex numbers). But on the other hand, the intensity of a traveling beam does oscillate with time.


title: Modelling radiance profile of a light bulb

text: For Graphics I need to model the shape and the radiance profile of an existing light bulb in order to use it in a ray-tracing application. I can freely choose the light bulb. The strategy must not be very detailed or very long, just a few lines. As most light bulbs don't offer the radiance profile, I wondered how to construct it from the given specifications. I'm allowed to make some assumptions. It would be of great help, if someone can guide me in a good direction.


title: Thermalising a sub-system of a larger, interacting system

text: I'm considering a joint system consisting of a spin-1/2 particle (qubit) and a spin-l particle (reference) coupled via a Hamiltonian $H_0$. At a certain point I want to couple the qubit to a bosonic bath. I tried to use the interaction Hamiltonian $$ H_{int} = \sigma_x\otimes\mathbb{1}\otimes\sum_k g_k(a_k^\dagger+a_k), $$ where $\sigma_x$ is the usual Pauli operator acting on the qubit, the identity acts on the reference, and the $a_k^\dagger$ and $a_k$ are the creation/annihilation operators of the bath. After setting up an appropriate master equation and numerically solving it, the problem is that this coupling, despite the identity on the reference, instantly affects the reference as well as the qubit, with the joint system approaching the Gibbs state given by $H_0$. What I was expecting instead was a much more indirect thermalisation of the reference, in which initially only the qubit is affected and then, with a delay, the reference reacts to the qubit's changed state. Now I'm wondering if there is a way in which I can couple the two systems to the bath which achieves such dynamics?


title: Error propagation estimations for sine and cosine

text: 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 $\alpha$?


title: effect of object mass in coeficient of restitution

text: I know the coefficient of restitution depends on the velocity before and after the impact of a bouncing ball. How does the mass affect it. Can't figure it out. ![formula](https://chart.googleapis.com/chart?cht=tx&chl=C_R%20%3D%20%5Cfrac%7Bv%7D%7Bu%7D) `v` is the scalar velocity of the object after impact `u` is the scalar velocity of the object before impact ![formula](https://chart.googleapis.com/chart?cht=tx&chl=C_R%20%3D%20%5Csqrt%7B%5Cfrac%7Bh%7D%7BH%7D%7D) and the H and h are related to the heights. How can I relate it to the mass of the object?


title: Electron speed in Atoms

text: Is the speed of electrons in Atoms consistent in all Atoms or clusters/groups/individual elements, and if it is consistent in any atom, is that speed constant?


title: Kaon reaction rates

text: I have a big difficulty in grasping weak interactions. For example, how would I go on about determining the following: $\frac{\Gamma(K^{+} \to \pi^{+}K^{0})}{\Gamma(K^{+} \to \pi^{0}K^{+})}$ I don't understand much of the reaction at all. I can find some standard decay reactions of Kaons, am I supposed to combine these from them? How? And for those reactions, "Cabibbo-angles" are defined. Are they related to some known properties, or defined experimentally?


If photons don't interact directly, how can electromagnetic waves interfere? I know that photons can scatter via higher order mechanisms, but not directly. Does those mechanisms explain the classical phenomenon of wave interference?


title: 1 Tesla electromagnet?

text: Is it possible to create a powerful electromagnet at home? With use of a ferromagnet it seems so... Using the following formula: $B(Tesla)= k\mu_0nI$. I understand some ferromagnet's like iron could have permeability above 10,000? That would easily boost the field above a 1 Telsa? Relative permeability of raw iron is: 200,000! There must be something wrong here, see this wikipedia table. **UPDATE: How much input power might be needed?**


title: Event horizons in Newtonian mechanics and Galilean relativity

text: I've been revisiting classical physics (in the sense of Newtonian mechanics and Galilean relativity) and I was thinking why can't we have an event horizon in classical physics? Is it because the structure of space is absolute, the Euclidean geometry, or because time is absolute, so simultaneity is absolute? Or both?


title: Cassini 's and Pioneers RTG power output datafile

text: 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.


title: Stability of square of masses on strings under rotation

text: 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)


title: Ideal gas system with diathermic piston externally manipulated

text: 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.


title: Why is the geomagnetic north pole at a different location from the magnetic north pole?

text: According to this site Magnetic North, Geomagnetic and Magnetic Poles _The Geomagnetic poles (dipole poles) are the intersections of the Earth's surface and the axis of a bar magnet hypothetically placed at the center the Earth by which we approximate the geomagnetic field. There is such a pole in each hemisphere, and the poles are called as "the geomagnetic north pole" and "the geomagnetic south pole", respectively. On the other hand, the magnetic poles are the points at which magnetic needles become vertical. There also are "the magnetic north pole" and "the magnetic south pole"._ ![enter image description here](http://i.stack.imgur.com/HnlCw.gif) But why does the compass needle not point to the geomagnetic pole? I mean why aren't they both at the same location?


title: Comparison between Cadabra and other Symbolic Computer Algebra software

text: Does anyone has some experience about working with Cadabra and it's (dis)advantage in comparison to other Symbolic Computer Algebra software such as Maple and Mathematica (physics package) in the field theory/gravity problems?


title: Derivation of Kepler's laws

text: How can analytically be derived the Kepler's laws? I found some extremely synthetic equations which from the Newton's laws (in particular $\mathbf{F} = m \mathbf{a}$) tried to obtain the Kepler's laws, but even if it seemed to be a nice procedure they were too much incomplete. Is there a book or a link where I can find an exhaustive treatise about this? I am looking for a (as complete as possible) set of hypothesis and the steps which lead to the laws. Could you suggest me some resources?


title: Electricity from lightning

text: 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


title: Wave functions for 2D potential with spin interactions

text: 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.


title: A rigorous treatment of distributions in quantum mechanics

text: In many introductory courses to quantum mechanics, we see $\delta$-functions all over the place. For example when expressing an arbitrary wave function $\psi(x)$ in the basis of eigenfunctions of the position operator $\hat x$ as $$ \psi(x) = \int\mathrm d\xi\, \delta(x-\xi)\, \psi(\xi). $$ In bra-ket notation this corresponds to $$ \left|\psi\right\rangle = \int\mathrm d\xi\,\left|\,\xi\,\right\rangle\\!\left\langle\,\xi\,\middle|\,\psi\,\right\rangle, $$ where $\left|\,\xi\,\right\rangle$ is the state corresponding to the wavefunction $x\mapsto\delta(x-\xi)$. Now the $\delta$-function is really not a function, but a distribution, that's defined by how it acts on test- functions, i.e. $\delta[\varphi] = \varphi(0)$. Do you know of an introductory text on quantum mechanics that stresses this point and uses the language of distributions properly, avoding any functions with seamingly infinite peaks?


title: Will a ferromagnet retain its magnetic properties after reaching the curie point?

text: If a piece of iron or any kind of ferromagnet reaches its Tc, it will lose all it's magnetic properties, and will not be affected by an external magnetic field, if it drops back to room temperature or below Tc will it regain its magnetic properties and get attracted without losing anything or get "damaged" from the heat? And will this always happen with all kinds of ferromagnets? Also, if a feromagnet's Tc is 500C for example, will it operate the same way in the presence of an external magnetic field at 450C just like in room temperature or will there be a weaker effect? Thank you.


title: Does distance traveled by a vehicle after its engine has been switched off depend on its mass at all?

text: A vehicle moving with some velocity on a rough horizontal road finally comes to rest after its engine has been turned off. Intuitively, it seems a vehicle with greater mass would stop first because it would experience a greater friction force, but if we go by the work-energy theorem as follows, it's clear that the distance covered does not depend on the vehicle's mass at all: $$K.E_{final} - K.E_{initial} = W_{friction}$$ $$KE_{final}=0\\\ KE_{initial}=\frac12MV^2\\\ W_{friction}=\mu Mgd\\\ \text{Therefore, }d=\frac{v^2}{2\mu g}$$ Is my notion valid?


title: Wave Function of Particle in Nuclear Reaction

text: I was thinking and came up with the question of what happens to the wave function of a particle that decays into energy, say a neutron in a nuclear reaction. I know that conservation of probability requires that the probability of finding the particle anywhere in space is 1. Therefore, it seems to me that the energy has to be dealt with in the wave function of the particle. I have a feeling that this has something to do with the Hamiltonian operator present in Schrödiner's equation, but I can't seem to find anything confirming or proving that incorrect. Anyone have an answer to how the wave function of a particle deals with decay into energy, or otherwise know of something I can read on the subject?


title: Calculate electric field induced by a changing magnetic field

text: I want to plot the electric field (as a vector field plot) which is induced by a changing magnetic field for some simple cases. Suppose for example that the magnetic field changes linearly (or quatratically) with t. Then you may calculate the curl $\nabla \times \vec{E}$ of $\vec{E}$ via: $$ \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} $$ as in this question. However this doesn't give you a unique solution for $\vec{E}$. I guess that the solution will become unique if I add some boundary conditions. But I don't know how to do this in detail. Suppose you have the experimental setup, that the magnetic field is (homogenous and) perpendicular to a given area $A$ in the $x_1$,$x_2$ plane (for example $A$ being a circular or rectangular area) and zero for every point $P$ which $x_1$ and $x_2$ coordinates are outside of $A$. What is needed for the experimental setup to make the solution unique. How to calculate the solution in detail, what will it look like? **Edit:** Since there are no charges in my example you have also the equation $\nabla\cdot \vec{E} = 0$. But I don't see how this helps. **Edit2** Here is what I tried so far: If you suppose that the electric field is symmetric around the $x_3$ axis (but **why** can I assume this?) one could procceed as follows: Let $\gamma$ be a circular path in the $x_1$-$x_2$-plane with radius $r$ and center $(0,0,0)$: $$ 2\pi r |\vec{E}| = \int_{\gamma} \vec{E} d\vec{s} = -\frac{d\Phi}{dt} = -\pi r^2 \frac{d |\vec{B}(t)|}{d t} $$ which implies: $$ |\vec{E}| = - \frac{r}{2} \frac{d |\vec{B}(t)|}{d t} $$ But this gives only the absolute value of $\vec{E}$ and not the direction. Furthermore it depends linearly on $r$ which would means that it goes to $\infty$ if $r$ goes to $\infty$. But that seems to be very unintuitive to me. Also there is no distinguished point, so I should perhaps better assume translation invariance... **Edit 3** Bonus question: How to solve it via differential equations (not using the integral form)?


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

text: 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...


title: Explanation of a proper time (an add)

text: I have read on web that: > Proper time $\tau$ is a time interval measured by a clock which at rest > relative to the observer. But lets consider this problem: > The plane is flying with a speed $u=300m/s$. How much time $\Delta t'$ would > have to pass for a passanger on that plane for time difference $1s$ to occur > between the time $\Delta t'$ on the plane and the time $\Delta t$ on the > Earth. We postulate that speed of light equals to the speed of sound > $c=343m/s$. * * * **Ok so lets say i set myself in coordinate system $x'y'$ and check the whole situation from a perspective of a passenger on the plane.** I first calculate the Lorenz factor: $$\gamma=\frac{1}{\sqrt{1-\frac{300^2}{343^2}}} = 2.06$$ If everything is checked from the perspective of the observer on the plane, he is not moving according to his clock so his time is the proper time and i set $\boxed{\Delta t'\equiv\tau}$. Because the proper time is the shortest it follows that: \begin{align} \Delta t = \gamma \Delta t' = \Delta t' + 1s \xrightarrow{\text{i use the RHS}}\gamma\Delta t' &= \Delta t' + 1s\\\ \Delta t' \left( \gamma - 1 \right) &=1s\\\ \Delta t' = \frac{1s}{\gamma - 1}\\\ \Delta t' = 0.94s \end{align} * * * **So now i reverse the situation and set myself in the coordinate system $xy$ on Earth.** Here the Lorentz factor is the same as before even if i had to use negative speed $u=-300m/s$ (squaring does its trick hehe). So it again holds $\gamma=2,06$. If now everything is checked from the perspective of an observer on Earth, he is not moving accordint to his clock which means now his time is the proper time and i can set $\boxed{\Delta t=\tau}$. Because the proper time is the shortest it follows that: \begin{align} \Delta t' = \gamma \Delta t = \Delta t + 1s \xrightarrow{\text{i use the RHS}}\gamma\Delta t &= \Delta t + 1s\\\ \Delta t \left( \gamma - 1 \right) &=1s\\\ \Delta t = \frac{1s}{\gamma - 1}\\\ \Delta t = 0.94s \end{align} * * * Well this means that the proper time definition needs an add which is: **"In the situation where we have more clocks the proper time is measured by the clock which is situated in the coordinate system from which we observe."** and **"we are free to choose to observe from whichever coordinate system we want."**


title: Conceptual doubts in EM waves and old quantum theory

text: I have a few questions.   1. I know that EM waves transfer energy. So when they are generated why do they form curves? Are energy packets moving in a curvy path, or energy packets (quanta) not considered when wave is propagating? OR are the electric and magnetic FIELDS curvy in nature? NUTSHELL: Why is the EM wave curly and how do we know that they are curvy?   2. When ever we heat an object, the energy supplied makes the electrons oscillate due to which they radiate electromagnetic waves of which frequency rises as temperature rises. At certain temp. First, the red colour comes and then as we heat further blue comes because frequency is rising, my interpretation says that if we heat it further the object would radiate waves with frequency higher than the visible range and we couldn't be able to see the object glow. Am I right? The main thing is that books write that waves are formed because electrons oscillate and they want to become stable but atoms are also moving when heated and so the positive nucleus is also oscillating due to which EM waves should be produced because whenever charged objects accelerate they radiate energy. First of all, is this true? Second of all, why do they radiate energy? Is to become stable or something like that?   3. I attended a lecture which said that every object is radiating energy But I don't understand why? Is this because of environment? Or is it because electrons are oscillating and radiating energy or something like that?   4. The famous question that electron have uniform centripetal acceleration so they would radiate energy and spiral in to the nucleus! Why can't a body moving at a uniform velocity in an Electromagnetic field radiate off its energy? What is so special about accelerated Motion?   5. The Bohr model was proposed to solve this major problem, by postulating that the energy is quantised. I don't understand this, So what if energy is quantised? How does this means that energy cant be radiated (or absorbed?)?


title: free energy and entropy of 2D soap froth

text: This is a (exploratory) computational project. The soap froth was created by injecting bubbles into a chamber formed by two rectangular plates which are 0.16cm. From the moment the soap froth was created, a camera is employed to take a snapshot of the froth every 10 minutes. The details of the experiment can be found here: http://link.aip.org/link/doi/10.1063/1.59943. I did not participate in the experiment. The data from the experiment include, for each bubble on the froth in a particular snapshot, area, perimeter, number of edges, the id of neighbors in unknown order (every bubble has a unique id, but the id was not tracked over time), lengths of edges in unknown order, xy coordinates of the vertices in unknown order. My task is to find a way to calculate (or estimate) the change of Helmholtz free energy as a function of time and hence also the change of entropy. The 1st problem is how to calculate the internal energy of the soap froth. An obvious part of the internal energy is the internal energy of the gas phase treated as ideal gas. (is it justified to approximate it as ideal gas?) The 2nd part in the internal energy of the soap froth should be the surface energy, which is $$ 0.5\sum_i\langle l\rangle_i\,z\lambda A_i, $$ where the sum is over all bubbles in the froth, $\langle l\rangle_i$ is the average edge length for an individual bubble $i$, $z$ is the space between the two plates, $A_i$ is the area of bubble $i$ (such that $zA_i$ is the volume of bubble $i$), and $\lambda$ is the surface tension per unit length. In the experiment, the excess fluid was allowed to flow into the slots of the chamber. I am not sure if the excess fluid contributes significantly to the internal energy. Is there any other significant contribution to the internal energy? Can it be computed from the data? The calculation of the entropy part sounds even more remote to me. I only know how to compute the ideal gas entropy. In the calculation, I assume the pressure is uniform inside individual bubble. Even then the estimation of the pressure is not easy. In principle, we can deduce the pressure difference any two neighboring bubbles given the length of the share curved edge and the two shared vertices. (of course one will also have to which bubble has higher pressure) But the resolution of the data is not very large, so there can be considerable error in the estimation, and what's worse is I have a hard time to identify (from the list of curved edge lengths) the correct edge length between two given bubbles. A very crude way is to assume the pressure outside a given bubble to be the atmospheric pressure, and the pressure inside the given bubble is larger than the atmospheric pressure if its number of sides is less than 6, the pressure lower than the atmospheric pressure if its number of sides is larger than 6. And the magnitude of pressure difference is estimated by its average edge length. That is, $$ P(i)=P_0+\dfrac{\lambda\alpha(n_i)}{\langle l\rangle_i}, $$ where $\alpha=\dfrac{2\pi}{n}-\dfrac{\pi}{3}$ and $n_i$ is the number of sides of bubble $i$. In summary, the non-computational questions are: 1) Is the internal energy calculation scheme reasonable? Are there any other contributions to the internal energy? 2) What contributes to the entropy of the soap froth besides the ideal gas entropy? 3) Is the pressure approximation reasonable? Or is there a way to tell whether the approximation scheme is reasonable or not?


title: When are leap seconds added in various time zones?

text: I understand that technically when a leap second is added, it is added after midnight UTC, but I'm unclear how the addition is handled in other timezones. For precise reckoning of course (e.g. astronomical ephemeris), it must be added at the same instant (e.g. at 1 AM in France, or 7 PM in in New York), but for many locations doing so could be disruptive and potentially confusing. Even in Britain, the addition at midnight on New Year's Eve could be confusing, since it would either extend the countdown by a second, or require it to be started a second "late". Are leap seconds in fact added everywhere, in all time zones, at the same UTC instant, or do different time zones (and perhaps even different countries and organizations) account for them at different, more "convenient", times.


title: Higher dimensions

text: 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.)


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

text: 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


title: Lots of little questions about radiation of LC circuits

text: I'm trying to get a more intuitive understanding of resonant inductive coupling. It's supposed be a more efficient way to transfer electrical energy wirelessly, because the coils are only coupled by near fields, and don't waste energy as far field radiation. The efficiency is supposed to increase as Q increases (the resistive component decreases).   ![LC circuit schematic](http://i.stack.imgur.com/1zhUl.png) Ideally, an LC circuit has a Q of ∞. If you charge up the capacitor in an LC circuit, and then leave it floating in space away from anything else, the energy will cycle back and forth between the L and C. The voltage, current, electric field and magnetic field will all cycle sinusoidally (right?) With ideal components, and no resistive component, this will continue cycling back and forth forever. Real LC circuits have some resistance, which wastes some of the energy as thermal radiation (at a frequency related to temperature), and the cycling eventually dies. I think they also have some other non-idealities that allow energy to escape as far field electromagnetic radiation (at a frequency related to L and C), correct? What are these non-idealities? Are they independent of the resistive component? If another identical LC circuit is brought near this oscillating circuit, will energy transfer from one circuit to the other due to the changing magnetic fields? Will this happen even with ideal components that don't have far field radiation? Is there no energy transfer if the coils are perpendicular and the axis of one is coplanar with the other, like this?: | — Will the energy distribute itself evenly among the circuits, so they reach an equilibrium where each is circulating half of the original energy? Or will the energy oscillate back and forth between the two circuits, with one alternately being depleted or full?


title: Lawrence Krauss' Theory of Origin

text: 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?


title: Does blackbody radiation work like this?

text: In a blackbody object the photons are reflected back and forward. If I heat up the the blackbody object with microwaves or other types of photons the electrons in the wall would vibrate. When the electrons absorbs the photons, does it take energy from the electron and becomes shorter in wavelength? And if you heat the blackbody even more, does the photons takes more energy from the electrons and gets even shorter in wavelength, leading to continuous spectrum ?


title: Rotational Constant and Moment of Inertia of Fluorine gas

text: 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...


title: A really simple question about heat transfer?

text: In rectangular coordinates, the heat conduction vector can be expressed in terms of its components $$\vec{\dot{Q_n}}=\dot{Q_x}\vec{i}+\dot{Q_y}\vec{j}+\dot{Q_z}\vec{k}$$ Please explain the meaning of this equation? I know about vector! I know mathematics, but not this physics terms! I know and don't know why you close my question since my issue is too simple to answer. Could any one see the difference between the concept of common vector and heat flux vector? I had edited my ques again because I want it to open again.


title: Light scattering vs roughness

text: I've been searching for an answer to this question in professional circles for a while; now I'd like to ask here, to find out why the answer is not there :) In my optical modelling (computer simulation), I would like to take scattering on rough surfaces into account. Professional software (like Zemax) can do it, provided that I know the scattering model (e.g. in terms of Gaussian scattering). However I cannot find the model parameters anywhere. It is possible to derive the model parameters if one knows the BSDF of the surface, which in turn can be measured. But I cannot find any standard values, like for example scattering by aluminium surface with a roughness of Rz=5. Photopia seems to include the measured values for some surfaces, but the range is very limited, and it's the only such database I could find. Others, such as Disney's BRDF Explorer, seem to be less accurate and are probably not applicable for design optimization. At the same time, refractive indices of dozens of glasses are readily available. Specialists gave me reasons why such information cannot be provided: one has to assume ergodicity of the manufacturing processes, which is not necessarily there. Then I wonder: from the general point of view, does an aluminium surface with a defined roughness really look very different depending on who made it? Any comments or additional data on the topic would be highly appreciated. I was actually asked to suggest an optimal shape of some element for an experiment, and it turns out it should be scattering. However, while I can optimize scattering in simulation, I will not be able to convert it to roughness, i.e. tell the guys: "sand-blast it to Rz=x". This is what I am missing, and I cannot understand why the situation is as it is.


title: Photon gas kinetic theory

text: Suppose a black body as an enclosure of volume $V$ with a hole of section $A$. In the interior there is a photon gas, whose energy density $u$ is, at temperature $T$. $$ u=cT^4$$ How can I show that the energy emitted per second is $$E=\sigma A T^4$$


title: Can you recommend me a book on the discoveries and properties of these particles?

text: > **Possible Duplicate:**   >  Good book about elementary particles for high school students? Some days ago, I've found this: ![enter image description here](http://i.stack.imgur.com/7LHxQ.png) When I was in high-school, They taught me about electrons, protons and neutrons, I've heard about other particles while reading stuff but I've read only their name, not their properties. I got curious about them and then I thought about searching a book (It could be something like a pop science book) with commentaries on the discovery and properties of these particles.


title: Non-diagonal elements when switching metric signature?

text: Considering a metric tensor with the signature $(-,+,+,+)$: $g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\\ g_{10} & a^2 & g_{12} & g_{13}\\\ g_{20} & g_{21} & a^2 & g_{23}\\\ g_{30} & g_{31} & g_{32} & a^2\\\ \end{pmatrix}$ If I switch the signature to $(+,-,-,-)$, what is the form of the new tensor ? $g_{\mu\nu}= \begin{pmatrix} c^2 & ? & ? & ?\\\ ? & -a^2 & ? & ?\\\ ? & ? & -a^2 & ?\\\ ? & ? & ? & -a^2\\\ \end{pmatrix}$


title: Electromagnetic wave propagation through two lossless dielectrics

text: 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.


title: Ultrarelativistic limit of helicity values

text: 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.


title: Schroedinger equation. Why Potential energy instead of Force?

text: What is the reason Schroedinger equation is quoted in terms of potential energy instead of force?


title: Schroedinger equation. Mass. Charge

text: 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?


title: Two different mediums can have equal index of refraction?

text: In other words: given a certain medium, is light's phase velocity unique?


title: Axial symmetry constraints on the metric

text: I am reading the following paper (link) Here is a quote(s) from that paper > Throughout this paper, we shall suppose that the 4-space is axially > symmetric and reflexion symmetrical.... > > ...... From the axial symmetry, the azimuth angle is readily defined. > Suppose we now put a source of light at a point $O$ on the axis of symmetry > and surround it by a small sphere on which we can produce the azimuth co- > ordinate $\phi$ together with a colatitude $\theta$ and a time co-ordinate > $u$. We then define the $u$, $\theta$, $\phi$ co-ordinates of an arbitrary > event $E$ to be the $u$, $\theta$, $\phi$ co-ordinates of the event at which > the light ray $OE$ intersects the small sphere. In other words, along an > outward radial light ray the three co-ordinates $u$, $\theta$, $\phi$ are > constant. If we wish to write down the metric for such a system of co- > ordinates (in which the part referring to the azimuth angle $\phi$ appears > separately) then we know that since only the co-ordinate $r$ varies along a > light ray, the term $g_{11}$ of the metric tensor must vanish, the four co- > ordinates $u$, $r$, $\theta$, $\phi$ being denoted by 0, 1, 2, 3 in that > order. Moreover, we must have $$\Gamma^0_{11} = \Gamma^2_{11}$$ **QUESTION: Why do the above Christoffel symbols vanish?** EDIT: **QUESTION 2: What does one mean by reflexion symmetry?**


title: What is a geometrical object?

text: 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?


title: Energy of an EM wave compared to energy of a photon

text: 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


title: Dimension of galaxy filaments

text: Galaxy filaments are > amongst the largest known cosmic structures in the universe. They [...] form > the boundaries between large voids in the universe. As their name suggests filaments are essentially 1-dimensional objects (significantly longer and wider than thick, and significantly longer than wide), but how can such objects be the (very) boundaries of 3-dimensional voids? In the same Wikipedia article, "supercluster complexes", "great walls", and "great attractors" are proposed as synonyms for "filaments" which suggest other dimensions (d):   * unspecified for "complexes"   * d=2 for "walls"   * fractal (1 < d < 2 or 2 < d < 3) for "attractors" My question is: > What is/can be said about the dimension of these cosmic objects?


title: Black magic "Hartree" approximation

text: 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


title: Why does pizza cheese seem hotter than the crust?

text: When I eat hot pizza or a melted cheese sandwich, the cheese feels a lot hotter than the crust or bread: in particular, the cheese might scald the roof of my mouth. but the crust will not. Is this   * my imagination, or   * because the crust cools a little faster than the cheese, so has already cooled a bit by the time I eat it, or   * because the cheese cools a little faster than the crust, so transfers heat to the roof of my mouth a bit more, or   * what?


title: Meaning of spin

text: I'm pretty astounded that I did not hear about this sooner, but in my course on QFT our professor told us that the concept of spin can be used to mean three things:   1. Mechanical spin (apparently a relativistic effect giving rise to classical spin-orbit coupling)   2. Magnetic spin (purely quantum mechanical)   3. Classification of representations of the Lorentz group (the manner in which the particles transforms under Lorentz-transformations) I take it that these meanings don't generally coincide since they don't seem to do so in the case of the photon: we describe this as a spin-1 particle (3rd meaning), though it has no intrinsic magnetic moment (2nd meaning). However, despite being swiftly explained in a few words in class, I cannot remember what exactly the first meaning is about. Furthermore I'd like to find out the exact relations between these three meanings. As the case of the photon showed the last two at least don't generally seem to coincide. Can anyone clarify? * * * Related questions are:   1. What is spin as it relates to subatomic particles?   2. What are some useful ways to imagine the concept of spin as it relates to subatomic particles?   3. Is Angular Momentum truly fundamental?   4. What does spin 0 mean exactly?   5. Spin of a particle and spin quantum number   6. Why do many people say vector fields describe spin-1 particle but omit the spin-0 part? but the answers there don't make the light go on in my head.


title: Units and Dimensions - Use of proportionality constant

text: > In units and dimensions we learn about **Establishing a Formula** : (example) : to establish a relationship between T (Time Period) , m (Mass) , l (length of the string) and g(acc. due to gravity) - Case of Pendulum : > $$T \propto m ,\quad T \propto l ,\quad T \propto G$$ So : $T\propto m^al^bg^c $. By taking proportionality constant, we get : $T = k \space m^a \space l^b \space g^c $ By using homogeneity principle , I get : $$a = 0 \quad , b = \cfrac{1}{2} , \quad c = \cfrac{-1}{2} $$ > So I established the formula as : $$T = k \sqrt{\cfrac{l}{g}} $$ To complete the formula, we put k(proportionality constant) = $2\pi$ . My question is that how we got the value for proportionality constant? Also, is it fixed or it varies according to the equation?


title: Spectrum of Free Strings

text: > **Possible Duplicate:**   >  Spectrum of Free Strings As far as I understand, both in bosonic and superstring theory one considers initially a free string propagating through D-dimensional Minkowskispace. Regardless what quantization one uses at the end one arrives at a spectrum, where the excitations are classified among other things by the representation of the _Poincaré_ group (or maybe a cover) they are in, this allows to speak of their mass, but seems to make the construction very dependent on the background. Do massless modes remain massless in an other background, for example? And how does one define the mass of excitations in this case? Moreover D'Hoker states in his lectures in Quantum Fields and Strings, that he is not aware of a proof that in a general background a Hilbertspace can be constructed. Is there still none?


title: Why does charge conservation due to gauge symmetry only hold on-shell?

text: While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge symmetry: to my understanding, it's a redundancy in the "formulation" of a theory itself. So, shouldn't it lead to quantities which are conserved irrespective of whether the field equations hold?


title: Infinity of running couplings

text: 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?


title: What is a "free" non-Abelian Yang-Mill's theory?

text: I hope this question will not be closed down as something completely trivial! I did not think about this question till in recent past I came across papers which seemed to write down pretty much simple looking solutions to "free" Yang-Mill's theory on $AdS_{d+1}$. The solutions looks pretty much like the electromagnetic fields!   * Aren't classical exact solutions to Yang-Mill's theory very hard to find? Don't they have something to do with what are called Hitchin equations? (..I would be grateful if someone can point me to some expository literature about that..)  I would have thought that non-Abelian Yang-Mill's theory has no genuine free limit since it always has the three and the four point gauge vertices at any non-zero value of the coupling however arbitrarily small. This seemed consistent with what is called "background gauge field quantization" where one looks at fluctuations about a classical space-time independent gauge fields which can't be gauged to zero at will since they come into the gauge invariant quantities which have non-trivial factors of structure constants in them which are fixed by the choice of the gauge group and hence nothing can remove them by any weak coupling limit. But there is another way of fixing the scale in which things might make sense - if one is working the conventions where the Yang-Mill's Lagrangian looks like $-\frac{1}{g^2}F^2$ then the structure constants are proportional to $g$ and hence a weak coupling limit will send all the gauge commutators to zero!   * So in the second way of thinking the "free" limit of a non-Abelian $SU(N_c)$ Yang-Mill's theory is looking like an Abelian gauge theory with the gauge group $U(1)^{N_c^2 -1}$. So is this what is meant when people talk of "free" non-Abelian Yang-Mills theory? (..which is now actually Abelian!..)  I would grateful if someone can help reconcile these apparently conflicting points of view.


title: Spekkens Toy Model, Internal Comonoids

text: I have been thinking about Spekkens Toy model in terms of interfaces. The Spekkens paper concerns a physics based on only being able to receive answers to half the number of questions necessary to specify the state of a system. This is something like having a limited interface to some kind of system. I take an apparatus as an internal category in a monoidal category and the apparatus is seen as some limited interface to an underlying quantum causal structure. Would it be possible to reformulate Spekkens' idea in terms of internal categories?


title: How accurate is Newtonian Gravity?

text: I know that really fast moving things need Relativity rather than Newtonian physics. I also know the quirk of the Mercury´s orbit. But just how much more accurate is General Relativity than Newton´s Law of Gravitation for predicting say the orbit of Earth or Neptune? Can the "slingshot" effect where we use another planet´s gravity to accelerate a space probe be done with Newton or does that require General Relativity? Is the speed of Jupiter (18 km/s I think) fast enough to make a difference in the accuracy of GR v Newton´s Law of Gravity?


title: How much does electromagnetic radiation contribute to dark matter?

text: EM radiation has a relativistic mass (see for instance, Does a photon exert a gravitational pull?), and therefore exerts a gravitational pull. Intuitively it makes sense to include EM radiation itself in the galactic mass used to calculate rotation curves, but I've never actually seen that done before... So: if we were to sum up all the electromagnetic radiation present in a galaxy, what fraction of the dark matter would it account for?