When can a surface charge density exist?

In my syllabus about electromagnetism, they state: "This surface charge density will not always be present, e.g. when considering two non-conducting dielectrics such surface charge density remains absent. However, at a perfect conductor, a surface charge density will be present. If one of the media (or both) carry a conduction current, a surface charge can also be present (explain why!)" I don't really see though, how this can be explained. They seem to imply that there is a relation between the conductivity and the possibility of a surface charge, but I can't figure how they are related. They also state somewhere earlier that there can't be a surface current between two lossy dielectrics. This seems to be analogous to my previous question. I thought that this was due to the fact that dielectrics don't conduct current, but I wonder if someone knows a better explanation here too (that perhaps explains why they explicitly mention that the dielectrics are lossy).


Hamilton's equations for a simple pendulum

I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$ Now Hamilton's equations will be: $$\dot p=-mgR\sin\theta$$ $$\dot\theta=\frac{p}{mR^2}$$ I know one of the points of Hamiltonian formalism is to get first order diff. equations instead of second order that Lagrangian formalism gives you, but how can I proceed from here without just derivating again wrt. $\dot\theta$ and substituting $\dot p$ to get the same equation that I get with the Lagrangian formulation? Or is that the way to do it? And how could I get the path of the system on the phase space with those equations?


Can x-ray radiation be compared to background radiation?

I've been trying to learn about the possible effects of x-ray radiation from dental x-rays and most of the resources I come across compare the exposure to that of natural background radiation. Here's an example (not specific to dental x-rays): http://www.hpa.org.uk/webw/HPAweb&HPAwebStandard/HPAweb_C/1195733826941?p=1158934607708 I know electro-magnetic-radiation comes in different wavelengths and some are ionizing and some are not. Is this a valid comparison? **Update:** Thanks for the helpful responses so far. Looking at the chart here from @anna v, is comparing x-rays with background radiation comparing EMR that is 10^-10 with wavelengths that are longer than infrared? That's what I still don't understand. Wouldn't those different types of waves have different properties?


Most natural tensor structure for a quantum field

A quantum field is described by a Hilbert space. In many instances, the chosen tensor structure on this Hilbert space corresponds to that of space-like separated regions of space-time. The justification to this is that we can only perform local measurements, so the tensor structure should correspond to what subsystems operationally are, e.g. space-like separated regions of space-time. Does any one know a paper where the choice of a tensor structure for quantum fields is better motivated, and alternative tensor product structure proposed (for example in the context of relativistic quantum information, where space- like separated regions should not be the fundamental unit anymore since communication is generally allowed)?


Boltzmann entropy and phase space volume

In Huang's book on Statistical mechanism and Statistical interpretation of Entropy, it is not mentioned that $\Omega$ is the phase space volume, but it is the states of the system. So, how does entropy be called as logarithm of phase space volume? A detailed explanation and reference where it is shown that Entropy as phase space volume will be appreciated. Where can I find the derivation that entropy is the phase space volume?


Covariant derivative with upper index

I just need clarification, that is, to see that I'm doing the right thing. When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have some expression, that contains terms like: $$\xi_\sigma D^\nu h^{\mu\sigma}$$ Where $D_\nu$ should be covariant derivative. Now, can I just say: $$D^\nu=g^{\nu\mu}D_\mu$$ And similarly for the $h^{\mu\sigma}$, and use the standard definition of covariant derivatives?


Possible Deflection Distance For Falling Object

I work for a company selling tethers that prevent tools on building sites from falling away from the user. One of the problems that has come up is that objects can deflect horizontally when they strike an object whilst falling. You can imagine that they've fallen and then hit a steel girder or something on the way down. Given that the weight of the object, the height dropped from and the impact height are known, how would I calculate the _possible_ deflection distance?


Lie algebra - basis for adjoint matrix products in SU(N)?

In $SU(N)$, the set of matrices in the fundamental representation plus the identity: $$ \left\\{ \mathbf{1}, t^a \right\\} $$ acts as a basis for generator products under matrix multiplication, such that any combination of the elements in the set can be written as a linear combination, i.e. for any $$ X_F = t^{a_1}\dotsm t^{a_n} $$ we can write $$ X_F = c^0 \mathbf{1} + c^a t^a \, .$$ It is easy to show that $$ c^0 = \frac{1}{N} \text{tr} (X_F) \, , \quad c^a = - 2 i\, \text{tr} (X_F t^a)\, . $$ We cannot do the same for the adjoint representation because the set $$ \left\\{ \mathbf{1}, T^a \right\\} $$ doesn't span the full product space, e.g. $T^a T^b$ cannot be expressed solely in terms of $\mathbf{1}$ and $T^c$. So my question is: can we add - and if yes which - a minimal number of $n^2\\!-\\!1$ matrices $\\{K^a\\}$ to $ \left\\{ \mathbf{1}, T^a \right\\} $ in order to make it a closed set? **EDIT** Where it comes from is that because in the fundamental representation we have additional anti commutation rules $$ \left\\{ t^a, t^b \right\\} = \delta^{ab} \frac{\mathbf{1}}{N} + d^{abc} t^c \, , $$ we can take the mean of these and the standard commutation rules to write $$ t^a t^b = \frac{1}{2} \delta^{ab} \frac{\mathbf{1}}{N} +\frac{1}{2} h^{abc} t^c \,,$$ where $h^{abc}=d^{abc}+i f^{abc}$. We chain this relation to write for any product of fundamental generators $$ t^{a_1}\cdots t^{a_n} = A^{a_1\cdots a_n} \frac{\mathbf{1}}{N} + B^{a_1\cdots a_n b} t^b \, ,$$ with $A$ and $B$ some constants. This is the same statement as above. However we cannot do the same in the adjoint representation, because then the anti commutation rules don't close, i.e. they cannot be written as a linear combination of $\mathbf{1}$ and $T^a$. So the question is, can we add some extra matrices $\\{K^a\\}$ such that we can write the anti commutation rules as $$ \left\\{ T^a, T^b \right\\} = \delta^{ab} \frac{\mathbf{1}}{N} + d^{abc} T^c + c^{abc}K^c\, , $$ which would give $$ T^{a_1}\cdots T^{a_n} = A^{a_1\cdots a_n} \frac{\mathbf{1}}{N} + B^{a_1\cdots a_n b} T^b + C^{a_1\cdots a_n b} K^c\, .$$


2nd - 3rd Year Physics Labs

I'm looking to propose a new physics laboratory experiment, does anyone have any particularly fond memories of an experiment they carried out whilst at University? I'd like to include a section that makes use of Mathematica, Arduino or Python. The current labs are lacking computational aspects. The lab would ideally run over three 4 hour sessions.


Equation of motion in quantum system

The commutation relation between H and $L_i$ is $[H,L_i]$. I want to show the equation of motion. **Is this right: $ [H,L_i] = -i \frac{dL_i}{dt}?$**


Why must these Spinors be normalized?

I have just begun studying spin and there are two spinors mentioned: The main spinor $\chi $ and the spin-up spin down spinors (eigenspinors) $\chi_+ ,\chi_- $. I learned that the main spinor is a linear combination of the two spin spinors and I understand that the main spinor must be normalized. I don't understand however why the spin-up and spin-down spinors must also be normalized. As far as I understand, they are not involved in probability calculations for example: $\chi_+^{(x)} = \left( \begin{array}{ccc} \frac{1}{\sqrt2} \\\ \frac{1}{\sqrt2} \\\ \end{array} \right)$ instead of just $\chi_+^{(x)} = \left( \begin{array}{ccc} 1 \\\ 1 \\\ \end{array} \right)$ What am I missing?


Space-time curvature creates gravity or is it (could it be) vice-versa too?

Mass (Energy) creates space-time curvature and thus it forms the reason for gravity. Can it be vice-versa too? Like, mass created gravitational field, gravitational field created space-time curvature? I would like to agree, because, in that sense, I can say electromagnetic field also produces space-time curvature! This can lead to a unification between them! (that's worth a question)


Is cosmic background radiation dark-matter and/or dark-energy?

Dumb question alert: Is it possible that the cosmic background radiation might be the source of dark-matter and/or dark-energy? What is the mass of the background radiation in the known universe?


Temperature vs AC energy consumption

I'm need to understand the following: to keep the room at confortable temperature (70 degree, for example), how does the amount of energy consumed by the AC grow as the outside air temperature rises in the summer over the same length of time (a day, say)? Is it more or less linear? quadratic? exponential? or anything else? I posted the same question on another StackExchange site, but I figured that physicist may provide more help because it seems to me that it is related to both heat conduction and the working mechanism of AC. Thanks.


why isn't the variance of Fano noise proportional to the number of photon arrivals?

I've been reading "Photon Transfer" by James Janesick and in Chapter 3 (http://spie.org/samples/PM170.pdf) he describes the various sources of noise in a CCD. Basically, incoming photons interact with the CCD and cause some electrons to jump the bandgap. Both of these processes have some inherent randomness and must be described statistically - the photon arrival rate's variability gives rise to the shot noise, while the variability in electrons produced from a given photon impact gives rise to the Fano noise. The photon arrival is a Poisson process, so if m is the expected number of photon arrivals during a given exposure, the actual number of arrivals is a Poisson distributed random variable N(m), and var(N) = m. Now here's the part I don't understand. I quote from Janesick, editing slightly for clarity (p. 26): > If all the energy of an interacting photon was spent in the production of > electron-hole (e-h) pairs, then there would be no variation in the number of > e-h pairs produced. On the other hand, if the energy was partitioned between > breaking covalent bonds and lattice vibrations, or if phonon production was > uncorrelated, Poisson statistics would apply. But neither is the case in > nature. The variance in multiple electron-hole charge generation, called > Fano noise, is empirically described by > > $$ \sigma_{FN}^2 = F_F \eta_i = F_F \frac{h_v}{E_{e-h}} $$ > > where [$\sigma_{FN}^2$ is the variance of the Fano noise], and $F_F$ is > referred to as the Fano factor, which is defined by the variance in the > number of electrons generated divided by the average number of electrons > generated per interacting photon. [notation: $h_v$ and $E_{e-h}$ were > defined to be the photon energy in eV and the energy required to create an > electron-hole pair, and $\eta_i$ is the ratio of the two. So $\eta_i$ is the > expected number of electrons per incident photon, if all photon energy goes > into production of electrons.] Based on this description of the process, I would assume that each incident photon liberates an uncertain number of electrons, and therefore each incident photon should contribute a constant amount of variance. So I think the Fano noise variance should be **proportional** to the incident number of photons. But according to Janesick, the Fano noise contributes a constant variance **independent** of the number of photon arrivals. To be more mathematically precise, I think that the whole process should be modeled as a compound Poisson process, since both the number of arrivals and the number of electrons liberated per arrival are stochastic quantities. The wikipedia article on the compound Poisson process shows that the variance of the compound process has two components (one for the randomness of arrivals and one for the random size of the arrivals), and both components are proportional to the mean number of arrivals. So can anyone explain the discrepancy between my expectations and the book's equations? I feel I must have misunderstood what Fano noise is -- because if it's what I think it is, the natural model of it contradicts Janesick's equation. By the way, I know that $\sigma_{FN}^2$ is **not** referring to the variance per incident photon. If it were, then Janesick would multiply it by the mean photon arrivals when calculating the total image noise, but he doesn't. Here is his calculation of the total noise variance (equation 3.17, p. 34, irrelevant flat field term removed): $$ \sigma_{TOTAL}^2 = \sigma^2_{READ} + \eta_i F_F + \eta_i S $$ where $S$ is the expected number of electrons from the exposure. The first term is from shot noise, the second from Fano noise, the third is from shot noise. Note that the shot noise term scales linearly with $S$ but the Fano noise term is independent of $S$.


Should a radiation-filled Universe be scale invariant?

Imagine a spatially flat Universe, without cosmological constant, filled only with EM radiation. As Maxwell's equations without charges or currents are scale invariant then should this Universe be scale invariant as a whole? i.e. should there be no length scale associated with such a Universe? Friedmann's equation for such a Universe is conventionally given by: $$\left(\frac{\dot a}{a}\right)^2 \propto \frac{1}{a^4}$$ The solution is: $$a(t) = \frac{t^{1/2}}{t_0^{1/2}}$$ where $t_0$ is the current age of the Universe. Now let us calculate the particle horizon distance $D$ given by: $$D = \int_0^{t_0} \frac{dt}{a(t)}$$ We find that: $$D = 2 t_0$$ Therefore conventional theory says there is a length scale associated with this Universe. But is that reasonable? PS If for some reason we had $\rho \propto 1/a^2$ so that $a(t) = t/t_0$ then we could have a truly scale invariant cosmology where both the particle horizon and the cosmological event horizon diverge leaving no length scale.


What's wrong with my calculation of gravitational potential for a uniform sphere?

This is really embarrassing, but I'm not quite sure where I'm going wrong here... Why is this calculation of the gravitational potential inside a sphere with uniform mass distribution incorrect? **Set-Up** Let's say the sphere has mass $M$ and radius $R$ (and uniform mass density $\mu$), and what we want to find is the potential at any distance $r$ from the center of the sphere, where $r<R$. We normalize the potential to zero at infinity. **Calculation** The potential $\phi(r)$ is equal to the potential right outside of the sphere, plus the potential difference between some point inside the sphere and a point right outside. $$ \phi(r)=\phi_0-\int_R^r \frac{\mu G}{r}dV $$ (Sorry for using $r$ for the upper limit of the integral as well as for the variable in the integrand. Hopefully this doesn't cause confusion.) Now to figure out the different aspects of the above equation equation. The potential right outside of the sphere is: $$\phi_0=-\frac{MG}{R}$$ The differential volume element can be expressed as the constant-potential spherical shell's surface area times the shell's differential width: $$dV=4\pi r^2 dr$$ And one final detail, the mass density of the sphere: $$\mu=\frac{3M}{4\pi R^3}$$ Using this information, $$\phi(r)=-\frac{MG}{R}-\frac{3MG}{R^3}\int_R^r r dr$$ $$\phi(r)=-\frac{MG}{R^3}\left[R^2+\frac{3r^2}{2}-\frac{3R^2}{2}\right]$$ $$\phi(r)=-\frac{MG}{2R^3}(3r^2-R^2)$$ **Conclusion** This result disagrees with a few places I've visited, like this one, which states that the correct result (in terms of the variables I've used) is $$\phi(r)=-\frac{MG}{2R^3}(3R^2-r^2)$$ Both results give the same potential at $r=R$, obviously, but my result starts to look ridiculous for values like $r=R/2$. The only part in my calculation that seems sketchy to me is that first equation, where I talk about the potential difference at points inside and outside of the sphere; I don't know if it's correct to be dividing by $r$ in the integrand... Or maybe I just made a stupid algebra mistake somewhere in there. Where did I go wrong?


What happens to "large objects" within very small time periods?

What can we say about the state of "large things" within very small time periods? While reasonable (or "useful") divisions between classical and quantum physics are usually made in terms of size, can a similar division be made in terms of time? If so, what scale of time would that be? For example, suppose someone is "sitting at a desk". During very small time periods we cannot specify exactly where all the particles of the body are, right? Does that mean that the body (and everything else for that matter) is statistically "spread out" in small time periods?


Understanding EM Fields

I am an electronics engineering major and some questions arise when studying communications technology that utilizes wireless technology. In particular, I am more of a complete picture kind of person, as such I like to see how various things are connected. I want to understand, at least on the surface level, how the following things are connected: Phonons, photons, electrons, charge fields. As far as I understand things you have phonons. These are quasi-particles representing the quantized energy packets of vibrations through a lattice structure. This seems intuitive enough, and after some further reading I see that photons can be spontaneously emitted from photon interactions resulting from absorption and re-emission of the quantized energy of the phonons. I assume the photon emission frequency/energy is directly related to the energy of the phonon. I believe that photons can also cause phonons when a photon is incident to a lattice structure, though I am not 100% on the details of this. Photons are actual particles of quantized packets of energy propagating through the EM field medium. As to what exactly constitutes the EM field I am not sure. I believe magnetic fields are due to electrons? I also believe that electrons have an electric field due to the inherent charge electrons pose? However, I believe the charges reach/influence is based on the inverse square law. As such, is the electromagnetic field simply composed on the electron charges as the medium? If this is the case, then how can the EM field be so far reaching? I apologize for the large amount of questions, and I am sure there are similar questions around, but none that seem to tie all of these things together. EDIT: To add, am I correct to assume the EM Field is modeled as a continuous distribution of charge, as such the inverse square charge distance is a non- factor? And to answer one of the comments in the question, I have taken classes on partial differential equations, but my physics did not require them, thus I have only indirectly utilized Maxwell's Equations. I assume those will help connect a few of my questions mathematically, so I'll be sure to take a look at those.


Explanation: Simple Harmonic Motion

I am Math Grad student with a little bit of interest in physics. Recently i looked into the wikipedia page for Simple Harmonic Motion. Guess, I am too bad at physics to understand it. Considering me as a _layman_ how would one explain:   * What a Simple Harmonic motion is? And why does this motion have such an equation $$x(t)= A \cos(\omega{t} + \varphi)$$  associated with it? Can anyone give examples of where **S.H.M.** is tangible in Nature.


Common false beliefs in Physics

Well, in Mathematics there are somethings, which appear true but they aren't true. Naive students often get fooled by these results. Let me consider a very simple example. As a child one learns this formula $$(a+b)^{2} =a^{2}+ 2 \cdot a \cdot b + b^{2}$$ But as one mature's he applies this same formula for Matrices. That is given any two $n \times n$ square matrices, one believes that this result is true: $$(A+B)^{2} = A^{2} + 2 \cdot A \cdot B +B^{2}$$ But eventually this is false as Matrices aren't necessarily commutative. I would like to know whether there any such things happening with physics students as well. My motivation came from the following MO thread, which many of you might take a look into:   * http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics


Current in a circuit with resistors and capacitors

Suppose we have a circuit with both resistors and capacitors and some voltage source. If we want to calculate the current in the circuit, do we apply Ohms law? My doubt is that this would only take into consideration the resistors and not the capacitors, so do the capacitors play any role in the current flowing through the circuit?


Are states from an unpolarized source beam distinguishable or not

I have seen this pop up twice so far in my reading of Feynman Lectures III. For example, in the first equation of section 5-8 (it may take up to a minute for the equation to load), where the beam is ejected from a furnace and so states are randomly produced, the probablities are added rather than first interfering the probability amplitudes and taking absolute-squared. But I thought nothing was done to distinguish the states $+S, 0S, -S$. Moreover, in the situation of the fancy "extended" Stern-Gerlach apparatus where I split the beam into three beams according to the $S$ states but recombine them without blocking any of the split beams, I perform future calculations by interfering the three amplitudes. I am convinced here that I have to do interference. But why is the first paragraph different?


Special Relativity Time problem

Suppose there are two observers, they each own a light clock which were synchronised in the beginning and are moving relative to each other. So each of them accuses that the other person's clock is running slow. Now they see a concrete event, like someone clapping or a bulb shining and note down the reading on their clock and send each other the information, now if the two pieces of information were different, which means one of the clock was actually running slow, so they will know who was moving actually. If the readings are same, then how will you justify Lorentz-Transformations for time ?


Can dark matter be relativistic dust?

As far as I know the mass of an observed object increases as it approaches the speed of light. Is it possible that the excess mass called "dark matter" is due to relativistic dust? Surely, stars cannot account for that excess mass since we can measure their speeds and they are not relativistic. However, can we observe and measure the velocity of dust in the galaxy? Maybe the dust could account for missing matter...


Why can we simply absorb the positive coefficient of $i\epsilon$ in a propagator?

As far as I know, absorbing of the positive coefficient of $i\epsilon$ in a propagator seems to be a trivial operation without even the need of justification. In Peskin page 286, he did this: $$k^0\rightarrow k^0(1+i\epsilon)$$ $$(k^2-m^2)\rightarrow (k^2-m^2+i\epsilon)$$ In M. Srednicki's Quantum Field Theory, page 51, > The factor in large parentheses is equal to > $E^2-\omega^2+i(E^2+\omega^2)\epsilon$, and **we can absorb** the positive > coefficient in to $\epsilon$ to get $E^2-\omega^2+i\epsilon$. Why and does this kind of manipulation affect the final result of calculation?   Although $\frac{1}{k^2-m^2+i\epsilon k^2}-\frac{1}{k^2-m^2+i\epsilon}$ is infinitesimal, but the integration of such terms may lead to divergences, and this is my worry. Also the presence of $k^0$ in the coefficient of $i\epsilon$ could potentially influence the poles of an integrand and consequently influence the validity of Wick Rotation.


What is the function of the top point of a bouncing ball?

![enter image description here](http://i.stack.imgur.com/fqSMt.png) A ball is thrown away as parallel to x axis from M(0,h) point with speed V . After each jumping on x axis , it can reach half of previous height as shown in the figure.(Assume that no any air friction and the ball is very small.) Question-$1$: Find f(x) that passes from of top points ? Question-$2$: I have just a claim about the system but I need to prove that if we throw away the ball from a point of $f(x)$ as parallel to x axis with the speed of $V_{initial}=V\sqrt{\frac{f(x)}{h}}$, the maximum points of the ball will be on same $f(x)$. How can the claim be proved or disproved? * * * My attempt to find f(x) $f(0)=h$ $f(x_1+x_2)=h/2$ $f(x_1+2x_2+x_3)=h/4$ $f(x_1+2x_2+2x_3+x_4)=h/8$ . . . $x_1=vt$ $mgh=\frac{1}{2}mV^2$ $h=\frac{1}{2}g t^2$ After first bouncing , On first top point: N $(x_1+x_2,h/2)$: $mgh/2=\frac{1}{2}mV_1^2$ $V_1^2=V^2/2$ $V_1=\frac{V}{\sqrt{2}}$ $h/2=\frac{1}{2}g t_2^2$ $t_2^2=t^2/2$ $t_2=\frac{t}{\sqrt{2}}$ $f(x_1+x_2)=h/2$ $f(Vt+t_2\frac{V}{\sqrt{2}})=h/2$ $f(Vt+\frac{t}{\sqrt{2}}\frac{V}{\sqrt{2}})=h/2$ $f(Vt(1+1/2))=h/2$ After second bouncing, On second top point: O $(x1+2x2+x_3,h/4)$: $mgh/4=\frac{1}{2}mV_2^2$ $V_2^2=V^2/4$ $V_2=\frac{V}{2}$ $h/4=\frac{1}{2}g t_3^2$ $t_3^2=t^2/4$ $t_3=\frac{t}{2}$ $f(x_1+2x_2+x_3)=h/4$ $f(Vt+Vt+V_3t_3)=h/4$ $f(Vt+Vt+Vt/4)=h/4$ $f(Vt(1+1+1/4))=h/4$ On 3th top point: $f(Vt(1+1+1/2+1/8))=h/8$ On 4th top point: $f(Vt(1+1+1/2+1/4+1/16))=h/16$ . . On nth top point: $n--->\infty$ $f(3Vt)=0$ How can I find $y=f(x)$ Thanks for answers * * * EDIT: I have found $f(x)$. MrBrody is right. The function is a line. my Proof: On nth top point: $f(Vt.a(n))=h/2^n$ $a(0)=1$ $a(1)=1+1/2=1+1-1/2$ $a(2)=1+1+1/4=1+1+1/2-1/4$ $a(3)=1+1+1/2+1/8=1+1+1/2+1/4-1/8$ $a(4)=1+1+1/2+1/4+1/16=1+1+1/2+1/4+1/8-1/16$ . . $a(n)=1+1+1/2+1/4+....1/2^{n-1}-1/2^{n}$ $a(n)=1+\frac{1-(1/2)^{n})}{1-1/2}-1/2^{n}=1+2(1-(1/2)^{n})-1/2^{n}=3-\frac{3}{2^n}$ On nth top point: $$f(Vt[3-\frac{3}{2^n}])=\frac{h}{2^n}$$ where $t=\sqrt{\frac{2h}{g}}$ $Vt[3-\frac{3}{2^n}]=x$ $3-\frac{3}{2^n}=\frac{x}{Vt}$ $1-\frac{1}{2^n}=\frac{x}{3Vt}$ $1-\frac{x}{3Vt}=\frac{1}{2^n}$ $$f(x)=\frac{h}{2^n}=h(1-\frac{x}{3Vt})=h(1-\frac{x}{3V\sqrt{\frac{2h}{g}}})$$ Now I am trying to prove or disprove my claim in second question. If I find the solution I will post it.


Finding out the potential

![enter image description here](http://i.stack.imgur.com/v5LA9.png) > According to me, if we want to find out the potential the the equation will > be, $$dV = \int \frac{dQ}{4 \pi \epsilon_0 x}$$. But the answer is given is on the basis of $$dV = \int \frac{dQ}{4 \pi \epsilon_0 \sqrt{(R^2+ x^2)}}$$. > Why the the distance x from the center of the radious is not allowed to get > the potential?


is it possible to have magnetic flux density B not in the same direction of magnetic field intensity H?

it is said that direction of magnetic flux density B in the same direction of magnetic field intensity H for isotropic media so what is isotropic media and is it possible to have B not in the same direction of H


What is a "measure equation" as mentioned by this TeX Users Group guide?

In this TeX Users Group (TUG) document, Typesetting mathematics for science and technology according to ISO 31/XI by Claudio Beccari, the author makes various typesetting recommendations including: > 9\. Measure equations should be absolutely avoided in professional > scientific texts; measure equations were somewhat popular before the SI was > universally adopted; now they should not be used any more. They survived in > those countries where the "English system of units" is being used, but, > since scientifically speaking this traditional system of units is "illegal", > measure equations have no reason to be used anymore. What are these "measure equations" he talks of? I take it that should he write say $7.25\,\text{cm}$, he would call $7.25$ the "measure" and $\text{cm}$ the "unit of measure". From this and the context of the quoted paragraph I would guess that a measure equation is a relationship between different units of the same quantity, such as " $1\,\text{in}\equiv2.54\,\text{cm}$ ". However I find such things handy, or at least not unnecessary or obsolete, especially when I'm working on atomic scales where SI isn't the most convenient. Is this what the author means and if so, are they really that bad?


Why are scattering matrices unitary?

In Griffith's QM book, he introduces scattering matrices as an end-of-the- chapter problem. For a Dirac-Delta potential $V(x) = \alpha \delta (x - x_0)$, I've derived the scattering matrix and observed that it is unitary $S^{-1} = S^{\dagger}$. I'm trying to explain why this is intuitively, but I don't really have an intuitive picture of what hermitian conjugation $S^{\dagger}$ is doing here. Thoughts?


Experimental evidence showing the kinetic energy of an electron changes in a static non-uniform magnetic field?

In a previous question, Does a magnetic field do work on an intrinsic magnetic dipole?, one highly rated answer suggested that static magnetic fields do work on intrinsic magnetic dipoles in a non-uniform magnetic field. I can visualise the change in kinetic energy of the nucleus of an atom coming from a change in the configuration of the electrons around the nucleus. But for an electron, since it's truly fundamental, I'm scratching my head over where the energy comes from to change its kinetic energy. If it does, then it really must come from the static magnetic field. So what is the experimental evidence that shows the kinetic energy of an electron changes in a static non-uniform magnetic field?


Michelson rotating mirror experiment

Could someone explain the calculation required to answer this question. It is from a text book and the answer is recorded as 585Hz but I cannot replicate the answer. > In 1931 Michelson used a rotating prism with 32 faces to measure the time > for light to travel through a 1.6km evacuated tube to a distant mirror and > back. From his results he calculated that the speed of light is 299774000 > m/s. > > Assuming the prism is rotated by only one face during the time of flight of > light calculate the rotation frequency of the prism. Is the book incorrect or am I? My attempt: $(1/(1600/299774000))/32=5855\:\mathrm{Hz}$ which is suspiciously similar to the suggested answer 585Hz. But then I remembered about the round- trip (3200m) which threw all the numbers again.


Renormalizing with external momenta set to zero

I've often seen in textbooks that authors renormalize diagrams by setting external momentum to zero. Under what conditions is this justified? An example of this is done in Manohar and Wise's book on Heavy Quark Physics after they renormalize QED and then calculate the operator renormalization, $ Z _S $, of \begin{equation} S = \frac{1}{ Z _S }\bar{\psi} _b \psi _b = \frac{ Z _\psi }{Z _S } \bar{\psi} \psi \end{equation} where $ \psi_b $ is the bare field. They calulate this through the diagram, $\hspace{5cm}$ ![enter image description here](http://i.stack.imgur.com/wWqzl.png) where the cross indicates an operator insertion. The authors then say ``The operator, $ S $, contains no derivatives (and $Z _S$ is mass independent in the $\overline{MS}$ scheme), so $Z _S$ can be determined by evaluating (the diagram) at zero external momentum (and neglecting the (fermion) mass).'' Are these the two conditions necessary,   1. The operator doesn't have derivatives   2. The quantity of interest has no mass dependence and if so how do we know that the quantity you want to calculate (in this case, $ Z _S $) is mass independent, ahead of time?


Nuclear fission mechanism: neutron capture

Consider this nuclear fission reaction: $\mathrm{^{235}U+{}^1n\to{}^{236}U \text{ (excited)}\to{} ^{92}Kr+{}^{141}Ba+3{}^1n}$ I have not understand why a thermal neutron ($^1$n with $E\simeq 0.025$ eV) has enough energy to start the reaction. I mean, to start the reaction the $^1$n should be captured by the nuclei of $^{235}$U. What are the conditions under which this neutron is captured? My reasoning is the following: the energy that the neutron must have to be captured by the $^{235}$U should be equal to the $S_\mathrm n$ (separation energy for a neutron) of $^{236}$U which I found is 6.34 MeV, higher than the energy of the thermal neutron. So my reasoning must be wrong but I cannot understand where I make mistakes... I looked on the Povh Rith Particles and Nuclei but it does not explain the reaction well. Could anyone give me a reference or explain here the fission reaction ?


phase transition by sublimation

Why do sublime solids sublimate in the first place? Is it that their melting point and their liquid's boiling point are the same so that they manage to magically skip the liquid phase? (if so, why does it have to be so?)And is the reverse possible?


Lagrangian and Equilibrium Points

I'm wondering whether you can tell quickly just from looking at a Lagrangian whether a given point $q^0$ is an equilibrium point. Obviously all you have to do is verify it satisfies the E-L equations, but they are messy in complicated situations. Is there a conceptual shortcut I could take, or am I expecting too much? Many thanks in advance!


Why does absorption cause seismic pulses to increase in length over distance?

The specific question I'm trying to answer is "How does the progressive loss of higher frequencies in a propagating seismic pulse lead to an increase in pulse length?" I understand how the higher frequencies attenuate faster than lower frequencies (through the absorption constant), thus resulting in a change of shape but not so much how this increases the pulse length? I've been researching for a while and have come across multiple ideas including: 1) I've read that in non-porous media velocity = freq x wavelength, as the frequency components reduce in value, the wavelength would increase to balance this? Though I'm not sure I can assume the media is non-porous. 2) I've been reading about wave dispersion which discusses how higher frequency components move faster than lower frequency components in dispersive media, thus the wavelength increases. My issue with this though is that the question specifically refers to the loss of higher frequencies influencing the pulse length, and just talking about different frequencies having different velocities doesn't seem to cut it. 3) Geometrical spreading could be increasing the wavelength? Though again this doesn't really relate to attenuation as the question is asking. Any help letting me know what I'm missing here would be fantastic. This question has been bugging me for a couple of days now. Cheers


Trying to understand Laplace's equation

I'm struggling here so please excuse if I'm writing nonsense. I understand that the gravitational potential field, a scalar field, is given by $$\phi=\frac{-Gm}{r}$$ where $\phi$ is the gravitational potential energy of a unit mass in a gravitational field $g$ . The gradient of this is (a vector field) $$g=-\nabla\phi=-\left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right)$$ And the divergence of this vector field is $$\nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho$$ and is called Poisson's equation. If the point is outside of the mass, then $$\rho=0$$ and Poisson's equation becomes $$\nabla\cdot\nabla\phi=0$$ My question is, how do I express $\phi=\frac{-Gm}{r}$ as a function of $x,y,z$ so I can then end up with $\nabla\cdot\nabla\phi=0$ in empty space $r\neq 0$? I would have thought that I could write $$\phi=\frac{-Gm}{r}=\frac{-Gm}{\sqrt{x^{2}+y^{2}+z^{2}}}$$ but when I try to calculate $\nabla\cdot\nabla\phi$ from this, I don't get zero for $r\neq 0$. I'm confused.


Give a description of Loop Quantum Gravity your grandmother could understand

Of course, assuming your grandmother is not a theoretical physicist. I'd like to hear the basics concepts that make LQG tick and the way it relates to the GR. I heard about spin-networks where one assigns Lie groups representations to the edges and intertwining operators to the nodes of the graph but at the moment I have no idea why this concept should be useful (except for a possible similarity with gauge theories and Wilson loops; but I guess this is purely accidental). I also heard that this spin-graph can evolve by means of a spin-foam which, I guess, should be a generalization of a graph to the simplicial complexes but that's where my knowledge ends. I have also read the wikipedia article but I don't find it very enlightening. It gives some motivation for quantizing gravity and lists some problems of LQG but (unless I am blind) it never says what LQG actually is. So, my questions: >   1. Try to give a simple description of fundamentals of Loop Quantum > Gravity. >   2. Give some basic results of the theory. Not necessary physical, I just > want to know what are implications of the fundamentals I ask for in 1. >   3. Why is this theory interesting physically? In particular, what does it > tell us about General Relativity (both about the way it is quantized and the > way it is recovered from LQG). >


How can you calculate air resistances at different speeds?

I've read that at 50mph air resistance to an average car is the equivalent of driving through water and at 80mph it's the equivalent of driving through oil. I can't find any references online to back up these figures. Is there a relatively simple way to explain the calculation to verify these figures, or are they incorrect?


Is the thermal time constant simply the heat capacity divided by thermal conductivity?

My professor gave me this temperature decay data to make an exponential fit and told me that the time constant tau should be heat capacity over thermal conductivity. I'm looking at the wikipedia page and that's not what I'm seeing. I don't know if there's some caveat to when we're doing 2D simulations but I just don't see how the time constant could be exactly heat capacity over thermal conductivity.


Physical interpretation of the change of diffusion term in navier stokes equations

In the Navier-Stokes Equations, there is one term accounting for convective flow and one term for diffusive flow. At high flow rates, the diffusive term becomes much smaller compared to convective term, therefore can be neglected, leading to Euler's equations. I can understand this in terms of mathematics: There is a multiplicative factor $\frac{1}{\text{Re}}$ in the diffusive term. With increasing flow rates, the inertial forces become larger than the viscous forces, so the Reynolds number increases and the term for diffusive flow becomes much smaller. But what is the physical interpretation of this? Does the influence of random motions decrease when the flow rate increases?


Lenz's Law and Eddy Currents

You can determine the direction of eddy currents according to Lenz's law. E.g. If a metal sheet is losing flux into the page, it will experience induced eddy currents in a clockwise direction to replace it. However, this doesn't make sense to me logically. :S Consider a straight current-carrying wire. It has equal flux on either side of it's length. Now warp it into a circle (to simulate eddy currents). It should have equal flux on both outside and inside the the circle. Thus, in the above example, the eddy currents ARE NOT replacing the lost flux into the page, as it is also producing EQUAL flux out of the page (which "cancels" the other flux into the page). Am I misunderstanding something about Lenz's Law/Eddy currents? :S


Why is honeycomb lattice the ideal shape for solar cells?

Quoting this web site (which I haven't thoroughly verified for credibility yet): > The material is graphene, also known as graphite, a naturally-occurring > mineral that forms a one-atom-thick sheet of honeycomb crystal lattice > material (the ideal shape for solar cells) that has surprisingly robust > electric characteristics. In what context is this ideal shape? In the nano or macro scale? Why?


How could I determine the temperature at which an egg explodes

A while back, some friends and I pondered "what will happen to an egg if placed very near a campfire". So we placed an egg on a grill, approximately 12 inches above a real campfire. Something like 7-10 minutes later, thankfully right as we walked away, the egg exploded in every direction in a gooey, golden glory. So I ask you, **how can I determine the internal temperature of an egg that will cause it to explode**?


Did a racer on a "restless" course race at some particular average velocity? (2: Starting gate and finish line being rigid to each other)

Consider two participants, a starting gate ($A$) and a finish line ($B$), in a (sufficiently) flat region, both undergoing uniformly accelerated motion, such that   * $A$ finds constant ping duration wrt. $B$: for any two (distinct) indications $A_J$ and $A_K$   $\mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }B}}} \\! \\! \\! \\! \tau_A[ A_J ] = \\! \\! \mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }B}}} \\! \\! \\! \\! \tau_A[ A_K ] = \\! \\! ({\small{\text{abbreviated as}}}) \\! \\! = \mathop{\Delta}\limits_{\small ABA} \\! \tau_A,$   * $B$ finds constant ping duration wrt. $A$: for any two (distinct) indications $B_P$ and $B_Q$   $\mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }A}}} \\! \\! \\! \\! \tau_B[ B_P ] = \\! \\! \mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }A}}} \\! \\! \\! \\! \tau_B[ B_Q ] = \\! \\! ({\small{\text{abbreviated as}}}) \\! \\! = \mathop{\Delta}\limits_{\small BAB} \\! \tau_B,$ and   * $A$'s ping duration wrt. $B$ is strictly less than $B$'s ping duration wrt. $A$: $\mathop{\Delta}\limits_{\small ABA} \\! \tau_A < \mathop{\Delta}\limits_{\small BAB} \\! \tau_B.$  (Evidently, therefore, $A$ and $B$ had not been at rest to each other; but they are rigid to each other.) Let there also be a racer ($R$) who passed the starting gate $A$ and (afterwards) passed the finish line $B$. For definiteness, suppose that $B$'s duration from having seen that $A$ and $R$ had passed each other until having been passed by $R$ equals $B$'s ping duration wrt. $A$.   (Consequently, $A$'s duration from having been passed by $R$ until having seen that $B$ and $R$ had passed each other equals twice $A$'s ping duration wrt. $B$.) **Question:** Can an "average velocity" of racer $R$ with respect to the race course, as far as it is specified by starting gate $A$ and finish line $B$ be defined, such that its magnitude could be expressed in terms of the given ping durations $\mathop{\Delta}\limits_{\small ABA} \\! \tau_A$, $\mathop{\Delta}\limits_{\small BAB} \\! \tau_B$, and the "speed of light, $c$"   (and where its direction is obviously "from $A$ to $B$") ?


Anti-Matter for Neutrons

The anti-particle corresponding to a proton or an electron is a particle with an equal mass, but an opposite charge. So what is the anti-particle corresponding to a neutron (which does not possess a charge)? And if it is just another neutron, will its collision with the original neutron be as destructive as the collision of a proton with an anti-proton or an electron with an anti-electron?


List of Scattering Phenomena

While at lunch with my lab group we got into discussion of the different types of scattering phenomenon that we encounter in everyday life and physical experiments. We ended up listing about a dozen or so types of scattering that we see constantly, but I left unfulfilled. I want to compile a list of scattering phenomenon and a brief description. At the very least this will be helpful for those taking the physics GRE in the near future, otherwise its a good exercise for anyone interested in physics, even those in professional physics. I'll Begin to give you the basic jist of what I'm looking for, feel free to elaborate on the description as much as you want. An equation is useful if it's easily expressed **Compton scattering** : Scattering of a photon off of a particle, photon imparts momentum to the particle and loses some of its energy, the change in wavelength is described by $$\Delta_{\lambda} = \frac{h}{mc}(1-\cos{\theta})$$ It is independent of the frequency of the incoming photon and inversely proportional to the mass of the particle scattered from. **Raleigh Scattering** : Elastic Scattering of photons off of molecules and other (micro and macro) particles, photons do not lose energy but change direction. Has a strong dependence on frequency and is the reason the sky is blue (and why you can see laser pointer beams in the dark). P.S. I don't have the reputation to create tags yet, and there are no tags relevant to this question (big-list scattering) please feel free to retag as necessary


A Musical Pathway

Using a small number of sound emitters, could you create a room where certain nodes emitted particular tones, but no meaningful sound was heard anywhere else. So, for example, by walking down a certain path, you could hear the tones for "Mary Had a Little Lamb." Is there a generalized algorithm to make particular paths for particular tone sets?


Shouldn't stars disappear from our view due to expansion?

The currently accepted theory of the expansion of the universe would suggest that for any point in the universe, there is a 'boundary' (somewhat like the Schwarzschild radius) beyond which it cannot see the stars that may be there. (since beyond that radius the expansion must appear to be faster than light). However this also implies that stars that are currently visible should disappear when the expansion takes them across that boundary (as some must since the expansion is ongoing). Now even allowing for a very long delay in this being observed from earth say, it still means that after billions of years we must be seeing stars "dropping" out of our visibility radius. Is this an observed phenomenon? or am I misunderstanding something?


How do you build a simple low cost single photon detector?

Rather than buying a 4K-10K single photon detector, I was wondering if there were simple ways to build a simple single photon detector, of decent quality...


Why do ice cubes stick together or to the edges of a drinking glass?

I was drinking iced-water from a drinking glass (made of glass) at a restaurant yesterday when I was taking a drink, I noticed that there is very little ice water coming out and then suddenly, the ice water mixture comes crashing down. With my open mouth overfilled with ice water, my shirt gets wet – I hate when this happens. After drying myself off, I started thinking why this does happen:   1. How does ice that starts off loose and separated in water fuse together to form one large clump?   2. What causes the ice-water mixture to stop flowing momentarily and then suddenly starting flowing again? Is it that the ice fuses together in the glass and forms one large clump, and then the friction with the edges builds a "dam-like" situation? Or are there individual pieces of ice fusing with the edges of the drinking glass damming-up the ice?


Will a diamond helm be more protective against bullets?

Will a helm made entirely of diamond be more protective against bullets than standard steel or kevlar helmlets?


What to show a beginner

At my new job, it's soon going to be my turn for doing night/graveyard shifts for a fair amount of weeks. Perfect excuse to buy a decent beginners telescope to replace the 4.5" 15 y/o Meade that doesn't work anymore with a 10" Dobsonian reflector! My girlfriend wants to join me (which I'm all for), and having grown up on a (relatively poor) farm, she's never had the pleasure of looking through a telescope. When I asked her what she wanted to look at first, she shrugged and said "I don't know. Show me everything!" The problem is, the stuff I generally stare at are Nebulae and star clusters (since that's all I really could see), despite knowing that there is so much more out there that are equally enticing to the eye. So my question is this: For any category of objects, what are the most popular? If you have gone to many star parties, what are the objects people are generally asking to see most of the time? What were the things that when you first saw them through a telescope made you go "Wow!"? Note: Do understand that the objects generally have to be viewable during Late September (new moon), but having new objects I can show her in the coming months would be awesome as well. I've already told her the journey of the grand tour will probably take 6 months to a year (due to earth rotation and all that,) and will require the joining of us to the local astronomy club.


Why does spin have a discrete spectrum?

Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) _continuous_ values, spin has a (probabilistic) _discrete_ spectrum?


Measure mass difference of an object without a scale

For a project i need to separate items based on their mass in real time. I like to explore measuring weight of an object (round metal ball) by taking photographs (several) during free fall and calculate mass from the pictures. Even if I cannot find the real mass I like to get a good estimate of the mass delta among many samples in real time. The basic logic is that I can measure velocity and acceleration from images with good accuracy. And use this data to calculate approximate mass. The density of the object is not controlled tightly neither the volume. Does this approach make sense?


Euler number of the world sheet

I have a question in the section 3.2 "The Polyakov path integral" in Polchinski's string theory p. 83. Given > $$ \chi=\frac{1}{4 \pi} \int_M d^2 \sigma g^{1/2} R + \frac{1}{2 \pi} > \int_{\partial M} ds \,\, k \,\,\,\, (3.2.3b) $$ It is said > We have noted $\chi$ is locally a total derivative in two dimensions and > therefore depends only on the topology of the world-sheet -- it is the Euler > number of the world sheet I have two questions related to this sentence... 1) Where did the book note that $\chi$ is locally a total derivative? 2) Does the Euler number for $\chi$ come from the Gauss-Bonnet theorem? I never learned that theorem in my differential geometry course :( and Eq. (3.2.3b) is in different form with wiki   by different factors $\frac{1}{4\pi}$ and $\frac{1}{2\pi}$ (why? convention of normalization?). Would you recommend any reference for the proof of Gauss- Bonnet theorem for physicist?


How to calculate the van der Waals force from the van der Walls equation?

Given the van der Waals equation $$\left(p+\frac{n^2a}{V^2}\right)\left(V-nb\right)=nRT$$ and the van der Waals constants $a$ and $b$, how can I find the van der Walls force between two atoms at distance $d$?


How do I calculate the perturbations to the metric determinant?

I am trying to calculate sqrt(-g) in terms of a background metric and metric perturbations, to second order in the perturbations. I know how to expand tensors that depend on the metric, but I don't know how to expand the metric determinant. Does anyone know how to to this or (even better) know of a source that does second order perturbation theory in GR?


Can an asteroid have a magnetic field?

Some M-type asteroids are believed to be metallic, even possessing Nickel- Iron. Given   * the proximity of the asteroid belt to Jupiter's magnetosphere,    * the regularity of motion of any given M asteroid relative to Jupiter   * the ease with which Nickel/Iron may be magnetized, and    * the strength of Jupiter's magnetic field Is it reasonable to expect an M-type asteroid that fulfills such conditions to exhibit a magnetic field?


Energy conservation at a the end of a hose pipe?

I am trying to figure out the energy balance at the end of a hose pipe with water pouring out. Applying Bernoulli's equation inside the pipe we have three energy terms: Kinetic energy, "pressure" energy and gravitational potential energy. This total energy is conserved unless its converted to heat or whatever through pressure losses. Consider the situation just inside and just outside the end of the hose pipe. Assume the pipe is horizontal so we can ignore the gravitational component. Let say just inside the end of the hose we have 50psi and 2 feet per second. Then consider the situation just outside the end of the hose. We still have 2 feet per second because the fluid stream has a constant diameter (its not a nozzle) and is virtually incompressible. However, we now presumably have atmospheric pressure in the fluid stream, i.e, 14.7psi. So what happened to the energy contained in the pressure difference, i.e, the difference in energy between the water at 50psi and at 14.7psi? Or is the pressure inside the fluid stream suspended in the air still at 50psi? Or does the water expand very slightly to account for the difference between 50psi and 14.7psi and the energy is released by the water doing work against the air to perform this expansion? (And maybe the expansion is so small that it is not visible so the fluid stream appears to be a constant diameter?).


is an electron gun dangerous?

Is an electron gun dangerous & what would be the repercussions to human tissue? If I were to do a DIY electron gun and put my hand in the beam would I regret it?


Clarifications about Poisson brackets and Levi-Civita symbol

I need some clarifications about Poisson brackets. I know canonical brackets and the properties of Poisson Brackets and I also know something about Levi-Civita symbol (definition and basic properties), but I have some doubts. I don't know how I could apply Poisson brackets properties if I have a summation, for example if in the case of a system of N particles I have to solve $[L_i, x_{\alpha j}]$. I know that a generical component of total angular momentum is given by $L_a=\sum_{\alpha=1} ^N l_{\alpha a}$ and also know that the components of algular momentum af a particle is given by $l_a= \epsilon_{aij} x_i p_j$. Now, if I have to calculate $[L_i, x_{\alpha j}]$, I have these doubts: 1) how can I decide the indices of Levi-Civita symbol that I'm going to use for solving the problem? 2) how can I use the property of linearity of Poisson brackets in this case? and an other (general) question: 3) If I have a Levi-Civita symbol that multiplyes a sum of two terms and each term is mulplied for a Kronecker delta, I have to follow these steps: a) multiply the Levi-Civita symbol for each term b) impose the condition thanks to each Kronecker delta isn't equal to zero c) eventually, substitute these conditions in the two Levi-Civita symbol, but I have to substitute in each Levi-Civita symbol the condition that I found for that Kronecker delta that at the step a) was multipling just that one Levi- Civita symbol Is it correct this way to go on?


Good references for specific heats

Does anyone know of a good reference for specific heats? It seems that it is no longer popular to publish values. I am especially interested in the specific heats of polymers.


When should we use the concept of Impulse/Momentum instead of Force?

In my notes it says > "The ideas of impulse and momentum is useful in solving problems where:- > > a) the force F is not easily calculable (e.g. sudden impact or blow) > > b) the impulse force is so large that the body suffers deformation (i.e. the > impact is no longer elastic) There is no explanation for what is said. When I think about it, force is the rate of momentum. How/why could we calculate momentum for small amount of time (sudden impact), but not calculate force?


Are scientists missing the point with distant cosmic objects, or is it just me?

I was reading this article this morning: http://www.bbc.co.uk/news/science-environment-13539914 Scientists have discovered a gamma-ray burst whose light has taken 13.14 billion years to reach Earth. This much is cool and interesting. However, the assumption is then stated that this is "the most distant single object yet spied by a telescope". But hold on a minute. It is also known that galaxies are moving away from each other at incredible speeds, most faster than the speed of light, because the space/time between them is expanding (or something like that... I'm no scientist!) So these so-called amazingly distant objects, well for starters, they don't exist any more... They are things that happened billions of years ago. But not only that, they are probably objects that were only a few million light years away from us when they actually took place. So surely then, the objects themselves aren't the most distant ones, but the light from them has been distorted such that the light has taken that long to reach us? Furthermore, if the galaxies are spreading out faster than the speed of light, who is to say this explosion actually happened 13.1 billion years ago? Isn't it possible that the light was created say 5 billion years ago, but has taken much longer to reach us because of the expanding space between the galaxies? I'm sure this stuff has already been considered by scientists, but I find it weird the way news articles always assume that just because light travels at a specific speed, that it's always going to take the same amount of time to reach us. Or am I getting it wrong? I'd love to know! EDIT: For people discussing the whole faster-than-light-speed thing, I came across this article: > As you look at galaxies further and further away, they appear to be moving > faster and faster away from us. And it is possible that they could > eventually appear to be moving away from us faster than light. At that > point, light leaving the distant galaxy would never reach us. When that > happens, the distant galaxy would just fade away as the last of the photons > reached Earth, and then we would never know it was ever there. And this one: > That mysterious dark energy force, which is accelerating the expansion of > the Universe is making the most distant galaxies move faster and faster away > from us. Eventually, they will cross an event horizon and appear to be > moving away from us faster than the speed of light. At this point, any light > emitted by the galaxy will cease to reach us. Any galaxy that crosses this > horizon will fade away from view, until its last photon reaches us. All > galaxies will disappear from view forever.


Looking for a way to simplify a physics formula

I have the following physics formula: $$d = \frac{1}{2} at^2$$ where d is equal to half (at) squared where:   * d is distance   * a is acceleration    * t is time I need to simplify this to get the acceleration. Can anyone think of an easy way to do this? I have been looking around but can't find a simple answer.


What is actually meant by 'sun set' and 'sun rise' times, when taking into account the mirage due to light bending in the atmosphere

I’ve heard from the likes of Brian Cox that what we see of the sun during a sunset and sun rise is actually the mirage of the sun. The Sun has actually set/risen and we see it due to the way light is bent across the atmosphere. Apparently due to coincidence of the size and distance of the sun, its exactly the same size - so if we see 50% of the sun, the sun is 50% below the horizon. So, I understand all this, so here is my question : If this is the case, then when we read things like what time sun sets and rises on websites, books, calendars, other official times, et al… does that mean when we see for example ‘sun set at 18:35’ is the time denoting the actual sun set taking into account of the mirage or what is visible to us. If I were to know the time and watched the sun against an accurate clock, would the sun not be visible before the actual sun set time? I would also like to know how this affects things like iphone apps that tells us what time sun sets are and where it is in relation to the horizon at a given time. So is the set/rise time taking into account of the mirage and the time is recalculated to give us when it has ‘actually’ set/risen or when the mirage has set/risen. Thanks Repost from : http://astronomy.stackexchange.com/questions/2190/what-is- actually-meant-by-sun-set-and-sun-rise-times-when-taking-into-accou


Church-Turing hypothesis as a fundamental law of physics

The Church-Turing hypothesis says one can not build a computing device which has more computing power (in terms of computability) than the abstract model of Turing machine. So, there is something in our laws of physics which prevent us from making devices which are more powerful than Turing machine, so in this respect it can be viewed as a law of physics. What is the physicists' view of Church-Turing hypothesis? Can the Church-Turing hypothesis be deduced from other fundamental law of physics?


How to measure Projective Symmetry Group in spin liquid?

Quasiparticles in spin liquid will no longer be the representation of symmetry group. So when group elements act on quasiparticles, there will be some phase factor. For example, in $\pi$ flux state, $T_xT_y=-T_yT_x$, which means if we move the quasiparticle around a lattice, and there will be a minus sign. However, because the quasiparticle are always emerging in pairs, when we move one quasiparticle, it cannot be avoided that we will also move another. In this case, the phase factor (minus sign) will be cancelled out. If we wants to move only one quasiparticle, the symmetry operator must be a local operator. But what does it mean for local operator of the global symmetry? How can we measure the PSG in general cases?


Lennard-Jones potential - equilibrium

If I have Lennard-Jones potential, how can I calculate equilibrium? $$ U(r)=\epsilon \left ( \left (\frac{r_m}{r} \right )^{12}-2 \left (\frac{r_m}{r}\right )^6 \right ) \ $$


Multivariable Dirac Delta and Faddeev-Popov Determinant

From this mathstack page and in particular Qmechanic's answer: >   1. There exists an $n$-dimensional generalization $$\tag{1} \delta^n({\bf > f}({\bf x})) ~=~\sum_{{\bf x}_{(0)}}^{{\bf f}({\bf > x}_{(0)})=0}\frac{1}{|\det\frac{\partial {\bf f}({\bf x})}{\partial {\bf x}} > |}\delta^n({\bf x}-{\bf x}_{(0)}) $$ of the substitution formula for the > Dirac delta distribution under pertinent assumptions, such as e.g., that the > function ${\bf f}:\Omega \subseteq \mathbb{R}^n \to \mathbb{R}^n$ has > isolated zeros. Here the sum on the rhs. of eq. (1) extends to all the zeros > ${\bf x}_{(0)}$ of the function ${\bf f}$. > Also from this page on the Faddeev-Popov procedure they say: > For ordinary functions, a property of the Dirac delta function gives: > $\delta(x-x_0) = \left|\frac{df(x)}{dx}\right|_{x=x_0}\delta(f(x))\,$ > assuming $f(x)\,$ has only one zero at $x=x_0\,$ and is differentiable > there. Integrating both sides gives :$$1 = > \left|\frac{df(x)}{dx}\right|_{x=x_0}\int\\!dx\,\delta(f(x))\,$$. Extending > over $n$ variables, suppose $f(x^i) = 0\,$ for some $x^i_0\,$. Then, > replacing $\delta(x-x_0)\,$ with $\prod_i^n \delta^i(x^i-x^i_0)\,$ :$$1 = > \left(\prod_i \left|\frac{\partial f(x^i)}{\partial x^i}\right|\right) > \int\\!\left(\prod_i dx^i\right)\,\delta(f(x^i))\,$$. Recognizing the first > factor as the determinant of the diagonal matrix $\frac{\partial > f(x^i)}{\partial x^i}\delta^{ij}\,$ (no summation implied), we can > generalize to the functional version of the identity: :$$1 = > \det\left|\frac{\delta G}{\delta \Omega}\right|_{G=0} > \int\\!\mathcal{D}\Omega\,\delta[G_a(\phi^\Omega)]\,$$, where > $\Delta_F[\phi] \equiv \det\left|\frac{\delta F}{\delta g}\right|_{F=0}\,$ > is the Faddeev-Popov determinant. What I don't understand is that it seems their function $f$ seems to be $f:\Omega \subseteq \mathbb{R}^n \to \mathbb{R}$. **How does the generalized Dirac formula $(1)$ work in this case?** I don't really understad their notation in: $$1 = \left(\prod_i \left|\frac{\partial f(x^i)}{\partial x^i}\right|\right) \int\\!\left(\prod_i dx^i\right)\,\delta(f(x^i))\,$$ What does $$\frac{\partial f(x^i)}{\partial x^i}$$ mean here?


Intuitive understanding of wavelength

Light is described as having wavelength. I can somewhat understand this in connection with for example the double slit experiment, that photons interact spatially in a wave like manner. But can the wavelength of light be described in a way which more intuitively relates to the waves of sound or oceans? Does "the length" have a specific orientation relative to the path of light? Doppler shifts seem to indicate that the wavelength is measured in the direction of light's movement, the speed stretches it out. But the double slit experiment, and the shapes of antennas, seem to indicate that the wavelength is meassured perpendicular to light's movement. How large is a photon and what shape does it have in terms of its "length"?


neutrinos by formation of "neutron pairs"

Here : http://www.newscientist.com/article/dn20084-neutron-star-seen-forming-exotic- new-state-of-matter.html are news on superfluidity in a neutron star. The necessary bosons they say are pairs of neutrons. So far, so good. But then they postulate the production of neutrinos in the formation of those neutron pairs. > When neutrons pair up to form a superfluid they release neutrinos which > should pass easily through the star,... What kind of reaction is that? n + n => neutrino + "n-n" + "XYZ" ?


Why don't we experience dilatations in Minkowski spacetime?

**The question** This question follows on from the use of projective coords for spacetime in Notation for Translation Group Generators . Under Felix Klein's Erlangen Program, Minkowski spacetime starts as a 4-d projective space. It is then made into an affine space (which has a notion of parallelism) by picking an invariant hyperplane at infinity; lines in the projective space which meet at infinity are parallel. An affine space admits dilatations which are expansions or contractions of space about the eigenpoint of the dilatation. When introduced projectively[1], dilatations are seen to be generalized translations because a translation is a dilatation in which the eigenpoint is on the hyperplane at infinity. Since one can smoothly change a dilatation into a translation by moving the eigenpoint onto the hyperplane at infinity, it is hard to avoid the conclusion that a dilatation is an active transformation on the same footing as translations, rotations and boosts. Yet we don't experience dilatations in everyday life. Why is this? **Some background observations** The general feeling in physics seems to be that dilatations are changes in the units of measurement, but this view is only possible because dilatations are introduced in an ad hoc way as scale transformations and the link with translations is missed. In order to see that dilations are active transformations like translations, the translation generators of Notation for Translation Group Generators become, $$ [P_{\Gamma}]^{\Phi}_{\ \Psi}=\delta^{\Phi}_{\Gamma}\delta^{4}_{\Psi} $$ where indices $\Phi,\Psi,\Gamma=0,1,2,3,4$. The translations are $P_{0},P_{1},P_{2},P_{3}$ and the dilatation generator is $P_{4}$. The attempt to construct classical relativistic Hamiltonian particle mechanics in Minkowski spacetime with dilatations fails unless the particles are massless. Massive particles only exist by removing the dilatations by hand in order to restrict the group to the ten parameter Poincare group. **Edit: An attempt to derive dilatations without having to read Coxeter [1].** Start with Minkowski spacetime as a 4-d projective space; points are rays in a 5-d vector space $x^{\Phi}\in V_{5}$ with upper case Greek indices like $\Phi=0,1,2,3,4$. The spacetime group is GL(5,R). Introduce parallelism by picking an invariant hyperplane $\delta^{4}_{\Phi}\in \tilde{V}_{5}$. Invariance means, $$ [D(g^{-T})]_{\Phi}^{\ \Psi}\delta^{4}_{\Psi}=\xi \delta^{4}_{\Phi} $$ where $\xi$ is a scale factor which is unimportant because the physical entities are rays. This implies that the group matrices are now have the block form, $$ \begin{equation} D(g)= \left[ \begin{array}{cc} \Lambda & a \\\ 0 & 1+a^{4} \end{array} \right] \end{equation} $$ where $\Lambda^{\mu}_{\ \lambda}$ is a 4x4 matrix and $a^{\mu}$ is a 4x1 vector and lower case Greek indices run $\mu=0,1,2,3$. These affine group matrices factor as, $$ \begin{equation} D(g)= \left[ \begin{array}{cc} 1 & a \\\ 0 & 1+a^{4} \end{array} \right] \left[ \begin{array}{cc} \Lambda & 0 \\\ 0 & 1 \end{array} \right] \ . \end{equation} $$ The first matrix is a dilation. In tensor notation, the pure dilatations are, $$ [D(g)]^{\Phi}_{\ \Psi}=\delta^{\Phi}_{\ \Psi}+a^{\Phi}\delta^{4}_{\Psi} $$ and a point transforms under a dilatation as, $$ x'^{\Phi}=[D(g)]^{\Phi}_{\ \Psi}x^{\Psi}=x^{\Phi}+a^{\Phi}x^{4} \ . $$ This shows that a dilatation acts like the identity on points in the hyperplane at infinity (the points which we can write as $x^{\mu}$ because $x^{4}=0$) and the eigenpoint of the dilatation is $a^{\Phi}$ with eigenvalue $1+a^{4}$. The last equation implies that a dilatation is an expansion or contraction of spacetime about the eigenpoint because a dilatation is a linear combination of the point and the eigenpoint. Consider two spacetime points $x^{\Phi}$ and $y^{\Phi}$. Transform them to $x'^{\Phi}$ and $y'^{\Phi}$ under a dilatation with eigenpoint $a^{\Phi}$. The dilatation moves the line joining $x$ and $y$ parallel to itself so that the line $xy$ (in an informal notation) is parallel to line $x'y'$. To see this, the point where the line $xy$ intersects the hyperplane at infinity is, $$ b^{\Phi}=\delta^{\Phi \Psi}_{[\Gamma\Delta]}x^{\Gamma}y^{\Delta}\delta^{4}_{\Psi} =x^{\Phi}y^{4}-x^{4}y^{\Phi} $$ and the dilatation acts like the identity on $b$ which is in the hyperplane at infinity, so line $x'y'$ also intersects the hyperplane at infinity at $b$ and so $xy$ and $x'y'$ are parallel lines. Now consider the quadrilateral $x'y'yx$. We have just seen that the sides $xy$ and $x'y'$ are parallel. The quadrilateral can be made into a parallelogram if the sides $xx'$ and $yy'$ are also parallel. These lines are parallel if they intersect at a point in the hyperplane at infinity. But, these lines both go through the eigenpoint $a$ of the dilatation. Therefore, $xx'$ and $yy'$ are parallel if the eigenpoint $a$ is moved onto the hyperplane at infinity. With the eigenpoint $a$ on the hyperplane at infinity, we've made a parallelogram by starting with a line $xy$ and moving it parallel to itself to $x'y'$ with $xx'$ and $yy'$ also parallel. This is how to construct a parallelogram with a translation. In other words, a translation is a dilatation with its eigenpoint on the hyperplane at infinity. This construction is why I cannot see a reason to exclude dilatations from the spacetime group, it seems dilatations must be just like rotations and boosts. [1] Coxeter, "Introduction to Geometry", chapter 5.


Is proper time an intrinsic value of Minkowski space?

What is proper time? Is it a part of Minkowski space (that is a mere spacetime interval)? Or is it an intrinsic characteristic of massive particles (a sort of "aging")? Example: In the following Minkowski diagram,   * A is the worldline of a particle moving with the observer's reference frame.    * B is the worldline of another particle with proper time between two events = 4 (space distance x= 3, time interval t = 5, proper time τ = 4, according to the equation $$ \tau^2 = \delta t^2 - \delta x^2 $$)   * C is just 2 events without any worldline between, and if a particle would travel through both events, its proper time would be 4 ![enter image description here](http://i.stack.imgur.com/3IKCG.jpg) Is there proper time in C even if no particle is travelling there? Or is in C only hypothetic proper time in case a particle is travelling this path?


1. How many photons can an electron absorb and why?   2. Can all fundamental particles that can absorb photons absorb the same amount of photons and why?   3. If we increase the velocity of a fundamental particle, do we then increase the amount of photons it can emit?   4. At constant temperature and mass of an isolated fundamental particle that does not and will not move (constant speed and 0 vibration ), is emitting a photon the only way to loose energy?


Under what conditions, will a projectile motion follow a half circle?

What came to my mind is to throw an object far from the earth with a particular speed equal to $$V_{sat}=\sqrt{\frac{G\times M_{central}}{R}}$$like satellites. However, it would follow a full circle.


Charge, velocity-dependent potentials and Lagrangian

Given an electric charge $q$ of mass $m$ moving at a velocity ${\bf v}$ in a region containing both electric field ${\bf E}(t,x,y,z)$ and magnetic field ${\bf B}(t,x,y,z)$ (${\bf B}$ and ${\bf E}$ are derivable from a scalar potential $\phi(t, x, y, z) $and a vector potential ${\bf A}(t,x,y,z)$), knowing that   1. ${\bf E}=- \nabla \phi - \frac{\partial {\bf A}} {\partial t}$   2. ${\bf B}= \nabla \times {\bf A} $   3. $U=q \phi - q {\bf A} \cdot{\bf v}$ ($U$ is the velocity-dependent potential) the Lagrangian is $$L=1/2 m v^2- q\phi + q{\bf A} \cdot{\bf v}.$$ Considering just the x-component of Lagrange's equation, how can I obtain $$m \ddot{x}=q(v_x \frac{\partial A_x}{\partial x} + v_y \frac {\partial A_y}{ \partial x} + v_z \frac{\partial A_z} {\partial x})-q(\frac{\partial \phi }{\partial x} + \frac{d A_x}{dt}) ?$$


Effect on length of day as the polar ice caps melt

If the polar ice caps of the earth melt, how will the length of the day be affected?


Scientist propose solution to quantum measurement problem, explain further

Link to Article The scientists > In their recent opus magnum, Theo Nieuwenhuizen (Institute of Physics, UvA) > and colleagues claim to have found a solution to the so-called quantum > measurement problem Their claim > After reviewing the literature on measurement models, they focus on the so- > called Curie-Weiss model for quantum measurements. The joint dynamics of the > tested system coupled to the measurement apparatus now produces results that > explain the measurement postulates. It shows that the off-diagonal terms > ("cat-terms") disappear in a physical process due to the coupling to the > apparatus, on a basis selected by the interaction Hamiltonian. The > registration of the measurement outcomes by the pointer of the apparatus > occurs because the measurement triggers the transition from the initial > metastable state to one of the stable final states; the apparatus being > large, this means an amplification of the signal. The transition of the > pointer variable from its initial metastable state to one of the final > stable states is a process with classical features, such as the dumping of > excess free energy in the bath. I am by no means a physicist, but I do enjoy trying to understand some of the issues they face. From what I've read this quantum measurement problem, well is a problem. Could someone explain why it is a problem, and how these scientist from the Institute of Physics propose to solve it? (as layman as possible)


N=4 SYM from Klebanov-Witten field theory

This is with reference to M. J. Strassler's lectures on "The Duality Cascade" pg. 46. I want to see how $\mathcal{N}=4$ SYM emerges when D3 branes, in the KW setup, are placed at smooth point of the conifold, purely from field theory point of view. That is: Given a $U(N) \times U(N)$ quiver with $A_i$ and $B_i$ in the bi- fundamental and the anti-bi-fundamental respectively and an $SU(2) \times SU(2)$ invariant quartic superpotential (which vanishes for $N=1$), in what manner do we higgs the theory so as to get $\mathcal{N}=4$ SYM? For the case of $N=1$ there is a comment in the lectures. It says, > ... suppose we just have one D3 brane and we allow $A_1B_1$ to have an > expectation value, so that the D-brane sits at some point away from the > singular point of the conifold. Then the gauge group is broken to $U (1)$, > and six scalars remain massless — the six possible translations of the > D3-brane away from its initial point — exactly the number needed to fill out > an $\mathcal{N}= 4$ $U(1)$ vector multiplet. I want to understand this statement. Please help me understand how to work out for case of $N=1$. I can try for the generic value of $N$.


how can you keep a gyroscope turning?

I'm trying to understand how do gyroscopes and cmgs used for navigating keep on spinning even though there's friction. Any engine/pushing mechanism I can think of will restrict the gyro's rotor to a fixed direction and defeat the purpose... how is it done without disturbing it?


Amplitude of power spectral density

Why is the amplitude of the Power spectral density higher for shorter period of time as compared to a longer period of time when calculated for any vibration data?


Potential energy of a rotating bar

I have a bar of length $L$, mass $m$, and negligible width. The bar can rotate along an axis through its center of mass (the rotation is in the $x$-$y$ plane), and also can move up and down (in the vertical axis). > My question is: What is the (gravitational) potential energy of this bar? I've found several exercises where $PE=mgy_{CM}$, where $y_{CM}$ is the position of the bar's center of mass, though I don't know why the assumption of just taking the CM. Since this is rigid body, I thought also in summing the potential energy of each particle (i.e, integrating the classical $mgy$ function with respect to the height $y$), but I don't know if this is correct, or necessary.


English translation of Heisenberg's paper ``Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik''

Is there any english translation of Heisenberg's paper _Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik_ online? I have found not. Thanks so much.


How do we calculate the Higgs potential?

What is the Higgs potential and how do we calculate it?


vacuum expectation value and Casimir effect

Is it correct to say that the VEV of the SM Higgs is 246 GeV? If so, is the VEV a reflection, or measure, of the Casimir Effect?


Relativity and Current in Wire

If an observer is stationary relative to a current-carrying wire in which electrons are moving, **why does the observer measure the density of moving electrons to be the same as the density of electrons if there were no current in the wire?** I read the explanation of magnetic force as a consequence of special relativity. That is, when the observer moves with respect to a current carrying wire in the same direction as the flowing electrons, then s/he observes the density to decrease due to Lorentz expansion and observes the density of positive ions to increase due to Lorentz contraction. The imbalance of charges results in a force that can be explained by Coulomb's law. That explanation mentions that when the observer is stationary, the density of positive ions and the density of electrons appears to be the same. That last point is what I am stuck on. Why does the density of _moving_ electrons appear to be the same as the density of _stationary_ positive ions? **Clarification:** I know that when the electrons are not moving, their charge density cancels out that of the stationary positive ions. When the electrons are moving, how could the charge density _still_ cancel out that of the stationary positive ions? **Important:** My assumption is that a current carrying wire will neither attract nor repel a charge that is stationary with respect to the wire. I am now not sure if that is right :-(


Energy used to stop / slow an object

I'm trying to workout how much energy (if any), I use (imagining me as an efficient machine rather than a complex bio-mechanical human) when I lower or catch a weight. I understand that when I push it up, I have used energy to do work as I have the force of the object (it's mass and gravity) and distance. So I've got two similar questions... If I am to stop a free-falling object, how much energy do I use? If I look at it from energy change I believe I need to use the same amount as the kinetic energy of the object at the moment before I catch? If I slowly lower an object how much energy do I use? In an ideal world would the total be the same as in the free falling situation, assuming the height at which the object fell was the same.


Hard disk rotational frequency

I noticed the other day that the rotational frequency of typical consumer grade HDD are either 5400rpm or 7200rpm. These correspond to 90Hz and 120Hz respectively. Given that in parts of the world the grid frequency is 60Hz, and 120Hz is the first harmonic. What effects does this have on the platters. It would seem to me that this choice in rotational speed (7200 rpm) would be problematic in these certain countries.


What is the best material and design for building a parabolic dish?

I'm currently using a homemade metallic spaghetti strainer with a wifi adapter inside it to increase my wifi adapter's signal reception, but would think that a different material, like the aluminum mesh used for the Arecibo Radio Telescope could be better? Why is this radio telescope's reflector spherical and not parabolic?


Addition of spin?

There is something I don't fully understand. When I have a problem like this: Let's look at a system with $j_1=1$ and $j_2=2$ spins. Let our system be in the state in which the total spin is $2$ and the projection onto the $z$ axis is 0 ($S_z$). What values can the $S_z$ of the $1st$ particle take? When I have a problem like this, I always start by creating this fig:![enter image description here](http://i.stack.imgur.com/JfWRc.jpg) It represents the eigenstates. It's easy, I just have to find the simpliest way to the $(J=2,M=0)$. But when the problem is that I just have to add, for example, 1/2 and 1/2 spins, then how can I find the eigenstates? There must be a basic formula for it, I've been looking for it but I can't find it.


Particle antiparticle annihilation-do they have to be of the same type?

I read that a particle will meet its antiparticle and annihilate to generate a photon. Is it important for the pairs to be of the same type? What will happen when for example a neutron meets an antiproton or a proton meets a positron? Are there any rules to determine what happens when such particles meet?


How is the atmosphere composition of astronomical objects in solar system determined?

How is the atmosphere composition of astronomical objects, for example triton, the moon of neptune, determined?


What shape of track minimizes the time a ball takes between start and stop points of equal height?

I was at my son's high school "open house" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end points of the second track were the same but it was bent into a dip with a decending slope and an ascending slope. The game was to guess which ball bearing wins the race. At first blush ("the shortest distance between two points is a straight line") one might think the straight wins, but in fact the bent track wins easily (because the ball is going very fast for most of the time). His point was that nature doesn't always work that way our antediluvian brains think it should. So my question (and I do have one) is what shape of track give the absolute shortest time? I think it's a calculus of variation problem or something. For simplicity, let's assume a) no friction b) a point mass and c) that the end point is a small delta below the start point. I played around with some MATLAB code and it seemed to give a parabola of depth about 0.4 times the distance between the start and stop point. The value of g and the mass didn't seem to matter. But I don't know if my code was right or wrong and I couldn't figure out the significance of 0.4. I Googled around for an answer but couldn't find one. Thanks for any pointers!


Effect of linear terms on a QFT

I was always told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \phi _i \phi ^\mu \phi _i - \frac{m ^2 }{2} \phi _i \phi _i - \frac{ \lambda }{ 4} ( \phi _i \phi _i ) ^2 \end{equation} with $m ^2 =-\mu^2 > 0$, adding a linear term in one of the fields $\phi_N$, does change the final results as you no longer have Goldstone bosons (since the $O(N)$ symmetry is broken to begin with). Are there any other effects of linear terms that we should keep in mind or is this the only exception of the "forget about linear terms" rule?


General Physics Short question

Does any one know where this formula come from ? $\alpha (t) = -\omega^2\ \Theta(t)$ Thanks


Asymptotic limit of the two kink solution of the sine-gordon equation

I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: $$\phi=4\arctan\left(\frac{\sinh\frac{1}{2}(\theta_1-\theta_2)}{(a_{12})^\frac{1}{2}\cosh\frac{1}{2}(\theta_1+\theta_2)}\right)$$ where $\theta_1=m\gamma_1(x-v_1t)+\delta_1$, and similarly for $\theta_2$. Here $a_{12}, m, \gamma, \delta$ are all constants. The author says that the limit as $t \to \pm \infty$, is $$\lim \phi(x,t) \to 4\arctan(\exp [\theta_1+\eta_1^{\pm}])+4\arctan(\exp[\theta_2+\eta_2^{\pm}]) $$ where $\eta_1^{\pm}=-\eta_2^{\pm}=\pm\frac{1}{2}\ln(a_{12})$ I am having a hard time proving this. I did the expansion of the hyperbolic functions in terms of the exponents, and tried to limit the appropriate terms to 1, but I am not getting an answer.


Why are generalized positions and generalized velocities considered as independent of each other?

I'm confused how $$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$ leads to the relation, $$\frac{\partial\dot{\mathbf{r}}_{i}}{\partial\dot{q}_{j}}=\frac{\partial\mathbf{r}_{i}}{\partial q_{j}}$$ Sources suggest that while differentiating the first equation the generalized velocity and generalized positions are considered as independent of each other, i.e., $(\dot{q_1},\dot{q_2},...,\dot{q_n})$ is independent of $(q_1,q_2,...,q_n)$. I don't understand how they are independent.


Feynman rule 4-point vertex WW -> ZZ

I am looking for the Feynman rule of the 4-point gauge boson interaction of W+ W- -> Z Z. I am guessing it looks like the Yang Mills 4-point vertex for gluons, but with helicity included. Equation 8.81 in my professor's lecture notes seems to imply polarization, or am I misinterpreting?


Thermal expansion is an expression of which conservation laws?

Many objects get larger as they heat up and contract as they cool down. Which conservation laws are applied to describe this phenomenon? How do they interact with each other to produce this effect?


Math needed for undergrad Statistical Mechanics/Thermal Physics

A professor recommended me to take a course on Statistical Physics as preparation for agent-based computing in social sciences. Now I have no experience in physics beyond basic highschool, and mathematics I have taken several discrete math courses (for computer science) and no calculus yet. I have had basic differentiation/trigonometry/integration in high school as well. What math/physics do i need to be able to understand undergrad stat mechanics? Thanks a lot for your answer! (btw. I am already planning to take classical mechanics, because that's what the university states as prerequisite)


What is three-photon interference?

Whilst reading this paper on a quantum processor that performs a type of matrix computation, I came across the concept of 'three-photon interference'. A quick Google search shows that this process is frequently referred to in quantum optical experiments but unfortunately, no wikipedia article exists for it, can anyone yield some insight on this? My own guess is that it is exactly what it says: the interference pattern that results when 3 photons are combined. Why is this distinction so important?, why 3 photons and not 4, for example?


Maxwells' equations and Coulomb's law

Coulomb's law and Maxwell's equations should be consistant as one can be derived from the other. Say we have a point charge with such a charge that $-kq=1$, meaning that at any point the electric field will have a magnitude of $$|E|=\frac{1}{r^2}$$ where $r$ is the distance from the origin (were we place our charge), and the vectors point towards the origin at all point. This would be equivalent to the following in cartesian co-ordinates: $$E=-\hat{x}\frac{x}{(x^2+y^2+z^2)^{3/2}}-\hat{y}\frac{y}{(x^2+y^2+z^2)^{3/2}}-\hat{z}\frac{z}{(x^2+y^2+z^2)^{3/2}}$$ We can verify that $$|E|=\frac{1}{\sqrt{x^2+y^2+z^2}}$$ Gauss's law in its differential form allows us to calculate a charge distribution that would give rise to such a electric field using the divergence operator: $$\nabla \cdot E=-\frac{1}{x^2+y^2+z^2}$$ from wolfram alpha: one, two Which absolutely doesn't make sense to me! Intuitively I would think it would be zero everywhere except (0,0,0). Or at least not to go of to infinity at any point. Could somebody please explain what's going on?


Ideal gas temperature and pressure gradients?

Consider an ideal gas in a $d\times d\times L$ box with the $L$ dimension in the $x$-direction. Suppose that the opposite $d\times d$ sides of the box are held at temperatures $T_1$ and $T_2$ with $T_2>T_2$ and that the system reaches a steady state. According to these notes, the thermal conductivity of an idea gas scales as the square root of temperature; $k=\alpha\sqrt{T}$ in which case by Fourier's Law one gets that the temperature gradient in the $x$-direction is $$ T(x) = \left[T_1^{3/2}+(T_2^{3/2}-T_1^{3/2})\frac{x}{L}\right]^{2/3} $$ What is the corresponding pressure gradient $P(x)$ in the steady state?


Sound Propagation using Finite-Difference Time-Domain (FDTD) considering wind effects

I am trying to plot the propagation of sound from a fixed source in a 2D environment using Finite-Difference Time-Domain (FDTD) Method taking into account the effects of the wind velocity. I came across the following set of equations for solving the equation for pressure with respect to time at different grid points in the 2-D environment. $$ \frac{∂p}{∂t} = -( v_x\frac{∂}{∂x}+v_y\frac{∂}{∂y})p - \kappa(\frac{∂w_x}{∂x}+\frac{∂w_y}{∂y}) +\kappa Q $$ $$ \frac{∂w_x}{∂t} = -(w_x\frac{∂}{∂x}+w_y\frac{∂}{∂y})v_x- (v_x\frac{∂}{∂x}+v_y\frac{∂}{∂y})w_x-b\frac{∂p}{∂x}+bF_x $$ $$ \frac{∂w_y}{∂t} = -(w_x\frac{∂}{∂x}+w_y\frac{∂}{∂y})v_y- (v_x\frac{∂}{∂x}+v_y\frac{∂}{∂y})w_y-b\frac{∂p}{∂y}+bF_y $$ where, $ b = \frac{1}{\rho} $ is the mass bouyancy. But, I am unable to figure out what $\vec{F}$ and $Q$ are? They are mentioned somewhere as dipole and monopole pressure sources respectively. What does they actually mean?


The Preference for Low Energy States

The idea that systems will achieve the lowest energy state they can because they are more "stable" is clear enough. My question is, what causes this tendency? I've researched the question and been unable to find a clear answer, so I was hoping someone could explain what's going on behind the scenes here (or if nothing is and it's just an observed law that isn't explainable).


Hard circuit problem

I want to find the p.d. between points A and B. I'm not sure how to approach this problem though, could I have a hint? ![enter image description here](http://i.stack.imgur.com/chmiO.jpg)


Is information about the speed of light hidden in its spectrum?

Can the speed of light in the vacuum (c) be inferred from the spectrum of light? If that is not the case is it possible to tell from lights spectrum that it has entered a different medium, e.g. can the correct fraction of c be inferred from the spectrum then?


Physics of a fixed wheel moving on a flat surface held at distance from a center point

Ok, so my boss is trying to make a car turntable. In essence, he has a two boards that sit atop a rotating ring. He wants to put two wheels at the end of each board (8 wheels total). He thinks that you can angle wheels properly (2d) such that they can be flat (axis is parallel to the floor) and the whole contraption move smoothly. Take a look at this picture (Edit: copied below - Mark) ![alt text](http://i.stack.imgur.com/YujYw.jpg) Ok, so the question, will this design move smoothly? My guess is "no." I reason that if the wheels are unbanked then they will move in a straight line when driven, and the ring will realign them periodically to its circular path (probably causing a skip at the wheel as the stress is released). He thinks it will just work.. Please use technical terms with very easy layman definitions so I can reference wikipedia and texts and learn more about this. Sorry, for the utter lack of useful physics terms.


Eye sensitivity & Danger signal

Why are danger signal in red, when the eye is most sensitive to yellow-green? You can check luminosity function for more details...


Levi Civita Symbol and contravariance vs covariance

I have a question regarding the Levi-Civita symbol and contravariance vs covariance. Some of this was asked in a previous post, but I think I need more clarification. Consider the magnetic field: \begin{equation} B^k = \tilde{\varepsilon}^{ijk}\partial_i A_j \end{equation} Qn 1: In the formula I wrote it as the Levi-Civita symbol $ \tilde{\varepsilon}$, is it right or should it be the Levi-Civita tensor $\varepsilon$ instead? Or does that not make sense since the formula $\vec{B} = \nabla \times \vec{A} $ is true only in cartesian coordinates and so we cannot use a tensor? (hope that made sense) Qn 2: I wrote $B^k$ with an upper index to indicate that it is a contravariant vector - so that means under coordinate transformations $\tilde{\varepsilon}$ transforms too. Except that I'm confused as to how it transforms? Carroll (in his book spacetime & geometry) says that the Levi-Civita symbol is defined not to change under coordinate transformations (so its entries remain +1 -1 0), yet goes on to derive a transformation law $\tilde{\varepsilon}_{i'j'k'\cdots} = |g| \tilde{\varepsilon}_{ijk} \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} \cdots$. So which is it? Qn 3: However in some books they write \begin{align} B_k = \epsilon_{ijk}\partial_i A_j. \end{align} In this case it seems like they are ignoring the tensorial nature of the object and just treating $\tilde{\varepsilon}$ as simply a symbol that gives the values +1, -1 or 0 in the summation. This occurs too, for example, in writing the commutation relations down for the spin operators: \begin{align} [J_i, J_j] = i \tilde{\varepsilon}_{ijk}J_k, \end{align} where I am pretty sure in this case the symbol is just thought of as just a number. (although $J_i$ are vectors of the Lie algebra...?). So, am I attributing too much meaning to the Levi-Civita symbol? Qn 4: In the course of my work I'm doing I have to contend with the angular momentum operator, which I wrote as \begin{align} L^k = \tilde{\varepsilon}^{ijk}g_{ia}r^ap_j. \end{align} In cartesian coordinates this reproduces $L^x, L^y, L^z$, since for example $L^z = x p_y - y p_x$. But is this formula right? Changing to spherical coordinates doesn't seem to give $L^z = \frac{\partial}{\partial \phi}$ and so on... Qn 5: Just a general question about the Levi-Civita symbol. Does it make sense to raise/lower indices in the Levi-Civita symbol? I know that you can do it for the tensor because the Hodge star operator uses it. sorry for the lengthy post and thanks in advance for the help!


Evolution of mass and velocity distributions under newtonian gravitation

Let $\rho(r,t)$ and $v(r,t)$ be mass and velocity distributions. Given $\rho(r,0)$ and $v(r,0)$ (initial conditions) what is the differential equation that describes the evolution of $\rho(r,t)$ and $v(r,t)$ in space and time, assuming Newtonian gravitation? You may assume $r$ to be one dimensional, if it makes writing the equations convenient. **EDIT** I think $v(r,t)$ should be replaced with $p(r,t) = \rho(r,t)v(r,t)$ and name it as momentum distribution to avoid some technical faults. **EDIT 2** It seems that to completely characterize an inviscid fluid flow, Euler equations are needed. The law of conservation of mass is already given in the answer. A component of pressure is missing in the equation related to conservation of momentum.(first equation). Lastly a third equation corresponding to law of conservation of energy should be introduced.(3rd one of the euler equations of inviscid fluid flow).For the third equation we assume that the internal energy per unit volume of the fluid is zero.


Is light particle of wave?

We know that Young's double slit experiment shows that light is a wave. On the other hand photoelectric effect shows that light is made up of photons. How can light be both at the same time?


What does the notion of basis sets for photons in number of particle picture?

Since $$|n.\rangle=|u_{k_1}\rangle\otimes...\otimes|u_{k_1}\rangle\otimes|u_{k_2}\rangle\otimes...\otimes|u_{k_2}\rangle\otimes...\otimes|u_{k_m}\rangle\otimes...\otimes|u_{k_m}\rangle$$(There are $n_{k_i}$ terms in $|u_{k_i}\rangle\otimes...\otimes|u_{k_i}\rangle$.) means in each modes there is a fixed number of photons. Does $$|\\{n\\}\rangle$$ means the set of all the possible candidates of $|n.\rangle$ ?


Is there a unit system, where all physical laws looks nice?

And by nice looking law I mean with no constants. I mean, what would we need to set, so all laws would without those nasty constants in front of them? (I mean all of them, also $\pi$!) What would it mean? What I am trying to say with this question is: Is there a universal language, do particles and gravity live in the same space? If gravity and quantum world should go together, there should be one basis, where you don't need to use any constants.


Can an LC oscillator be used to generate visible light?

The LC oscillator is most commonly used to generate radio waves for practical use and the frequency $\omega$ of the LC oscillator equals that of the electromagnetic wave so produced. So, can they in principle be used to emit _visible light?_ The frequency of visible light is on the order of a few hundred terahertz, and the frequency of an LC oscillator is $$\omega = \frac{1}{\sqrt{LC}}$$ I admit, the product $LC$ does become very small (on the order of $10^{-30}$) when the numbers are plugged in, but making an inductor and a capacitor with small values isn't difficult, is it?


Deriving Lorentz transformation - Serway

in Serway, the derivation of Lorentz transformation started from these two equations $$x′=\gamma(x−ut),$$ $$x=\gamma(x′+ut′).$$ The book derived the first equation and then deduced the second. Is there a proof that if the first is right then the second is also right? To be specific I don't understand:   1. How $t$ was converted to $t′$ in the second equation.   2. Why the position of $\gamma$ didn't change in the second equation.  Note: I understand the first equation well but I don't understand how to reach the second from the first.