title: Identifying the position of the moon

text: I encountered the following question in a previous year paper of a graduation- based test. > On a certain night the moon in its waning phase was a half-moon. At midnight > the moon will be (choose one of the following) > >   1. on the eastern horizon. >   2. at 45 degrees angular height above the eastern horizon. >   3. at the zenith. >   4. on the western horizon. > I remember looking out of my window and finding the moon on left and sometimes on right. I never imagined that it would be having a fixed pattern. This question is making me think otherwise. But I am unable to decide what that pattern is. Is the position of the moon really determine-able?


title: What is the correct arrangement of the elements of Pauli matrices?

text: I'm dealing with angular momentum, or particularly spin, on my quantum mechanics course; I guess the Pauli matrices thing is a more general one, but I'd like to illustrate my doubt with them (maybe get a deeper answer). Say, for the electron ($s=1/2$), why does $$S_z=\begin{pmatrix}\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle\\\\[0.1in]\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle\end{pmatrix}=\frac{\hbar}{2}\sigma_z$$ instead of $$S_z=\begin{pmatrix}\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},-\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle\\\\[0.1in]\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},-\frac{1}{2}\right\rangle&\left\langle\frac{1}{2},\frac{1}{2}\middle|S_z\middle|\frac{1}{2},\frac{1}{2}\right\rangle\end{pmatrix}=\frac{\hbar}{2}\begin{pmatrix}-1&0\\\0&1\end{pmatrix}=-\frac{\hbar}{2}\sigma_z$$ What I mean to say is, **why does the matrix elements go from $m_s=+s$ to $m_s=-s$ (left-upper corner to right-lower corner) instead of the other way, as usual**? We usually take rows and columns from smaller to bigger value, but why is this not the case? For instance, recently we've seen the matrix representation of the hamiltonian of a simple harmonic oscilator, with elements $H_{mn}=(n+1/2)\hbar\omega\,\delta_{mn}$ and it goes like $\hbar\omega\begin{pmatrix}1/2&0&\ldots\\\0&3/2&\ldots\\\\\vdots&\vdots&\vdots\end{pmatrix}$, not as, say $\hbar\omega\begin{pmatrix}\vdots&\vdots&\vdots\\\\\ldots&3/2&0\\\\\ldots&0&1/2\\\\\end{pmatrix}$, for example. I'm just trying to make clearer my question. Here ($s=1/2$) is only a difference of a minus sign, but when I did it for $s=3/2$, the ladder (spin) operators would interchange ($S_-$ would have non-zero values 'above' the diagonal and $S_+$ 'below' the diagonal). I tried to find the reason but found none; is it a mere convention or definition? Or am I missing something important here? Thank you in advance.


title: Why does shape of elements matter in finite elements analysis?

text: I have used FEA for a couple of years now, but using it and using it correctly are two different things, safety factor is not the solution to everything. I have the feeling I won't be using it right unless I have a clear answer to that question: I am aware elements must be close to their ideal shape (based on the Jacobian) in order to get accurate results.. But why? Since I understand it comes from a coordinate transform, unless two vectors of the element become colinear shouldn't the results be accurate no matter its shape? **A step-by-step answer based on n illustrated example** (arbitrary stress distribution) would be really appreciated, especially given that it is a relatively common question (but never well answered from what I have seen).


title: Bass and Treble-Car Steroes

text: In a car which phenomenon, diffraction or the resonant frequency of the car, lends itself more to the ability of bass to go farther? Related Answer: Why do bass tones travel through walls?


title: The discontinuity of Electric Field

text: ''electric field always undergoes a discontinuity when you cross a surface charge $\sigma$'' GRIFFITHS In the derivation; Suppose we draw a wafer-thin Gaussian Pillbox, extendind just barely over the edge in each direction. Gauss law states that: $\int_{S} E \cdot A = Q_{enc}/ \epsilon $ and so $E_{above}^{perp} - E_{below}^{perp} = \sigma/ \epsilon $ My question is why not $2A$ ? $\int_{S} E \cdot A = 2EA$ because the top area of pillbox and the bottom area of pillbox, just as because the 2 parts of the flux... SO.. WHY NOT : $E_{above}^{perp} - E_{below}^{perp} = \sigma/ 2\epsilon $ And why there is tangencial component of electric field; not just perpendicular to the surface, which can be seen as flat just looking very close to the surface.


title: Mass fraction of fuel on a fuel droplet surface

text: Concerning combustion of fuel droplets: Why is the mass fraction of fuel on a fuel droplet surface slightly less than one? It is known that the temperature is below the boiling point at the fuel droplet surface, so there is no evaporation at the surface and therefore my intuition tell me that the mass fraction of the fuel at the surface should be unity, but it isn't. So why is the mass fraction of fuel less than one, and what happened to that fraction of fuel mass?


title: How can I model a two dimensional and three dimensional equivalents of one dimensional delta dirac (impulse) function?

text: I just started to read the book 'A Brief History of Time' by Stephen Hawking. Actually When he was talking of the idea of infinite density 'thing' before big bang suddenly the mathematical function came to my mind was delta Dirac impulse function I am familiar at engineering classes. But that was normally either in time domain or frequency domain and also it was one dimensional. The Fourier transform dual of delta function is flat constant right from minus infinity to plus infinity . How can I model a 2D and 3D delta Dirac function in space and find its corresponding Fourier transform dual. Also what will be dual of space in the Fourier domain?


title: Is there any significance to 1.21 GW?

text: In connection with a related question on Science Fiction & Fantasy Stack Exchange page: does this number 1.21 GW make any sense? What is 1.21 GWatts? With what can you compare it? Can the lightning really produce this amount of power? Are there any idea of why this number was used in a physical sense?


title: Working with carbon as a material

text: I'm interested in building a robot with a carbon shell, and was wondering where I could find some ressources about how to work with carbon. For example, where to find it, how to shape it, and how to work with it. Thank you


title: Violating Cosmic Censorship

text: Let's say we try to remove the event horizon of a Kerr black hole by throwing in matter with some large angular momentum. If it starts with GM > a, could we increase a at all? Would such a particle be able to enter the black hole?


title: Potential difference between conductors

text: Reading some books there is something that i don't understand. Is the capacitors topics when they name a potential difference between conductors. As i know, Potential difference is the difference in electric potential energy per unit charge between two points. So, talking about capacitors, the potential difference between conductors is:               a) The difference between the potential of each conductor.     b) The difference of potential between those 2 points: (conductor)·     ·(conductor)


title: Quantum Theory as a framework for other theories of nature

text: We know that Quantum Theory should be considered as a framework in which all other theories/forces (Strong, Weak, EM and Gravity) exist. For example, we have the Quantum Chromodynamics, Quantum Flavordynamics (Electroweak), Quantum Electrodynamics (but still no Quantum GR). When I think about this, it then strikes me why gravity, and specifically special relativity, is part of the framework itself (because QFT is based on QM and SR). Why would a theory like SR (and maybe in future GR) be part of the framework? This looks like a circular logic. I would appreciate it if someone can explain? My understanding is that gravity is different because it exist everywhere, and are not a result of a charge (electric, color or flavor). That's why we cannot speak about a quantum theory of EM without considering gravity, but we can do it without considering strong or weak force. Is this correct? EDIT: it would make more sense to me to see Gravity/GR considered only as a framework (a geometric one) rather than a force, or it can be considered a fictitious force, as with the centrifugal force. see first paragraph of first answer here.


title: Upward bouyant forces?

text: When an object is submerged in a fluid (e.g., water), there is a pressure on the object due to weight of water by p=mgh, but why is there an upward buoyant force?


title: What are the prerequisites to study CP violation?

text: If one would like to study CP violation, what would be the prerequisites for it? For example, until now I have not studied quantum field theory and have done very little classical field theory, but I have done some quantum mechanics-- non-relativistic and relativistic. What else do I need?


title: Supersymmetry calculations using computer algebra

text: Already the early papers on supergravity were written using computer algebra software to do some calculations. What modern packages do people normally use for doing such calculations? Of course Mathematica and presumable a lot of other programs can be made to do such calculations, I am wondering if there are special packages or tutorials how to do such calculations in them.


title: Electric Current Power Calculations

text: With regards to the equations: $P=VI$ ; $P=V^2/R$ and $P=I^2R$, if you are given $P$, $V$, $I$ and $R$ for a circuit, how do you know which equation to use? Does the use of an equation have to do with whether the circuit is in series or in parallel?


title: Parton Distribution Functions, average cross section of the nucleon

text: Say we are given the scattering cross section for neutrinos from $d$ and $\bar{u}$ quarks as $\frac{d\sigma^{d}}{dQ^2}=\frac{G_F^2}{\pi}$, $\frac{d\sigma^{\bar{u}}}{dQ^2}=\frac{G_F^2}{\pi} (1-y)^2$, and the $u,\bar{d}$ cross sections are zero, how is the **average nucleon cross section** determined? The solution involves this expression: $\frac{d\sigma^p}{dQ^2}=\displaystyle \sum_q \int _0^1 dx\left(q(x) \frac{d\sigma^{q}}{dQ^2} + \bar{q}(x) \frac{d \sigma^{\bar{q}}}{d Q^2} \right).$ Why is this the cross section for the proton? What exactly is the meaning of those functions, $q(x)$? > $q(x)dx$ is the expectation value of the number of $q$ quarks in the hadron > whose momentum fraction is within $[x,x+dx]$. Why are we using $\frac{d\sigma}{dQ^2}$ and what does it represent? After this we can take $\frac{d}{dx}\frac{d\sigma^p}{dQ^2}$ do the same for the neutron and take averages. The result is: $\frac{d^2 \sigma^N}{dQ^2 dx} = \frac{G_F^2}{2\pi}\left(u^p(x)+d^p(x)+(1-y)^2(\bar{d}^n(x) + \bar{u}^p(x))\right)$ But I don't understand the origin behind the different quantities used here and the logic behind the derivation very well.


title: Representation of the velocity field

text: Im trying to understand this line given by our prof. :   "Representing fluid parameters as a function of the spatial coordinates($x$, $y$, $z$) and time $t$. For example: $$\vec{V} = u(x,y,z,t) \vec{i} + v(x,y,z,t) \vec{j} + w(x,y,z,t)\vec{k}$$ NOTE: $\vec{i}$, $\vec{j}$ and $\vec{k}$ are the unit vectors. So my question is how come the $x$ coordinate is represented as function of all spatial coordinates and time $u(x,y,z,t)$. (similarly for $y$ and $z$).


title: How the CMB anisotropy is linked to the existence of cold dark matter and dark energy?

text: After the data from the cosmic microwave background has been collected by WMAP or Planck, what types of analysis is needed to conduct in order to deduce the cold dark matter density and the distribution of matter in the universe? In other words - How the CMB anisotropy measurements giving us evidence for the existence of dark matter/energy? (Any further explanation on the analysis of data is also welcomed.)


title: When can a center of mechanical momentum frame be found for an electromagnetic system?

text: In classical mechanics, a center of _mechanical_ momentum frame can always be found for a system of particles interacting with one another locally. For an electromagnetic system where the charges interact non-locally via the electromagnetic field, a center of _total_ momentum can likewise always be found. I have two questions: When can a center of _mechanical_ momentum frame exist for an electromagnetic system? When is this frame also inertial?


title: Delta functional in path integral

text: I've recently encountered a path integral of the form $$\int \delta[a\phi+b\phi']\,L(\phi,\phi')\;\mathcal D\phi\mathcal D\phi'$$ (where $a$, $b$ are integers) and would like to eliminate one of the $\mathcal D\phi$ integrations. Is this possible? A change of variables doesn't seem to work since e.g. for substitutions $u:=a\phi$ the path integral Jacobian isn't well-defined if $a\neq 1$.


title: Does existence of magnetic monopole break covariant form of Maxwell’s equations for potentials?

text: Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\operatorname{div} \vec B = 0 \text{ (1).}$$ This equation allows us to introducte concept of vector potential: $$\vec B = \operatorname{rot} \vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = = - \frac{\vec j}{\epsilon_0 c^2} \text{ (2)}$$ $$ \nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0} \text{ (3)} $$ Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using dalambertian: $$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0} \text{ (4)}$$. If magnetic monopols exist, Maxwell's equation (1) will look as: $$\operatorname{div} \vec B = \mu_0 c \rho_{magnet}$$ As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduct concept of vector potential. Thus, the equation in the form of (4) will not be possible to express.


title: Why do we say that in Coulomb's law the force is proportional to $\frac{1}{r^{2}}$ and not $\frac{1}{r^{3}}$?

text: I am going over Coulomb's law and there is something that is a bit confusing for me: According to Coulomb's law, if I have a charge $q_{1}$ at a point $\vec{r_{1}}$ and a charge $q_{2}$ at a point $\vec{r_{2}}$ then the force that the first charge applies to the second charge is given by $$ F_{1,2}=\frac{Kq_{1}q_{2}}{|\vec{r_{2}}-\vec{r_{1}}|^{2}}(\vec{r_{2}}-\vec{r_{1}})=\frac{Kq_{1}q_{2}}{|\vec{r_{2}}-\vec{r_{1}}|^{3}}\hat{(\vec{r_{2}}-\vec{r_{1}})} $$ I see that in the first expression is does look like the force is proportional to $\frac{1}{r^{2}}$, but I don't understand why it isn't the second expression that matters, giving us that the force is proportional to $\frac{1}{r^{3}}$: Say both charges are on the $x-$axis, then $\hat{(\vec{r_{2}}-\vec{r_{1}})}=\hat{x}$. Increasing the distance between the charges to two times what is by moving the second charge on the $x$- axis was, would decrease the force $2^{3}=8$ times, since $\hat{(\vec{r_{2}}-\vec{r_{1}})}$ is still $\hat{x}$. What is the mistake in my reasoning ?


title: Atoms' excitation energies as derived from Frank-Hertz experiment data

text: I'm analyzing the experimental data obtained during Frank-Hertz experiment (conducted with Hg atoms): ![enter image description here](http://i.stack.imgur.com/D3K82.jpg) Accelerating voltage values were multiplied by 0.1 during measurement (i.e. the mean value of energies differences if not 0.508 eV, but 5.08 eV). The output voltage was measured on resistive load of the anode, therefore it is proportional to anode's current. The first five minimas in output voltage (top graph) have constant voltage difference - this value is used to calculate the first excitation energy of Hg atom (bottom graph). The sixth minima, however, can be seen to have larger increase in output voltage(top graph - the first five voltage peaks may be fitted by a linear graph, whereas the sixth is no longer fits to this graph) and a higher voltage offset from the previous minima (bottom graph). **Question 1:** Is the sixth minima corresponds to the higher order excitation of Hg, or ionization, or else? **Question 2:** The curve is clamped at $V_{out} = 5V$. Is there a physical effect which causes this, or it is just the limit of measurement equipment, or else?


title: How to work out momentum when there are velocity and mass changes

text: I have a pretty simple homework question, but I can't rap my head around it. In the question a swimmer of $55 \mbox{ } \mathrm{kg}$, jumps off a stationary raft of $210\mbox{ }\mathrm{kg} $. The swimmer jumps off the raft with a speed of $4.6 \mbox{ } \mathrm{ms}^{-1} $. I need to work out the recoil velocity of the raft. So because Momentum before = Momentum after, I went: $p_i = 0$ Therefore $0 = p_f$ and $p_f = mv$, $m = 210\mbox{ }\mathrm{ } $, so the $v$ would have to equal $0$. Making the recoil velocity equal $0$. However that doesn't seem right. Could use some clarification or help, thanks.


title: Why does a microwave oven affect other electronic devices

text: When I turn my microwave oven over the stove on, it will cause a motion sensor light in the hallway next to the kitchen to got off and on. This affect can be reproduced anytime. I did notice that we stand in front of the microwave instead of out of the room, the motion sensor light goes on then off then on repeatedly. How is the microwave affecting the motion sensing light that is about 4 ft. away behind a plaster wall? Is it dangerous?


title: Gravity: Why is the inverse square law exactly $1/r^2$ instead of just close to 2?

text: Is there something intrinsic about the structure of space that gravity is proportional to 1/r^2 instead of, for example, 1/r^2.143 ? What makes the exponent turn out to be a nice even number?


title: Differential cross section in momentum space?

text: Suppose I have a spherically symmetric potential and I can find its cross section in configuration space (i.e position-space), $d\sigma / d\theta$. Now I need to find its distribution $d^2\sigma / p_rdp_rdp_\theta$ in momentum space. How can I do this?


title: Can you launch an item farther by slamming into it or accelerating it along a length

text: I am trying to desgin a little penny balista toy (built with a 3d printer). The short description is that there is a track with a mass ("hammer") that slides along it. There is a stack of pennies in a hopper at the front. The hammer goes from back to the front, slams into the penny at the end and launches it. I had a debate with a friend about whether it would go farther if the penny started at the back and was accelerated along the full length of the track along with the hammer (a more difficult design). My question is, all else being the same, which method would impart more energy to the penny? (my suspicion, based on those "newtons cradle" toys, is that it would be the same).


title: Why is Titan able to maintain an atmosphere, and not Luna?

text: I just read stability of hypothetical lunar atmosphere. From the correct answer, i understand, the low escape velocity from Luna is part of the reason it is unable to retain an atmosphere. Titan apparently has a comparable escape velocity   * Titan = 2.65 km/sec    * Luna = 2.4 km/sec ; yet Titan maintains an atmosphere. Why? What have I missed? Does Luna's relative proximity to Sol make the difference?


title: Is a photovoltaic cell on an artificial satellite the same construction as one used on Earth's surface?

text: Are photovoltaic cell arrays on a satellite the same that are used within Earth, or is there some difference in their construction given the differing environment in which they are to operate? Does the efficiency/life of a photovoltaic cell change depending upon whether it is within/without Earth's atmosphere?


title: Rigid Body Mechanics In a vacuum (no gravity)

text: I am writing a physics engine to map the rotary and translatory movements of a uniformly dense solid cylinder within 3d space. If a vectored thrust is applied to one end of the cylinder at an arbitrary angle (say 30 degrees off center), I understand this will cause both a rotation and translation of the body. ROTATION....... Torque= radius*force*sin(angle) . moment of inertia(of a uniform cylinder) = 1/12*mass*(3*radius^2+height^2). Angular acceleration= torque/moment of inertia. TRANSLATION...... linear acceleration= force/mass. IF my thrust angle is 0 then sin0=0 so my torque=0. But my linear acceleration(in-line with thrust vector) is the same. Surely if some of the thrust is producing a rotation then I should have a lower translatory acceleration? I reason this as it takes energy to both rotate and translate, therefore I seem so be getting more energy with vectored thrust then when operating with a thrust angle of 0(through the cog). Maybe my intuition is wrong here.... thanks


title: Calculating acceleration function for given time, distance, initial velocity

text: An object has an initial velocity of **v** and should stop in **d** meters, in **t** seconds where **v*t >= d**. I can predict that this acceleration won't be constant but it is a function of time and this problem may not have one exact solution for given **v** , **t** and **d**. But I am not able to think of any solution, can any one help me to calculate this function?


About how fast can a small fish swim before experiencing turbulent flow around its body? An Engineering Problem! Please go through this question step by step. :D


title: Explain Turbulence

text: I'm a high school student. I still don't understand what turbulence is. Please can you explain what it really is? This is what I think it is: rotating motion of water when a particle travels at a velocity of $V$.


title: Consistency of equation with special relativity?

text: The following is the equation which, I want to know, if it is valid in relativistic domain. Consider two equal charges moving in same direction with velocity $v$ and charge $q$ at a separation of $d$. The magnetic force acting between them is $(1/4\pi\epsilon_0 c^2)v^2q^2/d^2$ and the electrostatic force is $(1/4\pi\epsilon_0 )q^2/d^2$.the ratio of the magnetic (attractive) force and the electrostatic (repelling) force is $v^2/c^2$ and therefore the conclusion provided was that two individual charges with same (or any) velocity can never attract. Is this equation valid relativistically. Moreover, since magnetic field is just the relativistic counterpart of electrostatic force, can we find a frame of reference wherein in the case stated, all the force is magnetic and must be attractive? I am comfortable only with basic special relativistic mechanics but an intuitive understanding would be better.


title: Do we have 2 minutes of extra morning?

text: my physics teacher told me about the refraction and its applications one of them was   **2 minutes of early sunrise** and after she explained this effect she concluded that days are   **2 minutes longer** than one would naively presume. However, I think that **her conclusion is wrong** because   **if sunrise is considered then sunset should also be considered** and according to me **sunsets should be 2 minutes late** therefore **the day time is increased by 4 minutes and not 2 minutes** over the naive calculation. According to me, the situation looks something like this:   ![enter image description here](http://i.stack.imgur.com/O1YUG.jpg)   Is this idea of mine correct or not? And if we both are wrong, then what should be the right conclusion and why ?


title: Conceptual difficulty in understanding Continuous Vector Space

text: I have an extremely ridiculous doubt that has been bothering me, since I started learning quantum mechanics. If we consider the finite dimensional vector space for the spin$\frac{1}{2}$ particles, I guess it is nothing but $\mathbb{C}^2$. Each vector has two components (which is why it is two dimensional right ?), each of which can be any complex number. Now coming to the case of position space (say one-dimension). I was taught this LVS is infinite dimensional (also continuously infinite, unlike the number operator basis). I am not able to understand this subtle thing that it is infinte dimensional (is it something like $\mathbb{R}^\infty$?). It is quite confusing every time I encounter this kind of space. Also in this **each component** (of the **infinite no. of them** ) can **take any real value (infinite number of them)**? I learnt that the way to represent these can be in term of complex-valued functions, I would like have it elucidated.


title: A paradox related to relative motion

text: I know that it is a very old question but still I don't find any satisfactory solution for Achilles Paradox. Please explain me the fundamentals of Achilles paradox in terms of stage wise distance covered. Note that it is easily solvable in terms of time, but if you start analysing this event in terms of time, then there is not at all any paradox. So please explain in terms of stage wise distances only.


title: Vector product in 2 dimensions

text: If I have a vector A=4i+3j and B=5i-2j, how can I find the vector product AxB? I know that given the angle, its C=AB sin theta, but how can I solve this without the angle?


title: Understanding gyroscopes

text: Considering the typical situation of a rotating bicycle wheel held by one end of its axle by a rope tied to the ceiling: gravity torque is the time derivative of the angular momentum, and in this case is perpendicular to the angular momentum vector which makes it revolve without changing its magnitude (it precesses). My question is the following: the wheel stops turning eventually because of the friction in the string (it also slows down its own rotation because of friction in the bearing but let's ignore it), but could you clarify how, step by step? Is it because the gyroscope creates a torque on the string axis, but it's reduced by the friction torque, and as a result the friction torque translates back into a residual "gravity torque" (in other terms, the gyroscope "gives up" on a certain amount of gravity torque it can't counteract because of the string)? To progress toward my actual problem in several dimensions, what if the friction torque is in fact torque given by another gyroscope on the same solid?


title: Group of translations in two dimensions - A weird treatment

text: Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$ G = p_1\delta x_1 +p_2 \delta x_2 $$ Now, we know clearly that this group is an Abelian group, therefore $ [p_1,p_2] = 0$. But the author suggests this is equivalent to saying that the commutator generates any unitary transformation without physical consequences, and goes on to write, $$ \hbar ^{-1}[p_1,p_2] = \mathbb{1} \tag{1}$$ Now, effect of an infinitesimal change in an operator($X$) is given by $\delta X = -i[X,G] $ where G is the generator of an unitary transformation ($U$) Hence I obtain, $$ \delta p_1 = -i[p_1,G] = \delta x_2 $$ $$ \delta p_2 = -i[p_1,G] = -\delta x_1 $$ Which then means, $$ x_1 = -p_2 \equiv q \:\:\:\:and\:\:\:\: x_2 = p_1 = p $$ Now, we see the association to the single particle's phase space state $(q,p)$. Further translations in this (phase-space) two-dimensional space are described by the three parameter group with the generator, $$ G = p\delta q -q\delta p + \delta \phi \mathbb{1} $$ My questions are : a) How is the choice of commutator in equation $(1)$? b) What exactly is the author trying to do here, or what is he motivating from this exercise? Why does he connect the translation group in normal 2D space to phase space of a single particle? c) What is this operator $\delta \phi \mathbb{1} $ in the last generator?


Is there any way to annihilate matter without the use of anti-matter? And vice versa? I mean, for example is it possible to totally convert the mass of a proton into "pure energy" without use an anti-proton?


title: What is drift velocity? And how to find the equation for it?

text: What is drift velocity? And why in some books it is expressed as **drift speed** and not **drift velocity**?Are these different ? Does it mean that the electron will have extra velocity opposite to the direction of electric field i.e **thermal(random) velocity + drift velocity=Total velocity** of electron,in the conductor with a specific direction (opposite to electric field)


title: What was Feynman's "much better way of presenting the electrodynamics" -- which did **not** appear in the Feynman lectures?

text: Does anyone know what Feynman was referring to in this interview which appears at the beginning of The Feynman Tips on Physics? Note that he is referring to something that did not appear in the Feynman lectures. > I didn't like to do the second year, because I didn't think I had great > ideas about how to present the second year. I felt that I didn't have a good > idea on how to do lectures on electrodynamics. But, you see, in these > challenges that had existed before about lectures, they had challenged me to > explain relativity, challenged me to explain quantum mechanics, challenged > me to explain the relation of mathematics to physics, the conservation of > energy. I answered every challenge. But there was one challenge which nobody > asked, which I had set myself, because I didn't know how to do it. I've > never succeeded yet. Now I think I know how to do it. I haven't done it, but > I'll do it someday. And that is this: How would you explain Maxwell's > equations? How would you explain the laws of electricity and magnetism to a > layman, almost a layman, a very intelligent person, in an hour lecture? How > do you do it? I've never solved it. Okay, so give me two hours of lecture. > But it should be done in an hour of lecture, somehow -- or two hours. > > **Anyhow I've now cooked up a much better way of presenting the > electrodynamics, a much more original and much more powerful way than is in > the book.** But at that time I had no new way, and I complained that I had > nothing extra to contribute for myself. But they said, "Do it anyway," and > they talked me into it, so I did. Did this approach to teaching electrodynamics appear in any of his later writing?


title: Rotating a complex number

text: Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\vec{V}(x,y)$ It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers "fundamental" – to combine the components into a complex number, $z=x+iy$. The two-dimensional rotations in $SO(2)$ are fully determined by the angle $δ$. And the matrix acting on the vector $\vec{V}(x,y)$ $M= \begin{bmatrix}+\cosδ & −\sinδ\\\\+\sinδ & +\cosδ\end{bmatrix}$ may be fully replaced by the complex coefficient $e^{iδ}$ that multiplies our complex coordinate $z$. Instead of $z$, however, we could have dealt with its power $z^p$. The rotation could be described by the complex transformation $z^p→z^{′p},z^{′p}=e^{ipδ}z^p$. In this context, the complex number $e^{ipδ}$ plays the role of the "transformation matrix" that rescales the one and only component of our tensor-spinor-whatever, $z^p$. In our overly trivial two-dimensional context, the coefficient or exponent $p$ may be anything you want. But the value $p=1/2$ may be identified with the spinors in two dimensions. The spinor in two dimensions may be represented as $\sqrt{z}$ where $z=x+iy$ encodes a vector; so the spinor $z^{1/2}$ is literally the square root of a vector (translated to a complex number) in this case. So I got all this from a blog that tries to explain spinors which unfortunately I don't remember. I don't understand why the $p$ in $e^{ipδ}$ has to be there since any complex number can be rotated without the need for that coefficient. Is there some rule for rotating complex numbers that are raised to higher powers $p=2,...,n$ or am i missing something?


title: Crystal Momentum in a Periodic Potential

text: I'm working through some basic theory on periodic potentials, and I would appreciate help in understanding the crystal momentum. Suppose we have a Bravais lattice with lattice vectors $\textbf{R}$. There is an associated reciprocal lattice with lattice vectors $\textbf{K}$ such that $\textbf{K} \cdot \textbf{R} = 2\pi n$ for $n \in \mathbb{Z}$. The relationship between these two lattices ensures that plane waves of the form $e^{i \textbf{K} \cdot \textbf{r}}$ are periodic in the direct lattice. A consequence of Bloch's theorem is that the states $ \langle x|\psi \rangle$ of a particle assume the form $$ \psi_{n\textbf{k}}(\textbf{r}) = e^{i \textbf{k} \cdot \textbf{r}}u(\textbf{r}), $$ where $$ u (\textbf{r} + \textbf{R}) = u(\textbf{r}). $$ For these wavefunctions, $\textbf{p} \equiv \hbar \textbf{k}$ is defined to be the crystal momentum. Canonical momentum is ill-defined for this problem since the crystal breaks translation symmetry. However, for any translation $T_{\textbf{R}}$ within a lattice vector, $[H, T_{\textbf{R}}] = 0$. My questions are:   1. In the first equation, I currently believe that $\textbf{k}$ can be any vector, and is not necessarily in the set of reciprocal wave vectors (i.e., $\textbf{k} \notin \\{\textbf{K}\\}$ necessarily). Since this is true, what is $\psi_{n\textbf{k}+\textbf{K}}?$   2. Suppose a particle has crystal momentum $\textbf{p} = \hbar \textbf{k}$. How do we interpret $\textbf{p}' = \hbar (\textbf{k} + \textbf{K})$?   3. Although there is no _continuous_ symmetry in the lattice, there is a discrete symmetry of the potential $U(\textbf{r} + \textbf{R}) = U(\textbf{r})$, and therefore of the Hamiltonian. If Noether's theorem does not apply here, what quantity is "conserved" in time, and how do we justify such a conservation in general?


title: Coordinates of the extra dimensions

text: If we live in more than three spatial dimensions, is it not right to conclude that all matter observable to us shares almost the same coordinates of extra dimensions. Or is it just that ordinary interactions (like electromagnetism) are simply independent of these extra dimensions?


title: Deriving conserved currents by promoting parameter

text: I currently reading Tong's text on String Theory. In Chapter 4.1.1 he alludes to a technique to derive conserved currents > Recall that we can usually derive conserved currents by promoting the > constant parameter $\epsilon$ that appears in the symmetry to a function of > the spacetime coordinates. Since I don't know what he refers to, could someone provide me a proper reference?


title: What is the 1/2 spin analog of the graviton called?

text: In some supergravities you have the gravition, gravitino, graviphoton and graviscalar. Each is analogous to each other in only sharing gravitational properties and nothing else. They differ by spin in the order of 2, 3/2, 1 and 0. What is the spin 1/2 analogue named?


title: Magnetic field exclusion and retention in superconductors

text: I'm studying the behaviour of superconductors in different cases, and I can't understand this. We have a superconducting cylinder with a coaxial hole inside (vacuum). When the cylinder is cooled below it's critical temperature, and later a magnetic field is applied (case a), the superconductor will exclude the field even from the hole (B=0 inside the hole). However, if we first apply the magnetic field and _then_ we cool down the cylinder (case b), the magnetic field will be excluded from the superconductor but not from the empty hole (as seen in the image below). Thus, depending on the procedure, we have two possible final states. My question is, why is there this difference? Why doesn't the superconductor allow a magnetic field inside the hole in case a? It would still be excluding it from all the superconducting volume. (Sorry, I don't know the source of the image, I took it from my professor's notes) ![Field exclusion, expulsion and retention for different geometry superconductors following different procedures.](http://i.stack.imgur.com/LXXup.png)


title: How jet engine works in vacuum

text: Its not about availability of oxygen for combustion of gases. I want to know since there is no air or friction providing things in vaccum, how the force applied by a jet engine get can make thrust to move the vechile


title: Textbook on the Geometry of Special Relativity

text: I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ? I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do **_not_** need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth. Thanks in advance.


title: Is this kids experiment a legitimate way to show that air has mass?

text: Consider the experiment in this link. The experiment includes using a ruler as a lever, with an inflated balloon on one side and a balloon which is not inflated on the other. The aim of the experiment is to show that air has mass. I have seen many kids performing similar experiments. But, if the air pressure inside the balloon is equal to that outside, then the buoyant force will cancel out the weight of the air inside the balloon, won't it?


title: Work in Newtonian Mechanics and Thermodynamics

text: Well, is the basic difference between the work that we learn in Mechanics and that in Thermodynamics? This is because in Mechanics, whenever work of magnitude $W$ is done on a system $S$, then the system $S$ also does a work of magnitude $W$ on its surroundings. If I take this consideration in Thermodynamics, then according to the sign conventions, the total work in any process would turn $0$ (once positive and once negative), which is not the case. I am really very confused regarding these. Please explain what exactly is work in Thermodynamics, and its differences with Mechanics.


title: Sampling a distribution (from a galaxy model)

text: I am reading the following article: http://www.kof.zcu.cz/st/dis/schwarzmeier/galaxy_models.html and am currently at section 5.6 (positions of bodies in a galaxy). I am trying to redo the simulations myself in Python, but I have a questions regarding sampling of the distribution. Given a distribution function (Hernquist distribution): $$\rho(r)=\dfrac{M}{2\pi}\cdot \dfrac{a}{r(a+r)^3}$$ the article states that to simulate the distribution, one has to calculate the mass within a circle of radius $r$ like follows: $$m(r) = \int_{0}^{r} 4 \pi r'^2 \rho(r') dr'$$ which is the cumulative mass distribution function. The article states that this formula represents the PDF. However, looking at the shape this appears to me to be a CDF. Approaching infinity the function approaches 1.0 which is for me a clear indication of a CDF. To sample this distribution the article cites the Von Neumann method where one has to generate an $r$ and a $m$ value, and scale them accordingly, and check whether or not they fall below the $m(r)$ graph. If they do they are accepted, else they are rejected. Am I completely off by thinking this is wrong? If I do this I end up with the majority of stars ending up at the higher radii. I have the feeling I am sampling an CDF here instead of a PDF. To get accurate results (e.g.: having the majority of the stars in the center) means that I have to perform the Von Neumann method with the $\rho(r)$ function. I am unable to contact the author of the article, so that's why I am asking here.


title: Connection between BAO und CMB Spectrum

text: I have a problem understanding the connection between the accoustic peaks in the CMB spectrum and the baryon oscillation picture. On the one hand it is stated, that the odd accoustic peaks (1,3,5..) are compressions peaks, i.e. peaks which have just reached maximum compression as decoupling occured. On the other hand there is the BAO picture, nicely illustrated for example here: http://astro.berkeley.edu/~mwhite/bao/ . An overdense region starts to expand, because of gas and photon pressure, right after inflation and stops at the moment recombination occurs. The postulated peak at the distance 150 Mpc (the sound horizon $c_{sound} t_{rec}$ at recombination) could be found in galaxy survey experiments. Furthermore it is stated this corresponds to the first accoustic peak in the CMB spectrum. Clearly this contradicts the statement made above: Here the first peak should be a maximum rarefaction peak. Besides that i don't know how to understand higher accoustic peaks in this picture. For me there should be no reason why the next peak, a maximum rarefaction peak, should be at approximately half the wavelength of the first peak. I can't see that we have here aboundary condition that would force a certain wavelength. Any ideas or references that enlightens this confusing connection a bit more would be much appreciated.


title: Preons as scalar-like fermions?

text: The symmetry of the standard model between the fermions and each generation of them is strongly suggestive of components as proposed in the Rishon model. Having discovered the Higgs boson, we know the Higgs plays a strong role in the mass of the fermions. Having heard of vector-like fermions, is there something that could be called a scalar-like fermion? Is it possible for Rishon fermions to exist with scalar properties to act like a Higgs boson and provide a substructure to fermions that obeys the standard model?


title: Why water vapour condense easier in contact with a solid?

text: Reading about cloud formation, I learned that to a droplet to form, the water vapour needs a Cloud Condensation Nuclei, which is an aerosol with a size in the order of 0.0001mm. And if no CCN is found temperatures as low as -13°C are needed to form the droplets. Why this is happening? What does the contact of the solid with water do to make it condense?


title: Direction of expansion of the universe

text: From what I understand the expansion of the universe has no "center". If we're flying through space away from the "center of the big bang", there's basically no way to tell. Every two given points in space gets farther away from each other, and we can pick any point as center if we like. I also understand that the speed of light is not relative to the speed of the source emitting the light. If I go on a train in $c/2$, turn on a flash light pointing forward, the light emitted from the flashlight will still travel at the speed of $c$. Now here's my question: Why can't we set up a sphere with photodetectors with synchronized clocks on the inner walls, turn on a light in the center, record the exact time at which each photodetector detects the light, and compare the times to figure out if the sphere was traveling in some certain direction? I mean if we turn on a lightbulb and light travels with the speed of $c$ in all directions at the same time, my intuition tells me that we should be able to figure out some form of "reference stand still". (Tagging this with general relativity because I _suspect_ that it's impossible to set up the experiment the way I like due to relativity.)


title: AC Electromagnets

text: Could someone help explain the uses of AC electromagnets. Wherever I look it says that DC electromagnets create stronger magnetic fields. I understand why AC electromagnets could be used in transformers but why use them in motors for example? Power stations also use electromagnets in the generator to create the magnetic field, would these be AC or DC? Finally, which type of current would electromagnets in household motors use e.g. food processors, blenders etc?


title: How much does the sound definition vary during an LP (Vinyl)?

text: This question came to me when I realized how the linear speed varies while listening to a Vinyl LP. The linear speed variation has to be compensated with a variation in the resolution of the grooves, that is, since the linear speed decreases, the groove resolution also has to decrease in some measure. What is this measure of reduction, or else, how much does the linear speed reduce? And how does that influence the sound definition?


title: Why is the inner product between divergence-free current$\vec{J}$, (satisfied $\nabla\cdot\vec{J}=0$) and a gradient field$\nabla \varphi$ zero?

text: I read a statement saying that the inner product between divergence-free current and a gradient field is zero. Divergence-free surface current is $\nabla\cdot\vec{J}=0$, and $\vec{J}$ could be represented as $\vec{J}=\nabla\times(\psi\hat{n})$, where $\hat{n}$ is the normal vector of the surface. So the statement becomes: $\nabla\times(\psi\hat{n})\cdot \nabla \varphi=0$. I think according to the identity: $$\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$$ we have $$\nabla\times(\psi\hat{n})\cdot \nabla \varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)+\psi\hat{n}\cdot\nabla\times\nabla\varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi),$$ but what next? **Update** Thank you Luboš Motl. I suppose I now understand why, but I have no enough points to reply below, so just update here my answer. Target is to prove $\int_s \vec{J}\cdot\nabla\varphi ds=0$ The whole process is as follows: First, $\vec{J}$cannot go across the surface edge, so $\vec{J}\cdot\hat{t}=0$, where $\hat{l}$ is the surface edge direction and $\hat{t}=\hat{l}\times\hat{n}$ is the edge out direction Second, according to the identity $\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$, we have $\vec{J}\cdot\nabla\varphi=\nabla\times(\psi\hat{n})\cdot \nabla \varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)+\psi\hat{n}\cdot\nabla\times\nabla\varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)$ since $\nabla\times(f\vec{A})=\nabla{f}\times\vec{A}+f(\nabla\times A)$ $\psi\hat{n}\times\nabla\varphi=-\nabla\times(\varphi\psi\hat{n})+\varphi\nabla\times(\psi\hat{n})$ Then $\nabla\cdot(\psi\hat{n}\times\nabla\varphi)=\nabla\cdot(-\nabla\times(\varphi\psi\hat{n})+\varphi\nabla\times(\psi\hat{n}))=\nabla\cdot(\varphi\nabla\times(\psi\hat{n}))$ finally, $\int_s \vec{J}\cdot\nabla\varphi ds=\int_s\nabla\times(\psi\hat{n})\cdot \nabla \varphi ds =\int_s \nabla\cdot(\varphi\nabla\times(\psi\hat{n}))ds=\oint_l \varphi\nabla\times(\psi\hat{n})\cdot\hat{t}dl=\oint_l \varphi\vec{J}\cdot\hat{t}dl=0$ I think here the important things are:   1. Generally speaking, divergence-free current usually can be expressed as $\vec{J}=\nabla\times\vec{T}$, and $\vec{J}=\nabla\times(\psi\hat{n})$ is specially for surface current.   2. the $\hat{n}$ is only valid on the surface(there is no meaning of $\hat{n}$ for point in side of a body). the integral is on the surface rather than on the body. According to the original article, it is just talking about PEC and surface current.


title: Why does adding red light with blue light give purple light?

text: Our eyes contain 3 photoreceptor cells (cones) to perceive three wavelength ranges of light. Here is a visual representation of the wavelengths by these receptors (S, M and L). ![Wavelength per photoreceptor](http://i.stack.imgur.com/5olj4.png) So if we have light of 440 nm, it results in the color blue. If we have light of 540 nm, it results in the color green. If we see light of 650 nm, it results in the color red. I think I understand our ability of the brain to mix the results of the signals of these receptors, producing colors like yellow. However, what I do not understand is how the color spectrum is displayed like this: ![Visible spectrum representation by wavelength](http://i.stack.imgur.com/IUxg1.gif) Given that spectrum, I would suggest that the color 'blue' is actually a mix between the receptors S and M. And the pure result of activating the S-receptor would result in the color 'purple' (I would describe the color in the left of the image as purple, right). Therefore, the receptor colors should instead be RGP (red, green, purple) instead of RGB (red, green, blue). However, there is one problem with this which I cannot explain. How come that mixing red light with blue light also results in purple light? How is it possible that purple light can be achieved through mixing (additively) blue and red light, just as going to the shortest wavelength boundary of what we can see (from blue to ultraviolet via purple)? So the actual problem here is:   1. Purple is the color at the very shortest wavelength we can see.   2. Purple is an additive mix between what we see as red light and blue light. That just doesn't make any sense. I don't see how our brain can possibly perceive this as being the same color. Shouldn't both purple colors actually be different colors (thus we would have a new different color for that)?


title: Is it possible to make a hydrogen-alpha solar scope?

text: Is the construction of an etalon / Fabry-Pérot interferometer within the reach of amateur telescope makers? Are there any resources pointing to such projects?


title: Emission line width units

text: What do these units mean: the large velocity widths of emission lines (in AGN) are 2,000 - 10,000 km s^-1? I've looked for the answer but keep getting swamped in myriads of details. I want to know what the term _velocity_ means in _large velocity widths_ and also why _km s^-1_. Thanks


title: Is acceleration an absolute quantity?

text: I would like to know if acceleration is an absolute quantity, and if so why?


title: Why are there not many detectable supernovas?

text: Astronomers estimate that there are between 200 billion to 400 billion stars contained within the Milky Way and Andromeda Galaxy probably has 1 trillion stars. There may be around 500 billion galaxies in the observable universe. So, my question is, statistically speaking because of the number of stars, there is should be lots of chances of stars going supernova; where at times they glow brighter than a whole galaxy. Why then do we not see many supernovas, for example like the 1987A? Why have we not been able to see one in our own galaxy since the SNR G1. Should there not be more supernovas in surrounding galaxies and even our own one as it has around 400 billion stars. I appreciate that there are different types of stars and various life span, but our galaxy being almost as old as the universe, surely there should be stars dying all the time. Thanks


title: Could spontaneous symmetry breaking happen again in our universe?

text: It is generally believed that $10^{-35}$ seconds after the Big Bang, the symmetry of a GUT was broken and after $10^{-12}$ seconds the electroweak force was broken: \begin{equation} \mathrm{SU(2)} \times \mathrm{U(1)} \rightarrow \mathrm{U(1)} \end{equation} This symmetry breaking is a result of the universe cooling down and undergoing a phase transition. I'm aware that the temperature of the universe it about $2.7$ Kelvin, so the temperature of the universe cannot decrease much more, but I was wondering if there is a chance that another phase transition might happen again in the future?


title: Brewster angle with diffraction propagation?

text: Diffraction theory is scalar. How you deal with beam propagation in fourier optics that is sensitive to the to the polarization? If I have linearly polarized gaussian beam incident on glass surface, how is the polarization included in the propagation code/theory ? What I am ultimately interested is to have let say linearly polarized input gaussian beam to propagate through a hollow fiber. I will use split step to propagate if that matters. And how is it done if thats not good approach ?


title: Is Kinetic Energy stored as mass?

text: I know that if some object acquires potential energy, it also gains mass- is it the same for kinetic energy?


title: Capacitance from Maxwell's equations

text: I am trying to get the definition of capacitance nailed down. This is my work so far: $$ \frac{\mathrm{d}U}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega_a}{ u(\mathbf{x},t)\mathrm{d}t} = \int_{\Omega_a}{ \partial_t u(\mathbf{x},t)\mathrm{d}t} = -\int_{\Omega_a}{ \mu_0^{-1}\nabla\cdot[\mathbf{E}\times\mathbf{B}] + \mathbf{E}\cdot\mathbf{J}\mathrm{d}t}\\\ U(t) = \frac{1}{2}C[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]^2\\\ \text{ Assuming C constant in time}\\\ \frac{\mathrm{d}U}{\mathrm{d}t} = C[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]\frac{\mathrm{d}}{\mathrm{d} t}[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]\\\ C\frac{\mathrm{d}}{\mathrm{d}t}[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)] = \frac{\frac{\mathrm{dU}}{\mathrm{d}t}}{[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]} = \frac{-\int_{\Omega_a}{ \mu_0^{-1}\nabla\cdot[\mathbf{E}\times\mathbf{B}] + \mathbf{E}\cdot\mathbf{J}\mathrm{d}t}}{[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]} $$ So where can I go from here? I have never seen anyone talk about electrodynamics and capacitance at the same time. I guess there is some kind of dimensional analysis in play that allows certain approximations of the Maxwell equations?


title: Computing Latitude Given Quiescent Gyroscope Data

text: Suppose I place a gyroscope in a theoretically perfectly quiescent, closed room. Let its output be given as a vector ${\bf v} = (v_x, v_y, v_z)$ indicating rate of rotation around three orthogonal axes, where the $x$-axis points towards true north and the $z$-axis points vertically. Given the data, how could I calculate the latitude of the test location? Safe to assume Earth is a perfect sphere. I reckon that it equals $\frac{180}{\pi}\arctan(v_z/v_x)$ should do it, because at the equator (where the latitude is zero), $v_z=0$, and as you increase the angle (move N, say) more and more of the magnitude of the rotation vector is accounted for by the $z$-axis. Am I thinking about this correctly? Can someone help me formalize the argument?


title: Controlling Neutrinos for Communication

text: Neutrinos travel straight through earth at the speed of light. Therefore, it seems that they could be great for intercontinental communication. Of course, I assume a lot still needs to be learned about detecting, producing and controlling neutrinos before they can be used for the practical purpose of communication. My question: In principal, could neutrinos be manipulated similarly to radio waves for the purpose of communication? I mean, modulation, filtering, etc. ?


title: Good introductory books on AdS/CFT correspondence

text: > **Possible Duplicate:**   >  Introduction to AdS/CFT Since my question in a similar topic was deleted, I'll ask away and hope ppl won't come here telling me: this was already asked! :\ I have a course in advanced gravitation, and I was thinking of doing this for a seminar, and maybe something about it for a masters thesis (but there is still time for that), so I'd need something that's readable and something that can introduce someone into this field of physics. I have some knowledge of particle physics (mostly what I've learned through Griffith's book), and I've started to read Introduction to AdS-CFT, lectures by Horatiu Nastase that I found on ArXiv, and so far I can follow what he's talking about (I am familiar with path integral notation, Feynman diagrams...). But I'd like some 'simple' books, so that when I'd be giving presentation, my colleagues won't be blindly looking at me and wonder what the hell I'm talking about :D Any suggestion is appreciated :)


title: AdS/CFT seminal papers?

text: I am about to begin my PhD in the applications of duality and holographic techniques to open problems in condensed matter physics. An area often called AdS/CMT. Having seen some relevant reviews, I realize that it is easy to find some nice descriptions and discussions concerning how holography is used to tackle such issues, i.e. holographic superconductors etc. However, in most such reviews the origin of the correspondence is only at best motivated, lacking any detailed discussion. I would like to ask your opinion about which two or three seminal papers regarding AdS/CFT should I focus on, in order to gain a deeper understanding on how the original correspondence was formulated on the first place. Brief remarks together with your suggestions are more than welcome.


In the weak interaction limit, behaviors of electrons can still be described by a 1-partitcle equation with a modified mass, where the change in mass can be understood as effects of other neighboring electrons as a single electron drags along. Since the cluster of electrons couples to gravity as usual, then the effective mass of a quasi-particle couples to gravity in the same way as that of a genuine electron. Moreover, the effective mass can be negative which is the case of holes which are “bubbles” in the electron liquid. In graphene, there are massless Dirac electrons. Do they couple to the gravity? **I think what I intended to ask should be "How to incorporate effects of gravity in a many-electron system?"** (Originally I asked "do effective masses couple to gravity?")


title: How to calculate speed given trajectory and acceleration?

text: I'm having real problems trying to solve the following: "The particle is moving in the plane. The trajectory is given by $y=cx-bx^2$, where $c$ and $b$ are positive constants. The acceleration of the particle is always constant and given by $\underline{a}=-\underline{j}$. Find the speed of the particle at the origin O (0,0)." I am aware that trajectory differentiates to velocity which differentiates to acceleration but I can't work out how to apply it in this case... I'm used to seeing trajectory given in terms of $r(t)=...$ and I can't deduce how to convert it into this form using the information given. I'm really hoping this is actually very simple and I'm missing one key idea!


title: Vibration of pulley and string system

text: So here's the statement: A pulley of a mass $M$ is hanged using a spring (stiffness of the string being $k_1$), as shown in the image. What is the frequency of the pulley's oscillation? ![](http://oi43.tinypic.com/255lxf5.jpg) So that's as far as I could get: Linear motion equation: $$T_2+T_1-mg=ma \tag{1}$$ Where $T_2$ is string tension on the spring side ($T_2=-kx$) and $T_1$ being string tension on the other side. Now for the rotational equation: $$R(T_2-T_1)=I\alpha \tag{2}$$ $$a=R\alpha\tag{3}$$ $$I=\frac{mR^2}{2}\tag{4}$$ Now I have a feeling that the problem should be solved using the equations: $$x'' + w^2x=0$$ $$T=\frac{2\pi}{w}$$ In this case $x''=a$. From the equations (2), (3) and (4) we derive that $$T_2-T_1=\frac{ma}{2}\tag{5}$$ And adding (5) and (1) we get $$-2T_2+mg+\frac{3}{2}ma=0$$ or $$2kx+mg+\frac{3}{2}ma=0$$ And there I'm stuck. Could anyone tell if at least I'm going to the right direction? Any help is appreciated!


title: electric potential energy symbol in Schrodinger equation

text: In my introductory physics class, $V$ is the symbol for electric potential (joules per coulomb) and $U$ is the symbol for electric potential energy (joules). Since the Schrodinger equation is the sum of Kinetic and Potential energies in the system, $V(r)$ must represent $U$... if so, is there any particular reason why $V(r)$ is used as opposed to $U$?


title: Why a system should be at its lowest energy state for its stability?

text: Every possible reaction in chemistry is to attain stability. In physics, the alignment of an electric dipole in an external electric field and in all other physical systems (at least those I study in high school) attains stability at the lowest energy state. But, why is it so?


title: References for ADM formalism and cosmological perturbation theory

text: What would you consider the best online resources for learning the 3+1 ADM formalism and gauge invariant perturbation theory in cosmology? (Assuming intermediate level GR and QFT familiarity)


title: What is the relationship between the verbal definition and the mathematical definition of some quantities?

text: I know this is probably an easy question, but it's been a while since I've studied physics and I've started reading some circuit analysis textbooks. I'm finding hard to understand the relationship between between the verbal definition of quantites and the mathematical definitions. For instance, in the Sadiku's book "Fundamentals of electic circuits", I've got the following verbal definiton for voltaje (literal) "Voltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V)." And then, it says that this mathematically "means" $$ v_{ab} \triangleq \frac{dw}{dq} $$ I can't understand well the relationship between this two "definitions" could someone explain further the relationship?


title: Understanding vectors in physics: notation

text: We have the formula for the Lorentz force $$\textbf{F} = q \space(\textbf{E} + \textbf{v} \times \textbf {B})$$ This is a simple formula you learn in high school, but I'm interested to self- study electromagnetism and I found out a different notation for this formula: $$ \textbf{F}(\textbf{r}, \dot{\textbf{r}} , t,q) = q[\textbf{E}(\textbf{r}, t) + \dot{\textbf{r}} \times \textbf{B}( \textbf{r}, t)]$$ I want to know where I can study this type of notation, as I've never encountered it during high school. I'm guessing the LHS needs to include every unit on the RHS, but I don't understand for example why we have $\textbf{E}(r,t)$, what is the significance of the position vector and time, and how can you know?


title: Kepler Orbital Elements to Cartesian (xyz)

text: I'm not really sure if this is the place to ask this, but anyway here is my question: Let's say I have the Kepler orbital elements of the ISS, for example, (NASA stuff). Now I want to compute the coordinates relative to the earth at a specific time so that it can be displayed, like in a sky map (I am actually experimenting with Google Sky Map). I found many websites discussing Kepler orbital elements, but I have only found 2 pdf's that talk about this conversion: http://www.phas.ubc.ca/~newhouse/p210/orbits/cometreport.pdf and I can't post a 3rd link yet. I do not fully understand which values I need to calculate, and which are given by NASA. I am also assuming that the data given by NASA is valid for the vector time provided, meaning that time obviously has to come into the calculations somewhere. Thanks, I would appreciate some clarification.


title: Is there record of a bosonic Stern-Gerlach measurement?

text: I cannot seem to find any peer-reviewed (or other) reference to an integer- spin Stern-Gerlach experiment. It shouldn't be too hard to do: just find you friendly neighbourhood Deuterium ion and shoot it through a Stern-Gerlach magnet. Can one devise a photonic Stern-Gerlach experiment, i.e. spatial seperation of polarization states? One should also see only two states in this case, because the spin-0 photon state is "reserved" for EM-interactions (this might be too simple a statement, but this is how I understand it currently). _EDIT_ it seems some of you are misunderstanding the question: I am inquiring about a Stern-Gerlach-like experiment, where spin states have been split, and by extension the perpendicular nature of non-commuting measurements. So only the concept of the S-G experiment as extensively described in introductory QM textbooks such as Sakurai.


I have four questions about black holes and universe formations.   1. Do new universes form on the other side of black holes?   2. Was our own universe formed by this process?   3. Was our big bang a black hole seen from the other side?   4. Are there solid reasons why this might not be the case?


title: Gauge Invariance of Yang Mills Lagrangian

text: I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$ under the following gauge transformation ($\theta$ being a rotation in color space and $g$ related to the structure constant): $$L \rightarrow -\frac{1}{4} \left( F^a_{\mu\nu} +\epsilon ^a_{jk} \theta^j F^k_{\mu \nu}\right)\left(F_a^{\mu\nu} + e_a^{jk}\theta_j F_k^{\mu\nu}\right) + \left( J_a^\mu + \epsilon_a^{jk} \theta_j J_k^\mu \right) \left( A_\mu^a + \epsilon^a_{jk}\theta^jA_\mu^k -\frac{1}{g} \partial^\mu \theta^a \right),$$ where each term is now transformed accordingly. I was able to simplify the above to and obtain: $$L \rightarrow -\frac{1}{4} \left( F^a_{\mu\nu}F_a^{\mu\nu} + \epsilon^a_{jk} \theta^j F^k_{\mu\nu} \epsilon_a^{j' k'} \theta_{j'} F_{k'}^{\mu\nu}\right) + J_a^\mu A_\mu^a - J_a^\mu \frac{1}{g} \partial^\mu \theta^a + \epsilon_a^{jk} \theta_j J_k^\mu \epsilon^a_{j'k'} \theta^{j'}A_\mu^{k'} - \epsilon_a^{jk} \theta_{j} J_k^{\mu}\frac{1}{g} \partial^\mu \theta^a.$$ How could I possibly reduce it to a form similar to the original, untransformed Lagrangian? There are about 4 terms I can't get rid of, though it has been suggested to me that I use the equation of motion of YM, which I have handy but can't seem to use them appropriately. Any help would be greatly appreciated. Also note that I may end up with a boundary term which would vanish when varying the action, thus possibly giving me say 3 terms instead of the original 2 (which is fine, though I can't identify them yet).


title: Opposing forces on an air cylinder -- half force each or full force each?

text: If an air cylinder is pushing two platens apart with a force of $100\: \mathrm{lbs}$, do the platens need to push back at $100\: \mathrm{lbs}$ or $50\: \mathrm{lbs}$ each to keep the cylinder from moving? Assume no friction and both platens are not fixed.


title: What is the length of 1 second in meters

text: If time is treated as a fourth dimension of spacetime, what is relation between length and time units? Or in other words, how can I convert time units to length units, for instance seconds to meters?


title: Why the energy of a marshmallow is so huge?

text: In this comment in a blog kudzu computes the energy of a marshmallow with mass $M=25 grams$ by using $E=mc^2$: $E=Mc^2 = 2.247\times 10^{+15} Joules$ I may be wrong but this seems like a huge energy for a marshmallow. Can anybody help me put this in context.


title: How is it possible that we see light from shortly after the big bang?

text: How can astronomers see light from shortly after the big bang? How did we get "here" before the light that emanated from our "creation"?


title: One-point Green's function

text: Why is one-point Green's function for scalar field equal to zero? Can one prove it using path integral formalism?


title: Conservation of momentum but not kinetic energy in inelastic collisions

text: In inelastic collisions, the kinetic energy of the system is not conserved but the momentum is. Kinetic energy is: $0.5 \times \text{mass} \times \text{velocity}^2$. Momentum is: $\text{mass}\times\text{velocity}$. I think that, considering that mass is constant:   * if Ke must be different also the velocity of the centre of mass of the system _must be different_ , after the collision. On the other hand:    * if the momentum of the system is conserved, the velocity of the centre of mass of the system _cannot be different_.  So, how can there be a change in kinetic energy of the system if there is no change in momentum? $mv = m_1v_1$


title: Intuitive meaning of the exponential form of an unitary operator in Quantum Mechanics

text: I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that > Every unitary operator $\hat{\mathcal{U}}$ can be written in an exponential > form as follows: > $$\mathrm{e}^{-i\alpha\hat{\mathcal{T}}}=\sum_{k=0}^{\infty}\dfrac{1}{k!}\left(-i\alpha\right)^{k}\hat{\mathcal{T}}^{k} > $$ Provided that I have no knowledge of Lie Group/Algebra, my questions are:   * Why a unitary operator can be always represented by an exponential form?   * What is the intuitive mathematical meaning of the exponential form/matrix?   * What is the relation between the operator $\hat{\mathcal{U}}$ and the operator $\hat{\mathcal{T}}$?   * What is its physical meaning?


title: Why is the local work function so much lesser than Work Function?

text: I have been trying to understand the difference between the Work function of a metal and the Local Work Function. I did some experiments to find the Local Work function of Graphite using an STM and found values ~0.3-0.7 eV. However, the work function of Graphite is ~4eV. Literature values indicate that my experimental results are correct. Some research leads me to these definitions on a website: Work Function: The work function corresponds to the minimum energy necessary to extract an electron from the metal Local Work function: The local work function, which is defined as the energy required to take an electron from the Fermi level to a specified position outside the metal. I frankly can't see a difference between these two definitions. Admittedly, my concepts in Solid State Physics are shaky. What is the difference between the two work functions? Can someone clarify, without assuming I know too much about Fermi levels.


title: Non-linear Wave Equation - Numerical Methods

text: **Motivation:** I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form $$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$ where $'=\frac{d}{dr}$ and $\dot{} =\frac{d}{dt}$. (The expression for the function $f$ is long to I don't include it, but it contains terms like $\dot{u}^nu'^m$.) I need to solve this numerically, and I have done this by defining the 'velocity' $Q = \frac{du}{dt}$ so that we can write the system as $$\dot{Q} = f(t, Q, Q', u' ,u'')$$ $$\dot{u} = Q$$ which I then discretize on a grid (for the space) and evolve in time using a a staggered leapfrog: $$Q_{n+1/2} = Q_{n-1/2} + f(t_n, Q_n, Q'_n, u'_n ,u''_n)\Delta t$$ $$u_{n+1} = u_n + Q_{n+1/2}\Delta t$$ This works fine for many applications, but I have some problems with getting this to work in all situations I'm interested in. **Question** I'm therefore looking for alternative methods to solve this equation. I welcome any suggestions. I have been thinking about Newton-Gauss-Seidel, but it's pretty computationally expensive so I'm hoping there are other methods I could try first. Also, if there is someone that has experience with numerically solving such equations it would be great to get some input on methods that are known to work well in practice.


title: Parallel electric and magnetic field in traveling wave

text: Can we have elctromagnetic waves which are characterized by parallel electric and magnetic field $\vec{E} || \vec{B}$ ? I am not talking here about free space, but maybe some kind of materials or waveguides or something else.


title: What is the relationship between string theory and quantum field theory?

text: Please forgive a string theory novice asking a basic question. Over at this question Luboš Motl gave an excellent answer, but he made a side comment that I've heard before and really would want to know more about: > Quantum field theory is the class of minimal theories that obey both sets of > principles [Ed: _SR and QM_ ]; string theory is a bit more general one (and > the only other known, aside from QFT, that does solve the constraints > consistently). What are the arguments that string theory is more general than QFT? I get that you can derive many different QFTs as low energy effective theories on different string backgrounds. But from my limited exposures to worldsheet perturbation theory and string field theory I can also see string theory as a very _special_ kind of QFT. I also realize these approaches don't fully characterise string theory, but I don't know enough about their limitations to understand why the full definition of string theory (M theory?) surpasses QFT. My naive guess would be that _no_ , string theory can't be more general than QFT because surely there are many more QFTs which are asymptotically free/safe than could possibly come from string theory. Think ridiculously large gauge groups $SU(10)^{800}$ etc.. Since these theories don't need a UV completion into something else string theory can't be a more general framework than QFT. Logically you could also have theories which UV complete into something other than string theory (even if no such completion is presently known). Alternately you could turn this around and say that string theory limits the kind of QFTs you can get in the low energy limit. This is saying that string theory is more predictive than QFT, i.e. _less_ general. I always thought this was the goal the whole time! If it is the other way around and string theory really is _more_ general than QFT, doesn't this mean that string theory is _less_ predictive than, for instance, old school GUT model building? So is the relationship between string theory and quantum field theory a strict inclusion $\mathrm{QFT} \subset \mathrm{ST}$ or more like a duality/equivalence $\mathrm{QFT} \simeq \mathrm{ST}$, or a more complicated Venn diagram? Note that I am _not_ asking about AdS/CFT as this only deals with special string backgrounds and their QFT duals. I'm asking about the general relationship between string theory and QFT.


title: Do we have control on what an electron transition emits: light or heat?

text: I don't know quantum mechanical model. So, I'm referring to just bohr's model of atom. Any atom emits energy when it makes transition from higher excited state to lower excited state. Now, some times they say that this energy is light energy and some times heat energy. I'm confused. What decides the emitted energy will be light or heat? Do we have control over what kind of energy it emits? I mean can I make an atom emit light and not heat? or heat and not light? What are the factors influencing this? **EDIT:** To clarify the issue of what exactly do I mean by heat. Well, I myself am not quite sure. I just thought heat as just heat. This question araised from my previous question where I discovered that in a filament bulb 98% of energy is converted into heat energy and rest into light energy. But where as in CFL about 40% to light and rest into heat. I know these conversion is because of electron transistions. But whats controlling light & heat components?


title: Confusion between Electric field and Magnetic field of a charged particle.

text: 1. Consider a charged particle (electron or proton) at rest. It is surrounded by its own electric field.   2. Now consider an electron moving with certain velocity (less than speed of light), Still is there electric field around it? i. If it has electric field around it, why is it that when electrons are moving in a conductor (i.e.. current if flowing in a conductor) there is no electric field outside the conductor? ii. Now, when a current is flowing in a conductor (I'm not sure what happens if the motion is not inside conductor) it produces magnetic field around it. I'm lost. What happened to the electric field? Is it still there? Are there both electric field & magnetic field? Why don't we discuss about it?   3. Hypothetically, If the electron is moving with speed more than that of light. What happens now?  I'm totally lost. Kindly some one clarify these issues.


title: why making a surface "super" smooth increases the coefficient of friction?

text: I read that: If you take a rough surface and make it smooth, the coefficient of friction decreases. But if you make it **super** smooth, then the coefficient of friction increases. How come?


title: Realistic projectile motion

text: I am working on a project involving a simulation of the motion of a projectile (in 3D) aimed at a moving target. The way projectile motion is analyzed in most introductory physics books is not accurate enough for this project. I would like to know what other influences on the motion of a projectile, including air resistance and spin, I need to take into account. What is a good book on this subject?


title: What is a black hole?

text: Is there a definition of a black hole in a generic spacetime? In some books, for example Wald's, black holes are defined for asymptotically flat spacetime with strong asymptotic predictability, although the definition makes sense without the second condition. Is there a notion of a black hole in general spacetime, not necessarily asymptotically flat? Or is it the case that there is not a "natural" or agreed upon definition?


title: How can you weigh your own head in an accurate way?

text: I read some methods but they're not accurate. They use the Archimedes principle and they assume uniform body density which of course is far from true. Others are silly like this one: _Take a knife then remove your head.   Place it on some scale   Take the reading   re-attach your head._ I'm looking for some ingenious way of doing this accurately without having to lie on your back and put your head on a scale which isn't a good idea.


title: Why are nuclear reactors dome/bell shaped?

text: Just what the title states. What is the reason that a Nuclear reactor has a characteristic dome/bell shape?


title: Fluid dynamics - Water level in a container

text: > An open container is being filled with water from the top at a constant > volumetric flow rate $300 \frac{cm^3}{s} $. At the same time, water is > leaving the container trough a circular whole at the bottom, whose surface > area is $0.6 cm^2$. What is the water level in a stationary state. First of all I do not understand this question. How is "stationary state" defined? Shouldn't the water level be increasing as time passes by? I don't want a solution to this problem but only a hint or an explanation of what this stationary state is.


title: How do adhesive and cohesive forces affect surface tension?

text: Surface tension appears at the interface of two immiscible fluids if the cohesive force of attraction is more than adhesive force. What will be the physical effect if the _adhesive_ force is more than _cohesive_ force?


title: Loss of atmosphere to a vacuum

text: I'm working to craft a space-horror themed RPG a la Dungeons & Dragons and I've run into a situation I can't resolve on my own. At what rate is atmosphere lost to space through various sized apertures? I'm looking generally at these sizes(in diameter) 10' 5' 30" 15" 7" 3" 1" Trying to determine how quickly the cabins/holds/etc will decompress. It is the deep space version of falling rocks :D


title: Exercise about Lagrange-Euler equations

text: I'm solving an exercise about the Lagrange-Euler equations, that states the following: > _Let $\gamma (t) = \\{ (t,q) : q = q(t), t_0 \leq t \leq t_1\\}$ be a curve > in $\mathbb{R} \times \mathbb{R}^2$. Further let $F(q,\dot{q},t)$ be the > function from $\mathbb{R}^2 \times \mathbb{R}^2 \times \mathbb{R} > \rightarrow \mathbb{R}$ for which the functional $\Phi = \int_{t_0}^{t_1} > F(q,\dot{q},t) dt$ is the length of the curve._ > > _(a) Which is the form of $\Phi$ in cartesian coordinates? Which is its form > in polar coordinates?_ > > _(b) Give the Euler-Lagrange equations in both coordinate systems._ > > _(c) Solve the differential equations in both coordinate systems and show > that the solutions are the same._ Now, my problem begins with giving the form of $\Phi$. I found that the element of length in cartesian coordinates is $ds = \sqrt{dx^2 + dy^2}$, so with $$\int ds = \int \frac{ds}{dt} dt = \int \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2} dt,$$ We find that $\Phi = \int_{t_0}^{t_1} ||\dot{\gamma}(t)|| dt$. Now, my plan is finding the element of length in polar coordinates, and plugging in the respective expressions in terms of $\gamma$. The problem is that I don't see how to find the element of length in polar coordinates. I looked it up on Wikipedia, and found $ds^2 = dr^2 + r^2 d\theta^2$. Now, for $dr^2$ I would plug in $||\dot{\gamma} (t)||^2$, for $r^2$ I'd set $||\gamma (t)||^2$, and for $d\theta$ I have no idea. Can you help me, especially with the derivation of the polar line element and the form of $\Phi$ in polar coordinates?


title: vector product between $ds$ and other portion of current loop

text: ![enter image description here](http://i.stack.imgur.com/WcibA.png) In the answer section it is written that,The magnetic field at O due to the current in the straight segments $AA^\prime$ and $CC^\prime$ is zero because ds is parallel to along these paths. > My querry is that "How ds is parallel to along these paths?" EDIT: ![enter image description here](http://i.stack.imgur.com/NfFgO.png)


**Why don't most physics programs study the primary sources?** In other words: Why don't they include Newton's _Principia_ , Lagrange's _Analytical Mechanics_ , etc., in the curricula?


title: Classical Information carrying capacity of two states

text: What classical information is carried by $\alpha|0\rangle+\beta|1\rangle$ and $\alpha|00\rangle+\beta|11\rangle$? How to quantify it? To be specific, A GHZ state, $\frac{1}{\sqrt 2}[|000\rangle+|111\rangle]$ can deterministically teleport both the states. Both the states appears to be same from the teleportation point of view. Does they carry same information?


title: Bekenstein entropy black hole v.s Hawking entropy black hole

text: Historically, Bekenstein estimated the entropy associated with a black hole in 1973, obtaining: $$ S_B = \frac{\ln(2)k_Bc^3}{8\pi\hbar G}A. $$ He already acknowledges in his article that his estimates are based on classical principles and that a quantum mechanical treatment will yield a different constant, though within a factor of order one the same. A year later Hawking derived: $$ S_H = \frac{k_Bc^3}{4G\hbar}A, $$ i.e. $S_B = (\ln(2)/2\pi) S_H$, such that $S_B<S_H$. I am wondering if there could have been examples, showing that $S_B$ was not correct. So, without knowing Hawking's results, can we see that $S_B$ cannot be correct, maybe by giving a certain counter example, or using the fact that $S_B<S_H$?


title: What is surface plasmon of metal nanoparticles?

text: I really want to understand about the Surface Plasmon of metal nanoparticles, if anyone can explain it to me i'll be grateful. I have checked Wikipedia for it but didn't get anything clear.


title: Weight on planet earth?

text: I was wondering : does the weight on the planet earth is equal over the years ? meaning : all the people , ground , water ,gas. does the weight stays the same over the years ?


title: What does really mean by- power of a number or an exponential function is dimensionless?

text: Is power of only a number or an exponential function is dimensionless? If power of any other thing can also be dimensionless then please explain with examples.


title: Calculate required voltage in mixed circuits

text: > **Q:** Calculate the voltage that V1 has to provide > > **Given:** The following circuit. The voltage through LAMP1 is 5.0V. A (U,I) > chart (not shown). LAMP1 and LAMP2 are identical, non-ohmic lightbulbs. ![circuit](http://i.stack.imgur.com/3tFZy.png) My attempt at this problem: LAMP1 and LAMP2 are two identical light bulbs. I know that $U(LAMP1) = 5.0 V$. According to the (U,I) chart, the value of $5.0$ V corresponds with $0.63$ A. If $U(LAMP1) = 5.0$V, $U(R1) = 5.0$V too. That value corresponds with $I(R1) = 0.5$A. $I(LAMP1 + R1) = 0.63 + 0.5 = 1.13$A. Because LAMP1 and LAMP2 are identical, they are both $P = 5.0 \times 0.63 = 3.15$W lamps. The voltage through LAMP2 has to be $U(LAMP2) = P \div I$. So $U(LAMP2) = 3.15 \div 1.13 = 2.78$V Hence the voltage V1 has to provide is $5.0 + 2.78 = 7.8$V Is this correct? Or am I using the wrong method(s)?


title: Link between anomalous dimensions and fractal dimensions

text: I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a scale transformation (renormalization group transformation in the QFT case and dilation of the mesh/ruler when computing the perimeter of a fractal). Is there a more profound link between the two? I have't read much about the subject but it seems that for any statistical model at criticality the fractal dimension of the clusters becomes a function of the full dimension of the field. Is it a general rule?


title: How did Newton discover his second law?

text: I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon? According to Wikipedia, the Principia reads: > _Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri > secundum lineam rectam qua vis illa imprimitur._ Translated as: > _Law II: The alteration of motion is ever proportional to the motive force > impress'd; and is made in the direction of the right line in which that > force is impress'd._ My question is how did Newton come to this conclusion? I get that he knew from Galileo Galilei the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this?


title: How to discover the $F = m\cdot a$ equation from scratch?

text: Is there a paper or a book that conceptually 'discover' the Newton's famous formula? I've been thinking about this equation, and I really don't know it for sure. Did Newton empirically noticed that when we apply an acceleration to a body, then the accelerationon the body that has contact with it, will be proportional either to the mass or to the acceleration? Then, he simply called it 'force'? OMG i'm so confused :c I can understand differential equations, integrals, and so, but I can't understand, from scratch, how to build up Newton's laws in my mind. I need good papers or books.


title: Does a particle annihilate only with its antiparticle? If yes, why?

text: Or to put the question another way - what is the result of a proton-positron collision, or an up quark-charm antiquark collision, etc.? As far as I know, annihilation happens only between particles of opposite charge and same mass, but perhaps I am wrong? And if the types of annihilation mentioned above cannot happend, what are the reasons? Thank you.


title: Electric field at a point from a square surface

text: I'm trying to determine the electric field at a point P (located on the +Z axis) due to a square of side length [L] and centered at the x-y plane origin. The square has a constant surface density [s]. I'm thinking that I should go about splitting up the square into four smaller squares (one in each quadrant) and then calculate the field from each on the point P. Is this the correct way of solving this type of problem, or should I be splitting the big square up into infinitesimal strips and calculating the field that way?


title: Problem in Youngs double slit experiment

text: ![double slit ray diagram](http://i.stack.imgur.com/Whguh.gif) This is from Young Double slit experiment. But How to prove the the two $\theta$ are equal, I meant, how $\angle EAD= \angle PEC$? I see from the both triangle have $90^0$ but what about others?


title: Abiotic oil vs the traditional theory of oil deposit formation

text: I am curious to see what people think of the abiotic theory of oil deposit formation versus the traditional theory. I have long wondered how enough organic material became trapped underground to degrade and form oil deposits. Especially when you consider the shear amount of oil that we have found underground over the decades. And considering how much oil we use. If you think about it, when most critters die the tend to decompose mostly or fully above ground. So I have trouble believing that such large amounts of oil, especially in such large concentrations, have formed from dying plants and animals over the centuries. Recently I heard about the theory of abiotic oil. My understanding of the theory is that oil deposits are actually chemicals and material that is formed at the mantle boundary that seeps up further into the crust where it is consumed by micro-organisms that process it and turn it into what we know as oil. My understanding may be a bit off I realize. But some evidence of this has been presented in the form of old, capped oil wells that after a few decades, when re-examined, seem to be full once more and are now run at full pumping capacity once more. There are other things I have read as evidence. It seems to me that this would be a better explanation for the large amounts of oil deep underground. Does anyone know enough about these subjects to say that one theory holds more water than the other? Has there been any new evidence pointing to one or the other theory more?


title: Future space-based telescope array

text: Radio-telescopes (e.g. the Very Large Array (VLA)) can simulate one gigantic dish by using separate smaller dishes. **Q:** Could such an array of _optical_ telescopes potentially see an exoplanet at say 20 LY away? Assumptions:   * The array is space-based, at an ideal location such as the Earth-Sun L2 Lagrangian Point   * Assume the exoplanet of interest is defined as a rockey planet up to 5 times the diameter of earth


title: Basic question on the Aharonov-Bohm effect

text: I have a very basic question on the Aharonov-Bohm effect. Why is the curve integral $\oint_\Gamma {A}\cdot d{r}$ non-zero ? $\Gamma$ is the "difference" of both paths $P_1$ and $P_2$. If the magnetic field is limited to the interior of the solenoid $\operatorname{curl} {A}=0$ along the integral path $\Gamma$, so I can conclude that I can write ${A}=\nabla f$. A closed curve integral of a gradient function is zero. I guess it is related with a possible singularity of $A$ in the very center of the solenoid. Nevertheless if I travel around a point-like source of a gravitational field and compute the integral $\oint_\Gamma {F}\cdot d{r}$ where $F=-\nabla V(r)$ the closed curve integral over an conservative force field is certainly zero, whereas $V(r)$ even has a singularity and $F$ consequently too. I would be very grateful for an explanation.


title: Is Sorkin's causal growth dynamics a Galois Connection

text: Sorkin is well known for his causal growth dynamics. I think it is a sensible question to ask if his growth dyanamics, which I think can be seen as a poset map (though his work is much more complicated than that), represents a galois connection.


title: Compactness of spacetime: experiment and math

text: It is common to find models built on a compact spacetime. In mathematics, compactness is a very nice property $-$ and lot of powerful results depend on it. But   * how _safe_ is assuming compactness of spacetime in physics?  Minkowski space is not compact, but, for instance, the treatment given to gauge theories in terms of bundles, assumes a compact basis $X$ for the principal bundle $G\hookrightarrow P \to X$ and the vector bundles containing matter $P \times_G \mathfrak{g} \to X$. This basis $X$ is thought of as Minkowski space-time (for sake of concreteness, assume an Euclidean spacetime $\mathbb{R}^4$, so one compactifies to $X=S^4$). Why can or cannot compactes be assumed? This one is perhaps model-dependent:   * is there an existing experimental validation of the compactness of spacetime?


title: Relation between $\mathcal{N}=2$ d=4 SUSY and $\mathcal{N}=4$ d=3 SUSY

text: What is meant with the fact that supersymmetry with $\mathcal{N}=4$ in three (2+1) dimensions is equivalent to supersymmetry with $\mathcal{N}=2$ in ordinary four (3+1) dimensions? Which way does one go from one formulation to the other?


title: Can photons and gluons be holes in an aether?

text: Consider photons and gluons have 0 mass and 0 charge. In many respects they're already understood as the absence of a particle by mathematical models. Couldn't this be interpreted to mean they operate by phenomenon similar to the charge carrying "holes" responsible for transmitting electro-magnetic waves (at the speed of light) in wires?   * http://en.wikipedia.org/wiki/Semiconductor#Holes:_electron_absence_as_a_charge_carrier   * http://en.wikipedia.org/wiki/Standard_Model (note photons, and "gluons" which are thought to make up protons and neutrons, have 0 mass and 0 charge) The implications of this model would seem to be:   * There is a relativistic aether that is similar to the concept of a theoretical inviscid in fluid dynamics (see en.wikipedia.org/wiki/Inviscid_flow).   * Differences in "pressure" at this fundamental level may be responsible for gravity, as well as providing possible explanations for a number of other phenomenon currently challenging physics.


title: Observable Operator on a Superposition?

text: I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across. $$ \hat{A}|\Psi\rangle = a|\Psi\rangle $$ The above equation implies to me that a single eigenket gives a single eigenvalue of $\hat{A}$. However Ket Vectors that are composed of superpositions have multiple possible eigenvalues. Which leads me to believe that that equation is only valid for eigenkets which are Basis States. However in the Schrödinger equation we have an Observable (Hamiltonian) acting on Wave Functions in Position Space which are composed of an infinite number of Basis States. In this usage is it somehow assumed that every Basis State in the Position Basis corresponds to a single Energy Eigenstate? (I wouldn't think this would be the case. But what is the point/result of applying the Hamiltonian to any given Wave Function then?) Further confusion arises from this because if the Energy is exactly known then shouldn't there be some sort of maximal uncertainty in time? As a final question is there any kind of useful interpretation of multiplying the eigenket by it's eigenvalue as appears in the above Observable Equation? In all treatments I've seen this multiplication is simply ignored and the eigenvalue itself is the only focus.


title: Lattice theory in mathematics and physics

text: I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in mathematics called lattice theory which deals with ordering. I am wondering whether the lattice theory in mathematics could actually help me in construct such "general algorithm" which can deal with any kinds physical lattice. If so, I will delve into this area. Please help me, thanks. If you know a more specific mathematical area that deals with general physical lattice ( I have some names in my mind like, mathematical crystallography, lattice graph theory). Please hint me so that I could move forward. Thank you.


title: Boundary conditions for the heat equation when solving a mass density gradient

text: I'm working with a mass density gradient with length $L$ and I'm trying to solve the heat equation in 1-D (mass diffusion equation, $\partial_t\rho(t,x)=D\Delta\rho(t,x)$), but I'm not sure which boundary conditions should I use and what would they physically mean. The starting mass density profile ($\rho(0,x)=f(x)$) is a step-function, where the lower half has density $\rho_1$ and the upper half has $\rho_2$. $f(x)=\begin{array}{ll} \rho_1 & 0<x<L/2 \\\ \rho_2 & L/2<x<L \\\ \end{array}$ Considering the boundary conditions, I find more difficult to interpret them. The mass inside the sample's volume is kept constant during the experiment as the sample is isolated from surroundings. For this purpose, would Neumann Boundary Conditions be appropriate? $$\partial_x \rho(t,0)=0=\partial_x \rho(t,L) $$ I'm unsure because all the examples I found do not treat the heat equation/diffusion equation as function of mass density ($\rho$). Would Neumann BC suggest No Mass Transfer between the sample's volume and the surroundings?


title: Massive particles and speed of their propagation

text: Can one show that in quantum field theory _at least some example_ massive particles propagate with speed less than speed of light, while massless travel at speed of light? Well, motion is a different thing in QM than in classical mechanics, and question might be formulated differently.


title: Does the proton have an equatorial bulge around its spin axis?

text: And if so, can we observe a difference in the electron scattering cross section with transversely polarized VS longitudinally polarized protons? P.S. Let me make my question more precise. Consider the charge shape of the proton. In the rest frame of a proton with spin in +z direction, what's the spatial dependence of the expectation of the electric charge operator $j^0(x)$, at some particular renormalization scale? Can this question be sufficiently answered by the currently available data from polarized proton scattering experiments?


Why do wind power plants have just 3 blades? It seems that adding more blades would increase the area that interacts with the wind and gather more energy. ![enter image description here](http://i.stack.imgur.com/KOC1w.jpg) (Image from Wikipedia.)


title: What is the shortest run needed for a "dropped skateboard" rider to reach 160 mph?

text: I just saw a great clip on the fastest skateboarder to date. He wants to reach 160 mph from a dropped board, but says his speed is limited by the roads currently available. What is the shortest run required for a "dropped skateboard" rider to reach 160 mph? Is it even possible?


title: Must all symmetries have consequences?

text: Must all symmetries have consequences? We know that transnational invariance, for example, leads to momentum conservation, etc, cf. Noether's Theorem. Is it possible for a theory or a model to have a symmetry of some kind with no physical consequences at all for that symmetry?