What is the missing proportionality constant in the magnetic levitation formula?

The formula for magnetic levitation is $$ B \frac{dB}{dz} = \frac{ \rho g }{\chi} $$ but as always, I have a hard time figuring the units in SI. The left hand side is $\mathrm{T^2 /m}$, while $\chi$ has units of $\mathrm{m^3\,mol^{-1}}$, which makes the right hand side with units of.. $$\mathrm{kg\,m^{-5}\,s^{-2}\,(mol)}$$ Obviously there must be some constant, but I don't know what is it, as all texts use this natural unit system. Help? **Update:** using the permeability of free space as reported by Wikipedia, which seems to be Tesla-meter per Ampere, leaves the right hand side as $$\mathrm{kg\,T\,m^{-4}\,s^{-2}\,Ampere\,mol}$$ still far from being recognizable to the left side


Is emission/absorption of a photon lossy?

I recall vaguely that energy is absorbed/radiated in packets called quanta. Quanta were what are now known as photons. What I'm curious about - Is absorption/radiation vis-a-vis photon lossy? Do the total number of photons exactly match the energy acquired/released?


X-Ray crystallography using Bragg's Law

I was looking up X-Ray crystallography using Bragg's Law: $2d\sin\theta = n\lambda$ and I can understand the values of everything except this integer value $n$. As far as my research got $n$ is used to describe the atom spacing in the crystal lattice, but I don't understand how you'd express $n$ or how it would describe it. Could someone please explain this to me please? Note: diagrams tend to be very useful in developing my understanding and if anyone has any reference to a video that might help as well. Thanks.


How to use The Schwarzchild Metric formula to get distribution representing "free-fall"

Given formula: ![enter image description here](http://i.stack.imgur.com/JsywY.png) How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to "free-fall path". As far as I know i should use square root of Determinant of given formula to do this, but i am not sure. Please, be gentle, I am computer scientist student that in process writing black hole simulation for academic purposes. NOTE: Simulation is in 2D. so some parameters may be irrelevant.


Variations of S-matrix functional and Feynman diagrams in Weinberg QFT

Weinberg on p. 287 of his QFT vol. 1 introduces the extended interaction operator: $$ \tag 1 \hat{V}(t) \to \hat{V}(t) + \sum_{a}\int d^{3}\mathbf x \hat{o}_{a}(\mathbf x ,t)\varepsilon_{a}(x). $$   Here $$ \hat{S} = \hat{T}e^{i\int \hat{V}(t)dt}, \quad \hat{o}_{a}(\mathbf x, t) = e^{i\hat{H}_{0}t}\hat{o}_{a}(\mathbf x , 0)e^{-i\hat{H}_{0}t}. $$ Then he says that S-matrix arbitrary element $S_{\beta \alpha} = \langle \beta | \hat{S}| \alpha \rangle$ after extension $(1)$ becomes the $\varepsilon $-functional, and after that he introduces generalized Feynman rules by adding new vertexes corresponding to $\hat{o}_{a}$ with $n_{a}$ lines (the number $n_{a}$ coincides with the number of fields in $\hat{o}_{a}$) and c-factor $\varepsilon_{a}$. After that he introduces variational derivative $$ \tag 2 \left( \frac{\delta^{r}S_{\beta \alpha}}{\delta \varepsilon_{a_{1}}(x_{1})...\delta \varepsilon_{a_{r}}(x_{r})}\right)_{\varepsilon = 0} = (i)^{r}\langle \beta | \hat{T}\left(e^{i\int \hat{V}(t)dt} \hat{o}_{a_{1}}(x_{1})...\hat{o}_{a_{r}}(x_{r})\right)|\alpha\rangle $$ and notices that all $n_{a_{1}}, ..., n_{a_{r}}$ lines correspond to $\hat{o}_{a_{1}},...,\hat{o}_{a_{r}}$ respectively are internal, i.e. in case when $n_{a_{1}} = ... = n_{a_{r}} = 1$ they are compared to the propagators. Finally, he says, that if we want to get Feynman diagram with $r$ external lines with types $a_{1},...,a_{r}$ in momentum representation we need to do following with $(2)$: 1) to throw out of propagators $D_{a_{1}a_{r}}(x_{1} - x_{r})$, 2) to apply the Fourier transformation, 3) to add corresponding coefficient functions $u_{a_{1}},...$. Here is the question: сould you make the sense of introduction of mechanism 1)-3) clearer for me? Why do we need additional $r$ external lines which with corresponding vertexes which aren't connected to other vertexes (so the diagram is non-connected), as I think? I need to rephrase the question. I know that n-point Green functions $\langle \beta | \hat{T}\left(\hat{o}^{H}_{a_{1}}(x_{1})...\right)|\alpha \rangle$ where the operators are in Heisenberg picture are equal to $(2)$. Also these functions are widely used for derivation of some non-perturbative results for scattering processes. But I don't understand how to apply these non- perturbative results for S-matrix, because we need to set $r$ to zero for obtaining the exact sets of Feynman amplitudes $S_{\beta \alpha}$, but I'm not sure that it is possible without loss of nonperturbative results. Here is some incorrect understanding of Weinberg method, so it will be very good if someone help me.


Longitudinal waves in a large (infinite) solid block

Specifically, I am trying to roughly determine the sound produced by a ball when it hits the floor and bounces. If the ball exerts a pressure onto the floor, then certainly this pressure will go on to create a sound wave in the floor (infinite block) and the air. I was just wondering if there is any way to find the sound wave (or the energy of the sound wave) released by the ball, thus finding the energy lost and the ball's approximate rebound height. Is the velocity of the sound wave given as follows?: ![](http://i.stack.imgur.com/r3bgq.gif) or is it more complicated than the basic equations provided in most physics textbooks?


shouldn't we add the oscillating terms into Bohr-Sommerfeld quantization formula

shouldn't be the quantization formula (in one dimension) equal to $ N_{smooth}(E)+N_{osc}(E) = \oint_{C}p.dq $ ?? where the Oscillating term is just the correction from Gutzwiller trace formula or a sum over Orbits why is just the oscillating term ignored ..


How to find the angle of elevation and launch speed?

A particle is projected from a point on level ground with a speed of $u$ meters per second and an angle of elevation $\theta$. The maximum height reached by the particle is $42$ meters above the ground and the particle hits the ground $196$ meters from its point of projection. Find the values of $\theta$ and $u$.


Four vector manipulation

I have started doing _Relativistic Quantum Mechanics_ from _Greiner_. I'm having difficulty understanding the following derivation for commutation relations (page 4): $$[\hat{p}^\mu, x^\nu] = i\hbar[\frac{\partial}{\partial{x_\mu}},g^{\nu \sigma}x_\sigma]=$$ $$= i\hbar g^{\nu \sigma}\frac{\partial x_\sigma}{\partial x_\mu}=$$ now this is the part I don't understand, how did we got from last equation to next one: $$=i\hbar g^{\nu \sigma}\delta^{\mu}_\sigma$$ Why is: $$ \frac{\partial x_\sigma}{\partial x_\mu}= \delta^{\mu}_\sigma$$ where $$\delta^{\mu}_\nu = g^{\mu \sigma}g_{\sigma \nu}$$


Relationship between Schrodinger equation and string/membrane

In Sakurai's _Modern Quantum Mechanics_ (2nd ed) p.99, he says > We know from the theory of partial differential equations that ( _time- > independent Schrodinger equation_ ) subject to boundary condition > ($\psi(\mathbf{x}')\rightarrow 0$ as $|\mathbf{x}'|\rightarrow\infty$) > allows nontrivial solutions only for a discrete set of values of $E$. It is > in this sense that the time-independent Schrodinger equation yields the > quantization of energy levels. Once the partial differential equation ( > _time-independent Schrodinger equation_ ) is written, the problem of finding > the energy levels of microscopic physical systems is as straightforward as > that of finding the characteristic frequencies of vibrating **strings or > membranes**. Although I've found a paper (http://www.scientificexploration.org/journal/jse_21_1_hocking.pdf) about this, the relationship is still not very explicit. So would you please explain briefly about the relationship between the Schrodinger equation and string/membranes? Many thanks.


Festive physics: gold flake vodka

I have a bottle of vodka that has a load of gold flakes suspended in it. It has been sat still for over 24 hours and the flakes are all still suspended within the liquid: they have not risen to the surface or sunk to the bottom. Any ideas as to the physics behind this?


Why is the chiral symmetry $SU(2)_A$ not anomalous?

Using Fujikawa's path integral treatment of the triangle diagram, one can show that $$\mathrm{Tr} \gamma^5 = \int d^4 x\ \partial_{\mu}j^{\mu} $$ Where $j^{\mu}$ is the Noether current of $U(1)_A$. Thus, the $U(1)_A$ anomaly can be traced to to fact that the trace of $\mathrm{Tr} \gamma^5 \neq 0$ at loop order. My question is, why isn't $SU(2)_A$ anomalous too? I fail to understand why this applies only to $U(1)_A$.


Orthonormality of Radial Wave Function in the Hydrogen Atom

`Question`: Show that the radial wave function for the hydrogen atom are orthogonal with no exceptions. `Attempt`: We know that the wave function for the hydrogen atom is orthonormal; that is, $$\langle\psi_{nml}\mid \psi_{n'm'l'}\rangle=\delta_{nn'}\delta_{mm'}\delta_{ll'}$$ Splitting up $\psi$ into a radial component $R_{nl}$ and a angular component $Y_{lm}$, we have $$\langle R_{nl}Y_{lm}\mid R_{n'l'}Y_{l'm'}\rangle=\delta_{nn'}\delta_{mm'}\delta_{ll'}$$ We know that the angular component $Y_{lm}$ is orthonormal, and thus $$\langle Y_{lm}\mid Y_{m'l'}\rangle=\delta_{ll'}\delta_{mm'} $$ I believe that, if we do not have orthogonality in the radial component, we will not have orthogonality in the wave function. For example, if we do not have radial orthogonality, then at $n=n'$, we might have the radial component be zero. Thus, the radial wavefunction must be orthonormal. Is this correct reasoning? i' worried I' missing something.


Large object is pushed through a small wormhole, what happens?

A simple question about wormholes: What happens if a large object is pushed through a much smaller wormhole? For example, what happens if I push an elephant through a palm-sized wormhole? My intuition is that depending on the magnitude of the force applied, either a part of the elephant is ripped out, or the elephant sort of gets stuck in the wormhole, as if hitting a wall. Is this correct?


Can Kirchhoff laws be applied to any circuit?

Kirchhoff's loop/current rule is just law of conservation of energy and Kirchhoff's junction rule is just law of conservation of charge.So, I think that these can be applied to any circuits unlike Ohm's law. Is there any exceptions?


Infinite period in Simple Harmonic Motion

I'm studying the Simple Harmonic Motion, and I am hesitant about, how to get mass values for infinite period?   * When mass is 0.   * When mass is infinite. With $\tau=2\pi/\sqrt{k/m}$.


Is there some other name used for "ping rigidity"?

In MTW, p. 398, "Box 16.4 (continued)", there's an interesting sketch (which can also be seen on   p. 15 of this excerpt (www.pma.caltech.edu/~ph236/yr2008/readings/MTW_Chapter16.pdf).   (It's not the only sketch of this kind in MTW; but it happens to be the only sketch on p. 398.)   Shown are (cmp. text of p. 397)   * two open-ended lines representing two " _particles_ " (They are supposed to satisfy additional relations which are not of immediate interest here, however.),    * several line segments criss-crossing between the two open-ended lines, representing " _light rays bouncing back and forth between [the two particles]_ ", and   * two of the criss-crossing segments being extended such that they meet, with the coincidence point of these two segments denoted as $\mathscr Q$. In order to describe what's represented by that third feature (which, unfortunately, MTW don't seem to spell out explicitly) it is convenient to refer to the two explicitly shown particles by explicit distinct names; say "Paul" for one and "Eric" for the other (i.e. by names which have been suggested through this sketch by M. Goossens). Accordingly, it may be said that the sketch of MTW p. 398 shows that `(a)`: Paul observed $1$ _light ray bounce to, and back from_ event $\mathscr Q$ (or: $1$ "ping wrt." event $\mathscr Q$), and `(b)`: for his same initial/signal indication, and his same final/reception indication, Paul counted $3$ consecutive _back and fourth bounces_ (or: "pings") with respect to Eric. And likewise: `(c)`: Eric observed $1$ _light ray bounce to, and back from_ event $\mathscr Q$ (or: $1$ "ping wrt." event $\mathscr Q$), and `(d)`: for his same initial/signal indication, and his same final/reception indication, Eric counted $2$ consecutive _back and fourth bounces_ (or: "pings") with respect to Paul. Now, rather than considering only one such event $\mathscr Q$, we may ask about, and Paul and Eric may under certain conditions even actually find, a set $\cal L$ with several **additional events** (and including event $\mathscr Q$) such that the conditions `(a)` \- `(d)` had been satisfied by **all events of** $\cal L$. Taking again the suggestion from the already mentioned sketch by M. Goossens it may be found that there can be (at least) one particle identified who participated in all events of such a set $\cal L$; let's refer to it by the name "Lutz". If "Paul", "Eric" and "Lutz" have been identified accordingly, as having satisfied conditions `(a)` \- `(d)` throughout some sufficiently extended trial, then obviously `(1)`: Paul found any $3$ consecutive pings wrt. Eric same as the corresponding $1$ ping wrt. Lutz; symbolically: " $\overset{\leftarrow ~ \rightarrow}{{}^EP{~}^L} = 3$ ", `(2)`: Eric found any $2$ consecutive pings wrt. Paul same as the corresponding $1$ ping wrt. Lutz; symbolically: " $\overset{\leftarrow ~ \rightarrow}{{}^PE{~}^L} = 2$ ", and apparently (without further conditions or reservations) it follows that `(3)`: Lutz found any $3$ consecutive pings wrt. Eric same as the corresponding $2$ consecutive pings wrt. Paul; symbolically: " $\overset{\leftarrow ~ \rightarrow}{{}^EL{~}^P} = \frac{3}{2}$". This relation between such three participants together may be recognized as a general kind of " _rigidity_ "; which I'd call in default of any other terminology: "(triplewise) ping rigidity"; in the above example symbolically: " $\underset{\\{P, E, L\\}}{\shortmid \leftrightarrows \shortmid} = \\{2, 3, 1\\}$ ". **My question:** Have such relations, of three (or more) participants " _bouncing light rays back and forth between_ " each other and counting **constant integer or rational ratios** for the numbers of (consecutive, or individual) "corresponding pings" been recognized and already named elsewhere?


How can I ensure my scale is not affected my surface beneath it

I have person weighing spring scale. I live in Japan where the house has wood floors which deflect significantly under weight. I weighed my self in the bathroom and in the tiled bath area. The difference was 2.5 kilograms greater in the bath area but it could still be affected by possibly an angled floor or deflection. So how you get as true a result as possible with that spring scale?


If $L$ is a matrix that represents real physical quantity, why is $L^2$ non-negative real physical quantity?

In my textbook, it says that when $L$ is a matrix that represents real($\mathbb{R}$) physical quantity, $L^2$ represents non-negative real physical quantity. What would be the proof of this?


My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why?

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it moves away from the "sun" and slows down as it approaches it. The best way I can describe it is that it's like the sun is at the wrong focus of the ellipse. I'm hoping someone can point me to what could cause this to happen? I've gone through all my code and I don't see any mistakes. My steps are:   1. Compute the Mean Motion of a satellite.   2. Use this to compute the Mean Anomaly   3. Calculate the Eccentric Anomaly from this (that was a doozy)   4. Calculate the True Anomaly from the Eccentric Anomaly   5. Calculate the Heliocentric Distance   6. Finally I take the Polar Coordinates and convert to Cartesian Coordinates which I then position over the top of my "sun" Somewhere in here I'm screwing up. I don't believe it's the final step as the calculated coordinates are derived from the angle and radius, which should be based on the correct focus. EDIT for the equations I'm using: **Mean Motion** $$ n = \sqrt{\frac{G(M+m)}{4\pi^2a^3}} $$ This is from Wikipedia. **Mean Anomaly** This is just $n \times t$ elapsed. **Eccentric Anomaly** I'm not really sure how to write this up for a non programmer. Basically the way I did this was based very much on some code that I found on the internet that I can no longer find. There's some recursion involved to gradually refine the answer. I'm convinced the issue isn't in here because I can enter my values into something like http://www.jgiesen.de/kepler/kepler.html and I have similar results. **True Anomaly** $$ \nu = 2\ \arg\left(\sqrt{1-e} \ \cos \left(\frac{E}{2}\right), \sqrt{1+e} \ \sin \left(\frac{E}{2} \right)\right) $$ $e$ is Eccentricity $E$ is Eccentric Anomaly from above Taken from here Wikipedia. **Heliocentric Distance** $$ r = a \frac{1-e^2}{1+e \ \cos(\nu)} $$ $a$ is my semi-major axis $\nu$ is the True Anomaly from above Taken from Wikipedia.


IIA and IIB Compact on 8D

How can compactifying IIA (non-Chiral) and IIB (Chiral) Superstring on $T^2$ (2-torus) gives rise to ($2$ dual descriptions of) the same $\mathcal N = 2$ supergravity in $8$ dimensions? I don't see it. Could you please explain it to me or recommend me some literature to read about it? Thank you!


Will perfect black hole apply force on matter?

While standing on a planet its gravitation force is noticeable and manifests only through the normal force, e.g. if jupiter had rocky surface, standing on the surface will apply about 2.5 times more force than when standing on earth. However sky diving on earth and on jupiter (without atmosphere) would apply the same amount of "destructive" force, that is 0. Thus gravitation force cannot really destruct matter without something for the matter to interact with.   If a black hole with no matter falling into it exists, will a person with space suite die if he would "jump" into it? he will have nothing applying "destructive" force on it, just kinetic energy that cannot manifest without something to hit.


Finding particles in the classically forbidden regions

Have particles ever been found in the classically forbidden regions of potentials? For example, in a square well: has an experiment been able to find an electron _outside_ the rectangular well (i.e. in the exponential fall-off regions) ? And more importantly, has anyone ever observed a particle **while** tunnelling?


Quantum mechanics potential barrier problem

While reviewing some quantum mechanics, I cam across a very interesting situation. For a potential barrier, if a particle has an energy $E$ less than the potential barrier $V_0$, it is possible to measure it inside the potential barrier or the _classically forbidden region_ quantum mechanically. But, if we calculate reflection and transmission coefficients for the wave, we find out $R=1$ and $T=0$ which means the wave will be fully reflected and no transmission will take place. But still there is probability of finding it inside the potential barrier. How?


What is the physical interpretation of force times area?

I know that $\text{Force} \times \text{Distance = Work}$.   But, what would be the physical meaning of $\text Force \times \text Area?$ Is such a quantity used in physics?


Commutator of Momentum with a Position dependent function

I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a derivation, or provide one here?


Classical dynamics with Schrodinger equation

What are some interesting classical systems for which the dynamics can be reduced to a many-body Schrodinger equation, at least in some useful regions of phase space, and in particular, with many variables.


It seems to me that superpotentials can be defined in a theory with or without supersymmetry. Is this true?

I recently read "An Introduction to Supersymmetry in Quantum Mechanical Systems" by T. Wellman (amongst other sources) in an effort to find out what a superpotential actually is and how it relates to the potentials of particles/fields). Here's the link: http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDkQFjAA&url=http%3A%2F%2Fphysics.brown.edu%2Fphysics%2Fundergradpages%2Ftheses%2FSeniorThesis_Wellman.pdf&ei=ulm- UPSCLZOY1AWjwYHYDw&usg=AFQjCNGrg_2jv5NZ7b6k4Fs7er34jgtw3w&sig2=yjQYy1Lf_gZVS- RRefUCsQ It occurred to me that the Fermionic Hamiltonian and the Bosonic Hamiltonian can be formulated without supersymmetry. On page 13, Wellman expresses these Hamiltonians in terms of the superpotential W in a way that is purely algebraic and doesn't require supersymmetry. In other words we have a bunch of terms that give us the two Hamiltonians; these terms are then simply replaced by W's (see equations 3.2 to 3.9). We only actually get supersymmetry when the separate assertion is made that Q operators exist that transform between our fermionic and bosonic states. So would it be true to say that non-supersymmetric theories contain superpotentials, W, within their Hamiltonians in the same way that supersymmetric theories do? If this is the case the superpotential is just a useful function that is especially helpfully when we consider supersymmetric theories? I.e. superpotentials exist with or without supersymmetry.


Dirichlet's work on gravity in non-Euclidean space?

In the book _The Norton History of Astronomy and Cosmology_ by the late John North I have found the following statement (page 514): "The German mathematician Lejeune Dirichlet studied the law of gravitation in non-Euclidean space towards the end of 1850." Does anyone perhaps know more about that?


Is it possible to build a huge spaceship like the one of of Star Trek or Prometheus?

Ok, so theoretically, is it possible to build a spaceship like the ones we see off of Star Trek or Prometheus or will physics not allow it? What are some challenge faces such an task. Also, is there other ways we can use to launch heavier payload from earth or are we just stuck with using tons of fuel to launch small object?


Dynamics of a Rocket

I am interested in modelling the trajectory of a rocket from the Earth to the Moon by solving a differential equation numerically. Below are some key facts and assumptions I am using. I want to make sure that I have not made any serious mistakes, nor disregarded any necessary facts. We will consider the following equation, $$ \vec{T} + \vec{c}(\vec{r})\dot{\vec{x}} + \vec{G}(\vec{r})= m(t) \ddot{\vec{x}}, $$ where $T$ is the _**constant**_ rocket thrust, $c$ denotes air resistance and is a function of radial distance from the earth, and the rocket has mass that drops at a rate that is _**constant**_ with respect to time (we are assuming that a constant amount of fuel is always used for constant rocket thrust -- is this a valid assumption?). Now a question:   * The trajectory of the rocket is not straight; how do we incorporate parabolic motion into the numerics?


Causal and Global structure of Penrose Diagrams

What kind of global and causal structures does a Penrose diagram reveal? How do I see (using a Penrose diagram) that two different spacetimes have a similar global and causal structure? Also, I have the following metric $$ds^2 ~=~ Tdv^2 + 2dTdv,$$ defined for $$(v,T)~\in~ S^1\times \mathbb{R},$$ e.g. $v$ is periodic. This is the according Penrose diagram: ![](http://www.matheplanet.de/matheplanet/nuke/html/uploads/8/8403_misner_penrose_2.jpg) Is the Penrose diagram that I have drawn correct?


Regarding formulation of a multipoint model of fluid dynamics

Suppose I am trying to formulate a multipoint model of fluid dynamics. I have a procedure for doing so the details of which is not important to this question, but only that it is based on a series expansion, the higher order terms in which series being dependent on larger number of spacial points. The equations for the first two terms in a Burger formulation is something like shown below: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 u^0}{\partial x^2}(x,t) \+ u^0(x,t)\frac{\partial u^0}{\partial x}(x,t)\\\ &\qquad\qquad\qquad\quad\;\;+\int_0^1 u^1(x,t;x_1)\frac{\partial u^1}{\partial x}(x,t;x_1) dx=0\\\ &\frac{\partial u^1}{\partial t}(x,t;x_1)+\frac{\partial^2 u^1}{\partial x^2}(x,t;x_1)+u^0(x,t)\frac{\partial u^1}{\partial x}(x,t;x_1)+u^1(x,t;x_1)\frac{\partial u^0}{\partial x}(x,t)=0 \end{align} it is however an odd formulation, the point $x_1$ appears as a mere parameter in the second formulation while it appears only in the integrand of the first equation, so that apparently there is no enough restriction on the behavior of the function $u^1$ with respect to $x_1$, so it seems there s no unique solution for $u_1$. If the flow is a homogeneous flow but $u^1$ can be written as $u^1=u^1(x-x_1,t)$, so that derivation with respect to $x$ will also take into play the point $x_1$, so that $x_1$ will now behave like it is a more important variable. This will get more importance if we further consider the higher order terms like $u^2(x,t;x_1,x_2)$ which in a homogeneous flow will be writable as $u^2(x-x_1,x-x_2,t)$ and whose $x$-derivative becomes: $$\frac{\partial u^2}{\partial x}=\frac{\partial u^2}{\partial(x-x_1)}+\frac{\partial u^2}{\partial(x-x_2)}$$ Such a treatment imports the variables $x_1$, $x_2$ and etc. in the second and higher order equations in a more proper manner, in such a way that hope to find unique solutions increase.   1. The problem is that how the first formulation which was more general is lame in giving such unique solutions as the second formulation is apparently capable of?   2. Or maybe there is a must in every multi-point modeling of fluid dynamics (actually turbulence) to consider one of the following two procedures? a- To expand the series in terms of functions of the form: $u^0(x,t)$, $u^1(x,x-x_1,t)$, $u^2(x,x-x_1,x-x_2,t)$ and etc. ? b- To write the equations once at the point $x$, then once at the point $x_1$ available also in the arguments of $u^1$ and higher order functions, then once at the point $x^2$ available also in the arguments of $u^2$ and higher order functions, and etc., then consider all those equations in one place, for example by adding them together. As the series is truncated somewhere at a function $u^n$ this will not contain infinitely of equations but I am not very hopeful that this is the right path to take.     * any idea about these or other methods for multipoint formulations?


This is a confused part ever since I started learning electricity. What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)? All of them have the same SI unit of Volt, right? I would appreciate an answer.


A water drop in a falling lift

Consider a lift, which is at rest in an homogeneous gravity field. There is a thin layer( **with thickness** $h$) of water on the floor of the lift. At some moment a single cable, supporting the lift, breaks and the lift begins free falling(forever). It is easy to describe qualitatively what happens with water, i think: the formation of a drop begins. During this process the drop jumps up from the floor and once the total kinetic energy of the drop is dissipated into the heat, the center of the drop stands at some height $H$ from the floor. (All the kinetic energy of the drop is coming from the difference in surface tension energy of water between initial and end moments.) Question: How to determine(approximately) $H$ ? For simplicity let's assume that the cross-section of the lift is a disc- shaped with radius $R$. $(h<<R)$ Remark: **A key point is to determine how the total kinetic energy distributes between internal kinetic energy and translational kinetic energy of the drop, i think.** With this the problem will be solved. So it is convenient at the early stage of formation of the drop ignore viscosity effects of water and air resistance. **Edit:** I would like to clarify that by internal kinetic energy i mean the kinetic energy of macroscopic motion of water inside the drop. At early stage there is no energy loss assumed, for simplicity.


What was so surprising about accelerating cosmological expansion?

The 2011 Nobel Prize in Physics, as far as I understand, concerns the expanding universe -- galaxies moving away from each other at ever increasing speed (that's what I think I read in newspapers). Now, the idea of expanding universe is not new. What is the novelty in this study? What was unexpected in it?


Energy conservation and interference

I have a problem with energy conservation in case of interfering waves. Imagine two harmonic waves with amplitudes $A$. They both carry energy that is proportional to $A^2$, so the total energy is proportional to $2A^2$. When they interfere, the amplitude raises to $2A$, so energy is now proportional to $4A^2$ and bigger than before. The equivalent question is what happens to the energy with the superposition of two waves that interfere destructively. Also, if someone could comment on the statement about this problem in my physics book (Bykow, Butikow, Kondratiew): the sources of the waves work with increased power during the interference because they feel the wave from the other source.


Does quantum mechanics prohibit extraterrestrial life?

There is a known thought experiment, connected to quantum immortality: a duel between physicist and a philosopher. Each turn physicist and philosopher fire at each other with a pistol. The quantum immortality predicts that each of the participants will find themselves alive and the opponent dead after a number of shots. We can modify the experiment in the following ways:   1. They do not fire at each other, but it is lightning that shots them. The outcome should not change.   2. There are not two but more participants.   3. They are located at separate islands in the ocean. It seems that even if there are hundreds of participants, each of them will eventually find all others killed by lightning. If he could not observe other isles directly, after he discovers and explores other isles, he will find them uninhabitable. Now pretend that the participants actually live on separate planets, and in each million years there happens disaster that kills all inhabitants on 1% of all planets. It seems that after years the physicist that lives on one of the planets should discover that all other planets are uninhabitable. Even more: there is even no chance that organic matter on each other planet could actually evolve into anything resembling actual life. That said the physicist will find out that there is no other inhabitable planet in the Uinverse. On the other hand, this does not mean that extraterrestrial life does not exist. It actually cannot not to exist. But it exists in parallel universes (in terms of MWI) and thus unobservable. Both physicist on planet A and philosopher on planet B exist, but will never meet in the same universe. Thus, QI it seems predicts one concrete observable consequence that extraterrestrial life can be never detected. What is your opinion?


Could moving land mass alter Earths gravity?

This is potentially a very stupid question but I'm going to ask it anyway. With all these huge buildings such as the Abu Dhabi Mosque, where an unbelievable amount of materials such as marble was moved from one side of the earth to the other, is it possible that if we 'shift enough stuff' that we could change the earths centre of gravity, and potentially alter its orbit?


Does the weight of fluid in a conical container act entirely on the base?

# Scenario description Lets assume both containers have a capacity of 300 litres. One is a vertical tube as shown in pic 1 Other one is more or less a V shaped containers as shown in pic 2 Both are at ground level # Question No doubt that whole weight of 300 litres will act on the base of the tube (pic 1). But will the weight of the whole 300 litres act on the base of the container as shown in pic 2? I.e if you keep both the containers on a weighing scale, for sure, it will show a weight of approx. 300 kg each. But what I'm asking is whether the weight of whole 300 litres acts on the **base** of container 2 (since the container's walls are slanting, as opposed to the walls of the tube in pic 1). If the whole weight will not act on the base of the container, can you state the reasoning and also the principle, so that I can further get to know about it? Pic 1                          |         |                        |         |                |         |                |         |                |         |                |         |                |_________|      Pic 2                        \              /               \            /                \          /                 \________/


Was the firewall paradox resolved?

For quite sometime there has been a claim that the firewall paradox has been resolved (via lasers). For instance, http://global.ofweek.com/news/Lasers-to- solve-the-black-hole-information-paradox-9867 . I was wondering what the physics community thought about it? I haven't heard much of a reaction of the physics community. Why is this? His paper: http://arxiv.org/abs/1310.7914


Does the entropy of an adiabatically isolated system stays the same after we decrease/increase the pressure?

> Even if the system is isolated and there is no heat exchange with > surroundings, shouldn't the decrease/increase of pressure result in > increase/decrease of entropy? > > Does this property of an isolated system means that increase of pressure > means equivalent decrease in volume, so after the process the $pdV$ i.e. the > work is still the same? > > Can someone explain this in more detail? Quoted text above was original question. Please read on further, I clarified the question. To be more specific, situation is this. We have an adiabaticaly isolated system and (for example) a piston inside it. If there is piston, that means that volume work or $PdV$ can be done. **So if we do work on the system, by some force from surroundings which pushes piston(compresses the gas) we would get increase of the pressure inside the system... right?** **If we increase the pressure in the system, the entropy will decrease as well, altough we haven’t bring in the heat .. right?** So if we increased the pressure and mutually decreased the volume, the molecules of the gas are packed more tightly and they have lesser space to move in...This is followed by system having more internal energy now, then it had before process...Since the activity of molecules inside is higher, they are moving at higher speeds(cause of smaller volume)... **That means that system should have more internal energy after the process, right**? Taking that into consideration, and if we follow the rule that increase in pressure is also increase in temperature, that is molecules have higher activity on higher pressure and that kind of activity is measured with temperature... **that means that system also has more amount of heat?** _Correct me if I am wrong._ **And even if it is adiabatically isolated (dQ=0) shouldn't the internal energy increase and thus amount of heat in the system increase as well when we do the work on the system (move the piston to compress the gas )?** I am talking that higher activity leads to more heat inside the system even if it is prevented (isolated) to exchange heat with its surroundings. I hope you understand me now.


About binary stars and calculating velocity, period and radius of their orbit

I saw somewhere about being able to measure the velocity, period and radius of a binary star orbit by looking at red shift and blue shift. I understand it but can someone give me an example of calculations etc done to calculate the velocity, period and radius of a binary stars in orbit?


Could the collision of two pairs of quantum entangled protons cause a temporary "wormhole"?

I recently read this article from MIT News. I then started thinking about how a particle accelerator creates a temporary microscopic black hole. My question is: > _If quantum entangled pair $A$, consisting of $A_1$ and $A_2$, and a second > pair $B$, consisting of $B_1$ and $B_2$, were collided, $A_1$ collides with > $B_1$, and $A_2$ collides with $B_2$, would this create a temporary > wormhole?_


What is an "Interaction Hamiltonian"

I'm an undergraduate reading up on some quantum physics so that I can help out more in the lab that I'm working in this summer. In the book I'm reading (Shankar's "Principles of Quantum Mechanics") I just came across the term interaction Hamiltonian in describing how orbiting electrons interact with a magnetic field. I have an idea of what it might mean, but I can't find a good explanation anywhere. What is an "interaction Hamiltonian", and how does it differ from a standard Hamiltonian?


Dark energy before radiation or after?

In the cosmic history of the universe, does the dark energy comes before the radiation epoch, or only now, in the 'matter' universe (matter dominated era)? Because, we now know that like 75% of our visible universe is made up of dark energy, about 20% is dark matter and the rest is the 'ordinary matter', but does that mean that this dark energy came after the radiation and matter domination of the universe or was it there in the beginning, at the BB, and then was just spread out via inflation, while radiation era and matter domination era followed?


Basic Interpretation of Compostion of Observables and their Measurement

Given two (or more) observables $A, B$ which commute one can construct a third observable $C= A \circ B$. If $\psi$ is a common eigenvector of $A, B$ with eigenvalues $\lambda_1, \lambda_2$ then it is clear that the measurement of $C$ of the state $\psi$ gives the measurement result $\lambda =\lambda_1 \lambda_2$, i.e. the result of the measurement of the observable $C$ is the result from the measurement of $A$ times the result from the measurement $B$. But what if $\psi$ is an eigenvector of $C$, but not of $A$ and $B$? Is there any connection between the measurement results of $A$, $B$ and $C$? Example: Let there be three observers which measure a spin state with the corresponding observables $A = \sigma_x \otimes \mathbb{I} \otimes \mathbb{I}$, $B=\mathbb{I} \otimes \sigma_y \otimes \mathbb{I}$ and $C=\mathbb{I} \otimes \mathbb{I} \otimes \sigma_y$. They commute and we can construct $D= A \circ B \circ C = \sigma_x \otimes \sigma_y \otimes \sigma_y$. Now the GHZ-state $\psi = \frac{1}{\sqrt{2}} ( | +z, +z, +z \rangle - | -z, -z, -z\rangle)$ is an eigenvector of $D$ with eigenvalue $\lambda =-1$ but it is not an eigenstate of $A, B$ or $C$. Each of the observers will get a result $\pm 1$. Is there any connection between this individual results and the eigenvalue of $\psi$ (respectively the expectation value $\langle \psi | D | \psi \rangle = -1$)? Intuitively I would say that the product of the results should give the eigenvalue of $\psi$ but I can't see how this should follow from any quantum mechanical postulate or mathematical reasoning like in the case of the common eigenvector.


Counting of brownian particles: Point Process

Imagine a point process defined by the passage time of purely brownian particles through a given point (in 1D), line (2D) or plane (3D). I'm interested in the variance of the counts (number of particles passing the points) as a function of sampling time. Unlike brownian motion, I expect my signal to show non-vanishing correlation, because a particle that just crossed the point, has a high probability to cross it again. Thus the correlation should look like a dirac delta function (classic for discrete point process) plus a decreasing time function. However, I'm a bit confuse with the way to calculate it analytically. I search in areas of photon counting, geiger counter, and gaz dynamics, but I didn't find any explicit calculation of the variance. Two idea I had: 1/ Studying the Spatio-Temporal correlation function of a field of Brownian particles (see Gardiner, Stochastic Methods, eq. 13.3.22 for instance): $G(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-x^2/(4Dt)\right)$ in 1d. We should have: $Var(T)=\lim_{x\rightarrow 0} \int_T\int_T G(x,t) dt dt'$. However, the last expression is hard to compute. 2/ Using the probability of return time to origin of Brownian motion (the correlation arise from successive passage of the same particle) Any idea or suggestions would be welcomed !


Quantum Locking Superconductors on Earth's Magnetic Field

Is the earth's magnetic field strong enough to hold a quantum locked superconductor, or the superconductor wouldn't be able hold its own weight? How strong should be the earth's magnetic field to hold quantum locked superconductors?


Ascent rate and size of balloon

I am part of a school project, Project Stratos to send a balloon to the edge of space (the closer side :P) and was wondering how you would work out the accent rate of a large balloon (roughly 1m^3 of helium with 100g of mass) and the size of it as it increases its Altitude. I am creating a live map (that will be based on predictions rather than its actual location) and want to know the speed it will float up into the atmosphere. Currently we are assuming the ascent rate will be about 5m/s but I doubt that is very accurate and would this speed increase as it gets higher? Edit: I would also quite like to know the burst height of the balloon.


Understanding the cause of the big bang

Ok, as I understand the expansion of the initial singularity was caused by quantum fluctuations like the ones predicted by the Heisenberg Uncertainty Principle. But how can these fluctuations occur within a singularity? And how can they cause the expansion of that singularity? Or do I have a terrible misunderstanding of the theory?


Does a Lorentz-contracted object float or sink?

Consider the following thought experiment: Imagine an object of a certain mass density which allows it to float in water. Now if this object is viewed from a moving frame with high speed, it will look Lorentz contracted hence its density will look greater than the rest density. So from the rest frame it floats but from the moving frame it sinks! What am I missing?


Is it possible/correct to describe electromagnetism using curved space(-time)?

Comparing the simples form of the forces of both phenomena: the law of Newton for gravitation $V\propto \frac{1}{r}$, and the Coulomb law for electrostatics $V\propto \frac{1}{r}$, one might think that if one can be extended relativistically to a curvature in space-time, that the other one would lend itself to a similar description. Is this so? Thanks for any useful thoughts and/or suggestions!


Is there any relationship between Gravity and Electromagnetism?

We all know that the universe is governed by four Fundamental Forces which are The strong force , The weak force , The electromagnetic force and The gravitational force . Now, is there any relationship between Electromagnetism and gravity?


Theories that Relate Gravity, Electricity, and Magnetism

There are some people who (without having a stated theory that I know of) insist that Gravity, Electricity, and Magnetism are related. Some point to symmetry in Maxwell's Equations as a potential indication of this (intuitive) connection. Is it possible that there are "Maxwell-like" equations that relate Electricity and Magnetism directly to Gravity? All forms of electromagnetic energy do interact with gravity. Are there any physicists who are working on this sort of theory?


Additive Trichromatic Color: Seeking Answer for a Fun Application

I'm not the brightest person and I was wondering if it was possible, in the same way that it's possible to make ink that's visible under UV light, if it's possible to make ink that requires a distinct combination of colours of light to work. If there are different coloured UV lights would it be possible to make an ink (or buy one) that reacts only when all three are present and overlapping?


Lorentz Transformation via Geometry

Today, I was tutoring and explained the space-time. I explained how one can convert North-South into West-East by rotating, and how you can convert time into space with velocity. Below the Energy-Momentum stuff the book had some problems. One was the following: > Given an event at (1 Lightsecond, 2 s), give it's coordinates in a frame of > reference that is moving with 0.6 of speed of light to it. This the diagram I came up with, in blue there is the event. (Not really to scale.) ![](http://wstaw.org/m/2011/11/11/m8.png) Then I projected the point onto the orange axes, and I get smaller values on each. This makes sense to me, as a moving observer would see thing smaller and slower, resulting in less values in either measurement. But how do I get the orange lines? I cannot see how I get this angle. If I got them, I could tell you the speed of the reference frame, but the speed is given here … And did I do the projection of the speed right?


Stern-Gerlach-Experiment with j=1 atoms?

Suppose you do a Stern-Gerlach experiment with atoms in a $j=1$ state. There would be three separate beams ($m_j = -1, 0, 1$) coming out of the apperatus. But what would be the relative distribution of atoms in these beams? On first thought I would say that the distribution should be $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{3}$. But considering the "classical" distribution of the projections of the angular momentum onto one axis I could imagine that there could be more atoms in the $m_j = 0$ state. This would still be compatible with symmetry considerations. To state the question differently: how does the density matrix of an unpolarized beam of $j=1$ atoms look like?


Why is a coherent state an eigenfunction to the annihilation operator?

In class when we talked about the harmonic oscillator in QM we noticed that the eigenfunctions to the annihilation operator are coherent states in the sense that they have minimum uncertainty in momentum and position. My question is: What is the physical meaning behind this? Why are coherent states eigenfunctions to the annihilation operator and: Was this only a lucky punch and a special property of the harmonic oscillator or is this true for more physical problems?


Is carbon dioxide a greenhouse gas?

> **Possible Duplicate:**   >  What experiments prove the greenhouse effect? I am seeking for a proof that CO2 is a greenhouse gas. I posted this on **Skeptic.SE** recently but found no help in seeking for proof: > I assisted to a physicist conference in my university a few years ago > against the case that carbon dioxide was a cause of global warming. The main > point was that CO2 is **not** a greenhouse gas. I did a research to find > evidence for either side and found absolutely nothing. > > So, is carbon dioxide a greenhouse gas? If yes, has it been demonstrated in > a scientific paper? > > Here are some discussion articles describing arguments against CO2 being a > greenhouse gas: Home experiments Ph.D. Tim Ball Evidence from temperature rise through history I understand that the sources do not prove that CO2 isn't a greenhouse gas...but I can't find evidence proving the contrary. Since a real physicist with many years of experience a demontrated to me that it is not possible by definition, I need to find proof to accept the other premise. I found this post on physics.SE, but the asker do not seek proofs and simply accept what the others are saying. Baically, to have a greenhouse effect, a gas needs to absorb only a portion of a wavelength and let pass another portion. Since CO2 absorbs some IR, a part of it is remitted to earth. The spectrum is absorbed by the earth (everything) and it is remitted on another wavelength which allows a different proportion to pass through the CO2 layer. The question is, why the heat would be remitted toward earth (it should go in the colder direction)? And also, if CO2 is all around the earth and blocking a certain wavelength, why would increasing it warm the earth? Am I missing something?


What are the temperatures of objects in Low Earth Orbit (LEO)?

What is the temperature of objects in Low Earth Orbit? Consider LEO to be 600km to 800km.


Optical refocusing efficiency

What is the material for optical focusing that produces the less diffraction losses? Suppose one have a sequence of serial optical elements $R$ that keep refocusing a beam of collimated light, each a distance $D$. Now suppose that the wavelength $\lambda$, the optical element and the distance are such that the $\epsilon$ is the amount of radiation that falls outside the next optical element, which i'll take to be small. I'm concerned with other losses in this system, for instance, the optical material can have some intrinsic diffusion, or it absorbs a portion of the radiation What are the main losses that an optical element can have of this kind, and which are the best refocusing optical elements that minimise those losses?


How will the super massive black hole affect our galaxy?

I've recently learned that the general consensus is that several (if not, most) galaxies have super massive black holes in their center, in particular the Milky Way. This, at least to me, makes perfect sense seeing as we are in a spiral galaxy which means we need something to "spiral" around (a large body or a bunch of mass). But seeing as we're rotating around this massive black hole, won't we inevitably end up sucked in by it? Aren't we spinning towards the center of the galaxy, or are we staying steady where we are?


Reachable area of cannonball given fixed initial speed

The trajectory of a cannonball fired from the origin with initial speed $v_0$ at an angle $\theta$ is given by $$y = x\tan\theta - \frac{g}{2v_0^2 \cos^2\theta}x^2.$$ For fixed $v_0$, at what angle $\theta$ should the canon ball be fired so that it will hit a target located at $(x_0,y_0)$? What is the area of the region in which the target can be located such that it is possible to hit the target?


Suppose we fill a membrane with ferro fluid, and alter the magnetic field around it in specific ways, what dynamics can come into play?

im trying to develop a flexible robotic arm filled with ferro fluid. Different flexible motions can be performed using this concept.


How does one subtract two light beams?

From what I understand, it seems like you can only "add" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a local oscillator beam to mix with the signal beam so you can decipher the difference in phase change which would give you information about the system your signal beam is probing. Homodyne detection can give information regarding the position of an atom. Say I wanted to detect the relative distance of two atoms. Each atom/ion is stuck inside an electric harmonic potential. The location of the harmonic potentials are known, (if this is possible) Now, let's say I turn off the harmonic potential trap and at the same time I use the homodyne measurement technique to determine the position of both of the atoms. This means shining a laser to detect the atom's position. (The light is absorbed and re-emitted?, or reflected?) In any case, light that comes out of the atom/ion cavity is related to the atom's position/momentum and it is combined with a local oscillator beam to determine the position and momentum of the atom. The light hits photodetectors which then integrate the signals together producing a photocurrent. The analysis of this photocurrent will allow one to figure out the atom's position relative to the center of the harmonic trap. To figure out the relative position of the atoms from each other, I'll need to "subtract" the position of one atom with the other. If one atom's position is labeled x1 and the other is labeled x2, than wouldn't we simply do x2 - x1? If so, how is one to do this?! How does one "subtract" the position of one from the other using light beams?


Could the outer structure of the Universe be made from Antimatter?

Cern recently stated that antimatter may be repelled by matter, much like the opposite effect of Gravity. So is it possible that antimatter is actually repelled to the edges of the Universe to create a sort of outer-shell, something that allows the expansion of the Universe into nothingness..


Is restoring force a particular type of force?

I have a question about the restoring force in elastic band or rope which confusing me for a long time. As I was told in high school physics, for an elastic band (or spring), if Hooke's law holds, we have $F = k\Delta x$. What's confusing me is: should F be the total force acting on the object or the restoring force only? Or I ask this way: is there anything called "restoring force" existing independently, just like gravity, friction or tension? To my understanding, restoring force should be the total force which is pointing to the equilibrium point. For example, if we consider a bungee cord, we should always count the tension of the cord as well as the gravity so the restoring force at any time should be the total force of tension and gravity; hence, when we apply Hooke's law, we should always have $F$ being the total force, not just the tension. Is that correct? This is pretty confusing to me because there use many terms in the book. Sometimes they said it is the tension in Hooke's law, sometimes they say the restoring force and sometimes the total force....


Does changing the electric / magnetic field cause self-reinforcing induction of the other?

I understand that changing electric field produces magnetic field and changing magnetic field produces electric field. Are these produced magnetic and electric field produced due to one defined to be constant or variable? If these are defined to be variable then do they continue to produce one another? By this I mean if changing electric field produces changing magnetic field, does this changing magnetic field produce a new electric field or the same one again?


Do radio waves go faster than the speed of light?

My science teacher used to say a lot of weird stuff, but I'm just making sure on this one.


Does a car consume more fuel when it's raining?

Yesterday my wife asked me that question, and I couldn't answer. Consider a car, in a sunny day, and that is consumes x gallons per mile. Considering that everything is equal, except that it's traveling in a rainy day, but at the same temperature as the sunny day so that air density is the same. Will the lower friction of the tires make it consume more or less fuel? And the fact that rain drops are falling over and in front of it ? I answered that it'll consume more fuel, since friction is what makes car move and that the rain will act against it... but I'm not sure ?


How do we determine the mass of a black hole?

Since by definition we cannot observe black holes directly, how do astronomers determine the mass of a black hole? What observational techniques are there that would allow us to determine a black hole's mass?


Elements of a Planet reveals nearby supernova remnant?

During a random reading through this site, I found this one: Origin of elements heavier than Iron (Fe)... The answer was "The formation of many elements in earth was due to Supernova nucleosynthesis" as told by some guy. Here, A question crosses my mind:   * If the elements were formed due to the explosion of a supernova, then there should be a **remnant** like a black-hole or a neutron star nearby... Were there any nearby? Or, the _famous_ **Big Bang** is responsible for this? ( _Down-voters - leave comments_ )


Partial waves and the velocity expansion of a scattering cross section

I'm confused about the relation between the velocity expansion of a scattering cross section and the angular momentum (partial wave) expansion. For example, for dark matter annihilation, we write $\sigma v = \sigma_s + \sigma_p v^2 + \cdots$, where $v$ is the velocity and $\sigma_s$ corresponds to $s$-wave scattering (orbital angular momentum $\ell =0$) and $\sigma_p$ is the $p$-wave scattering ($\ell =1$). **Question:** : why does the $v^{2\ell}$ term correspond to initial states in an $\ell$-wave configuration? * * * I understand that in these calculations we treat the scattering states as plane waves, and that these can be expanded as a series of partial waves ($\ell$ eigenstates). And I can see that $\langle \mathbf x | E,\ell,m\rangle \sim j_\ell(kr) \sim k^\ell$. Is there more to this story?


What is the particle residence time for given flow rates of gas mixture components?

For two different chemical substances there are two open valves with flow rates $$Q_1=a\frac{m^3}{h}\ \text{ and }\ \ Q_2=b\frac{m^3}{h},$$ leading into seperate cables. Next, the cables join, the substances mix perfectly and they flow along together. Then during the route, there is an observational volume $V$ (e.g. a cylinder or length $d$ and cross section $A$). > How long is the residence time $\Delta t$ of a particle of the mixture > inside the volume?


A relative time dilation paradox.

Let us assume that there are two astronauts A and B who are floating in space. A sees B passing by and vice versa. A sends signals to B every minute. According to A since B is moving his clock will be slower. So B will receive the signals prior to the appointed minute. The same argument can be applied for B who will conclude A's clock is running slow. Who is right?


Does gravity limit the number of bosons that can occupy the same single-particle state?

QFT says that an unlimited number of bosons can occupy the same "state" (what I mean by that is that the whole system's wavefunction is composed of a product of many identical wavefunctions). However, gravity increases monotonically with energy density. It seems that at some point, one additional boson would create a high enough energy density to create a black hole. Is this true? Could I calculate the number of bosons necessary to cause this?


Ward Takahashi identities from Z invariance

I'm trying to get Ward-Takahashi identities using the approach used in Ryder's book (pages 263-266). I like that he starts from demanding gauge invariance of Z in a explicit way and them explores the consequences of that to functional generators of vertex functions. But the actual calculation is bugging me out. The author seems oblivious to the fact that the fermionic fields and sources ($\psi$ and $\eta$) are Grassmann variables and keeps commuting them out with no regard for my sanity. For instance, equation (7.102) has a term: $$i e (\bar{\eta}\psi-\bar{\psi}\eta) $$ that promply becomes (exchanging the fields by ${1\over i}$ times the derivatives on the sources, acting on Z to the right): $$ e (\bar{\eta}{\delta \over \delta \bar{\eta} }-\eta{\delta \over \delta \eta }) $$ In my opinion that should be: $$ e (\bar{\eta}{\delta \over \delta \bar{\eta} }+\eta{\delta \over \delta \eta }) $$ This one has no consequences because he commutes them again right after. But when I try to calculate it being careful with the Grassmann variables, I can never get the right signs in (7.111). I'm specially troubled by the derivatives below (this is what I'm getting, but one of them should have a different sign in order to get the right WT identities): $${\delta \over \delta \bar{\psi}(x_1) } {\delta \over \delta \psi(y_1) } \left[{\delta \Gamma \over \delta \psi(x) }\psi(x)\right]_{\psi=\bar{\psi}=0}=-\delta^4(x-y_1){\delta^2 \Gamma \over \delta \bar{\psi}(x_1) \delta\psi(x) }$$ $${\delta \over \delta \bar{\psi}(x_1) } {\delta \over \delta \psi(y_1) } \left[\bar{\psi}(x) {\delta \Gamma \over \delta \bar{\psi}(x) } \right]_{\psi=\bar{\psi}=0} =\\\=-\delta^4(x-x_1){\delta^2 \Gamma \over \delta \psi(y_1) \delta\bar{\psi}(x) } = \delta^4(x-x_1){\delta^2 \Gamma \over \delta \bar{\psi}(x) \delta\psi(y_1) }$$ Does anybody ever did this calculation in detail and has some pointers? Are there any other references that follow this same approach? EDIT: Just a shameless bump: I still looking for some light on this. Any reference on where this is done in detail would help.


Noether theorem, gauge symmetry and conservation of charge

I'm trying to understand Noether's theorem, and it's application to gauge symmetry. Below what I've done so far. First, the global gauge symmetry. I'm starting with the Lagragian $$L_{1}=\partial^{\mu}\Psi\partial_{\mu}\Psi^{\ast}-m^{2}\left|\Psi\right|^{2}$$ with classical complex fields. This Lagragian is invariant with respect to the global gauge symmetry $\Psi\rightarrow\tilde{\Psi}=e^{\mathbf{i}\theta}\Psi$, ... such that I end up with $$\delta S=\int dv\left[\dfrac{\delta L_{1}}{\delta\Psi}\delta\Psi+\dfrac{\delta L_{1}}{\delta\Psi^{\ast}}\delta\Psi^{\ast}+\mathbf{i}\left(\Psi\partial^{\mu}\Psi^{\ast}-\Psi^{\ast}\partial^{\mu}\Psi\right)\partial_{\mu}\delta\theta\right]=\int dv\left[\partial_{\mu}j^{\mu}\right]\delta\theta$$ provided the equations of motion ($\delta L / \delta \Psi = 0$, ...) are valid. All along I'm using that $$\dfrac{\delta L}{\delta\phi}=\dfrac{\partial L}{\partial\phi}-\partial_{\mu}\dfrac{\partial L}{\partial\left[\partial_{\mu}\phi\right]}$$ and that $\int dv=\int d^{3}xdt$ for short. The conserved current is of course $$j_{1}^{\mu}=\mathbf{i}\left(\Psi^{\ast}\partial^{\mu}\Psi-\Psi\partial^{\mu}\Psi^{\ast}\right)$$ since $\delta S / \delta \theta =0 \Rightarrow\partial_{\mu}j_{1}^{\mu}=0$. **Here is my first question:** Is this really the demonstration for conservation of charge ? Up to now, it seems to me that I only demonstrated that the particle number is conserved, there is no charge for the moment... Then, I switch to the local gauge symmetry. I'm starting with the following Lagrangian $$L_{2}=\left(\partial^{\mu}+\mathbf{i}qA^{\mu}\right)\Psi\left(\partial_{\mu}-\mathbf{i}qA_{\mu}\right)\Psi^{\ast} -m^{2}\left|\Psi\right|^{2} -\dfrac{F_{\mu\nu}F^{\mu\nu}}{4}$$ with $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$. This Lagrangian is invariant with respect to the local gauge transformation $$L_{2}\left[\tilde{\Psi}=e^{\mathbf{i}q\varphi\left(x\right)}\Psi\left(x\right),\tilde{\Psi}^{\ast}=e^{-\mathbf{i}q\varphi\left(x\right)}\Psi^{\ast},\tilde{A}_{\mu}=A_{\mu}-\partial_{\mu}\varphi\right]=L_{2}\left[\Psi,\Psi^{\ast},A_{\mu}\right]$$ Then I have $$\delta S=\int dv\left[\dfrac{\delta L_{2}}{\delta\Psi}\delta\Psi+\dfrac{\delta L_{2}}{\delta\Psi^{\ast}}\delta\Psi^{\ast}+\dfrac{\delta L_{2}}{\delta A_{\mu}}\delta A_{\mu}\right]$$ with $\delta\Psi=\mathbf{i}q\Psi\delta\varphi$, $\delta A_{\mu}=-\partial_{\mu}\delta\varphi$, ... such that I end up with $$\dfrac{\delta S}{\delta\varphi}=\int dv\left[\mathbf{i}q\Psi\dfrac{\delta L_{2}}{\delta\Psi}+c.c.+\partial_{\mu}\left[j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu}\right]\right]$$ with $j_{2}^{\mu}=\partial L_{2}/\partial A_{\mu}$ and $F^{\nu\mu}=\partial L_{2}/\partial\left[\partial_{\nu}A_{\mu}\right]$ Then, by application of the equations of motion, I have $$\partial_{\mu}\left[j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu}\right]=0\Rightarrow\partial_{\mu}j_{2}^{\mu}=0$$ since $\partial_{\mu}\partial_{\nu}F^{\nu\mu}=0$ by construction. Of course the new current is $$j_{2}^{\mu}=\mathbf{i}q\left(\Psi^{\ast}\left(\partial^{\mu}+\mathbf{i}qA^{\mu}\right)\Psi-\Psi\left(\partial^{\mu}-\mathbf{i}qA^{\mu}\right)\Psi^{\ast}\right)$$ and is explicitly dependent on the charge. So it seems to me this one is a better candidate for the conservation of charge. NB: As remarked in http://arxiv.org/abs/hep-th/0009058, Eq.(27) one can also suppose the Maxwell's equations to be valid ($j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu} = 0$, since they are also part of the equation of motion after all, I'll come later to this point, which sounds weird to me), and we end up with the same current, once again conserved. Nevertheless, I still have some troubles. Indeed, if I abruptly calculate the equations of motions from the Lagrangian, I end up with (for the $A_{\mu}$ equation of motion) $$j_{2}^{\mu}-\partial_{\nu}F^{\nu\mu}\Rightarrow\partial_{\mu}j_{2}^{\mu}=0$$ by definition of the $F^{\mu \nu}$ tensor. So, my **other questions** : Is there a better way to show the conservation of EM charge ? Is there something wrong with what I did so far ? Why the Noether theorem does not seem to give me something which are not in the equations of motions ? said differently: Why should I use the Noether machinery for something which is intrinsically implemented in the Lagrangian, and thus in the equations of motion for the independent fields ? (Is it because my Lagrangian is too simple ? Is it due to the multiple boundary terms I cancel ?) Thanks in advance. PS: I've the feeling that part of the answer would be in the difference between what high-energy physicists call "on-shell" and "off-shell" structure. So far, I never understood the difference. That's should be my last question today :-)


Could fast vibrations cause us to travel forward in time

Assuming it's possible to vibrate a human at near light speed without harming him, would a few minutes of this from his point of view be much longer from a stationary observer's point of view? In other words do vibrations work the same as normal movement with regards to time dilation? So a person could walk into such a machine, and walk out hundreds of years in the future, even though a much smaller amount of time would have passed from their perspective?


What is bulk viscosity and how does it affect the flow? Explain the idea of introducing such a term in the Navier-Stokes equation. What are the consequences if not taken into account?


Vibrating system at angular frequency in a vibrating system

Can anyone clarify what does mean the angular frequency of a system in case of the vibrating membrane. Angular frequency is measured in radians per second, what does this have with the vertical displacement in case of the wave equation.


Does turning sharply on a bicycle conserve more energy than a wide turn?

I use a bike to commute, so I spend a lot of time thinking about how to get the most bang out of my momentum. Aside from the extra distance traveled in a wide turn, does making a sharp turn save you any energy? My guess is no, because these things tend to even out, but it definitely _feels_ like I'm going much faster. I've even considered that maybe taking a turn sharp is worse, because the extra pressure will cause more friction in the bearings and the tires. Either way it's more fun, though that probably doesn't get a term in the equations.


Shor's algorithm and Bohmian Mechanics

Do quantum computer's tell us anything about the foundations of quantum theory? In particular Shor argued in the famous thread by 't Hooft Why do people categorically dismiss some simple quantum models? that quantum computation was at odds with 't Hooft's ideas. Does quantum computation tell us anything new about hidden variables like Bohmian mechanics (which, at least so far, is 100% in agreement with everything we know about physics, contrary to what some people (e.g. Motl) claim)?


Name of battery voltage when load connected/disconnected

If I had a 3V battery, and when no load connected it reads 3.2V, and with a load 2.8V (just a hypothetical example), what is the name for these two terms, with a load or no load? I know the voltage drop occurs due to its internal resistance when a load is connected, however I still am not sure of the terms. Here's a few which may be of help:   * emf ($\mathcal{E}$)   * Potential difference   * Terminal voltage Any help would be appreciated in what these terms are called, this is one of those questions where it almost seems too simple to find this answer online.


Scattering of two particles - phase factor

I did see some posts on stackexchange on this matter, but I find them to be beyond my scope or not directly related to what I am looking for. I am reading Feynman Lectures III, chapter 4. It talks about scattering of two particles $a$ and $b$ and it defines $f(\theta)$ to be the probability amplitude of particle $a$ scattering at angle $\theta$. Then it says "You might also think that the amplitude for the second process (where particle $b$ instead enters the detector placed at angle $\theta$) is just $f(\pi-\theta)$. But that is not necessarily so, because there could be an arbitrary phase factor." I'm not questioning why things behave the way they are in quantum mechanics. But, isn't it **_by definition of $f$_** that the probability amplitude of particle $a$ scattering at angle $\pi-\theta$ is $f(\pi-\theta)$? I don't see why there is a phase factor. If there's a phase factor, it should already be part of $f$.


Gauge invariance of Rarita-Schwinger action in curved spacetime

The Rarita-Schwinger action in curved $n$-dimensional spacetime is $$ \int \sqrt{g} \overline{\psi}_a \gamma^{abc} D_b \psi_c $$ Here $g = \det(g_{\mu \nu})$, and the indices $a, b \dots$ are 'internal' indices that transform under e.g. $\mathrm{SO} (3,1) $ in $3+1$ dimensions. $\gamma^{abc} = \gamma^{[a} \gamma^{b} \gamma^{c]}$ with the gamma matrices obeying $\gamma^a \gamma^b + \gamma^b \gamma^a = 2 \eta^{ab} $, and $\eta^{ab}=\mathrm{diag}(1,1 \ldots 1,-1,-1 \ldots -1)$ is the 'internal metric'. $\psi_{\mu} = \psi_{c} e^{c}_{\mu} $ is a spinor-valued one form. Spacetime indices $\mu, \nu$ can be 'converted' to internal indices using the frame field $e_a^{\mu}$, and vice versa. The covariant derivative is $D_{\mu} \psi_{\nu} =\partial_{\mu} \psi_{\nu} + \frac{1}{4} \omega_{\mu}^{ab} \gamma_{ab} \psi_{\nu} $. Here $\omega$ is taken to be the torsion free spin connection, and $\gamma^{ab} = \gamma^{[a} \gamma^{b]}$. In flat space, the covariant derivative becomes a normal derivative, and the action then has a symmetry $\psi_c \rightarrow \psi_c + \partial_c \phi $, with $\phi$ an arbitrary function. This freedom can be used to eliminate some of the degrees of freedom from the field $\psi_c$ which correspond to lower spin. However, in curved space there is no corresponding symmetry under $\psi_c \rightarrow \psi_c + D_c \phi$. For this reason, it is said that the Rarita-Schwinger action in curved spacetime is inconsistent. My question is, what goes wrong when you don't have this extra symmetry? And do the problems manifest at the classical level or only at the quantum level?


Why is the blue color of sky darker in high altitude regions?

I read that as the air density is lower at high altitudes, the scattering is less, hence the darker blue but then other colors are scattering lesser than blue, why don't we see a darker shade of the other colors?


Why does a rod rotate?

I'm a physics tutor tutoring High School students. A question confused me a lot. Question is: > Suppose a mass less rod length $l$ has a particle of mass $m$ attached at > its end and the rod is hinged at the other end in vertical plane. Another > point object of mass $m$ is moving with velocity $v$ and hits the rod at its > end and continues in its path with velocity $v/2$. The rod gets enough > velocity that it can complete a vertical circular path. What would be the > force due to hinge on the rod just after collision? Assume that time taken > for collision is very small. From conservation of momentum, the particle attached to rod gets velocity $V' = v/2$ which is equal to $2\sqrt{gl}$ and therefore $$T-mg=\frac{mv'^2}{l}$$ The tension in the rod is the Normal reaction due to hinge on the other end of the rod. Hence, the force exerted by rod is $$\frac{mv'^2}{l}+mg$$ But other teacher claims that I'm wrong. He says there will be horizontal force due to hinge on the rod. I say, there won't be any horizontal force. He gives an example that suppose there is a rod and you flick it at one end with finger then won't the other end move (or tend to move)? Well it does. **EDIT:** I've understood your answers mathematically.I still don't know how to counter my friend's argument. I'm convinced by it. Because it sounds intuitive. He says, suppose you have a rod (massless or with mass) in space, and you flicked at one of its end, then the other end surely will have velocity.Similarly, in the above problem, the collision will impart velocity to one end, so other end also must move but it is hinged and can't move because of hinge. So, there is a horizontal force due to hinge on the rod.


Idealized trajectory from sloped surface

I am a GIS programmer implementing a visualization. I am modeling the idealized trajectory of a particle ejected from a volcanic vent using: $$\text{distance} = \frac{(v^2 \times sin(2\theta))}{g}.$$ Where $g = 1.62\:\mathrm{m/s^2}$, $v$ is velocity, and $\theta$ is ejection angle. $g$ is the lunar gravity constant I was supplied. How can I incorporate the slope of the underlying surface assuming a single point of ejection? $$$$$$$$ EDIT: My current workflow is to compute total travel distance, extract a topographic profile along the total theoretical travel distance and then check the height of the projectile to the height of the actual surface at 100m intervals. In this way I can compute the landing site for the projectile. EDIT 2: I updated the question with the correct formula. Apologies for the incorrect transposition. My implementation now assumes a completely flat surface. What happens when the ejection surface is sloped either uphill or downhill?


Quantum Teleportation Fidelity

I understand that quantum teleportation fidelity is the overlap of the initial quantum state with the teleported quantum state. If the teleportation is perfect, then the fidelity would equal 1 or 100% successful of retrieving the quantum state at the desired location. In all real experiments I've glanced at and read thoroughly, the fidelities are never 1. Doesn't this suggest that what the experimenters "teleported" was really just a different state? Doesn't the fidelity have to be exactly 1 in order to really remove a quantum state from one location and make it appear at another location? My concern is that either I've missed something, or that we aren't really teleporting at all because the final state achieved on the other side is not equal to the initial state!


Can a divider "laminarize" turbulent flow and thus reduce friction?

Looking at the Moody chart I think to myself, the _friction factor_ doesn't decrease much at all with Reynolds number after a certain point. I wonder if laminar flow is more efficient in a sense, and what sense would that be? ![Moody diagram](http://i.stack.imgur.com/1bIeg.jpg) I understand that in laminar flow you have clear lines that the fluid doesn't cross, whereas that wouldn't be true in turbulent flow. We could imagine a pipe with a divider placed in the middle to keep the fluid from mixing in eddies, but that would just create more friction with the divider. There are thinkable cases, however, where you could introduce a divider that moves along with the fluid, in particular, the Taylor-Couette flow... http://en.wikipedia.org/wiki/Taylor%E2%80%93Couette_flow This setup describes basically one cylinder rotating within another cylinder with a fluid in-between them. ![Concentric rotating cylinders](http://i.stack.imgur.com/zA1r3.png) Let's say that you kept the fluid the same, and the distance between the inner cylinder and outer cylinder the same. In that system, let's say you insert a divider at a radius in the middle of the annular area, and this divider was mostly buoyant in the fluid, so it's not experiencing friction on the edges, and it also is free to rotate with the fluid. Would doing so actually reduce the frictional torque on the rotating inner cylinder? If you could introduce an infinite number of infinitely thin dividers is there a theoretical limit to how much you could reduce the retarding torque? Would that just make it laminar, or laminar-ish?


Why do liquids boil when their vapor pressure equals the ambient pressure?

Given that the boiling point of a liquid is the temperature at which the vapor pressure is equal to the ambient (surrounding) pressure, what significance does a liquid's vapor pressure have in the formation of bubbles that happens at and above the boiling point? The definition of boiling point seems to imply that the pressure inside of the bubbles must be at least as great as the liquid's vapor pressure in order to balance the outside pressure, but is there any particular reason why the pressure inside of the bubbles is related to the vapor pressure? The vapor pressure seems to be a measurement describing the tendency of the molecules to escape from the surface of the liquid, but I don't see how that relates to bubble formation within the liquid. This question has bothered me for a while, so any help would be much appreciated.


How do I go from exponents to a formula?

This is a continuation of this question. http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics- fall-1999/video-lectures/lecture-1/ skip this lecture to around 25:50. After doing dimensional analysis on $t\propto h^\alpha m^\beta g^\gamma$ Lewin concludes that: $$\alpha = \frac{1}{2}, \gamma = -1/2, \beta = 0$$ This is all fully understood, but he then goes to conclude from this that: $$t = C \sqrt{\frac{h}{g}}$$ How did he get to this? And why is he allowed to just assume that there is a constant, C, there when he doesn't even know its value or what it is? Keep things as simple as possible please, I'm 16.


Current formula and form factor

I am currently struggling with the formula for an exact current in QFT, a fermion with an upcoming momentum $p$ and an outgoing momentum $p'$. My problem is to show whether or not a term of the form $\bar{u}(p')F(q^2)i\sigma^{\mu\nu}(p+p')^{\nu}$ where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. I would like it to be forbidden by current conservation but I can't prove it, is my intuition wrong?


Classical Viewpoint on Electromagnetism

**Note:** This question may be difficult or impossible to answer within the rules of these forums due to its philosophical nature. I will delete the question if I am violating the rules. Onto the question! Recently I have been curious whether classical electromagnetism is fully solved (up to the divergences). Specifically, can we completely mathematically describe and then interpret the classical world of charges and currents and them the associated fields. Let us assume that our world is classical and that electromagnetism is a complete theory even though there are certain inconsistencies (self-interactions, infinite energy of point charges, etc). A common description of $\textbf{E}(\textbf{r}, t)$ and $\textbf{B}(\textbf{r}, t)$ among physics textbooks is that changing electric fields induce magnetic fields and vice-versa. This is assuming that there are no external fields or sources of field present. In symbols, $$ \partial_t\textbf{E} \neq \textbf{0} \implies \exists \textbf{B} \quad (1) $$ $$ \partial_t\textbf{B} \neq \textbf{0} \implies \exists \textbf{E} \quad (2) $$ Many physicists come to this conclusion from Maxwell's equations. Specifically, they argue that Faraday's law, $$ \nabla \times \textbf{E}(\textbf{r}, t) = -\partial_t\textbf{B}(\textbf{r},t), $$ implies (1), and that Ampere's law (with Maxwell's correction term and no currents), $$ \nabla \times \textbf{B}(\textbf{r}, t) = \partial_t \textbf{E}(\textbf{r},t), $$ implies (2). Note that we are using natural units with $c = 1$. However, these equations do not have any obvious causal connection. While we may like to pretend that _right_ implies _left_ , this is purely notational convention. Who is to say from these equations alone that one field having a nonzero curl doesn't produce a changing dual field? One attempt at reconciling this problem seems to be in Jefimenko's equations. I will state the equations without derivation, but the fields can be solved for completely in terms of the source charges and currents (I'm lazy and the following equations are in mks units from Wikipedia): $$ \textbf{E}(\textbf{r}, t) = \frac{1}{4 \pi \epsilon_0}\int [\frac{\rho(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^3} + \frac{1}{c}\frac{\partial_t \rho(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^2}] (\textbf{r} - \textbf{r}') - \frac{1}{c^2}\frac{\partial_t \textbf{J}(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^2} d^3\textbf{r}', $$ $$ \textbf{B}(\textbf{r}, t) = \frac{\mu_0}{4 \pi}\int [\frac{\textbf{J}(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^3} + \frac{1}{c}\frac{\partial_t \textbf{J}(\textbf{r}', t_r)}{|\textbf{r} - \textbf{r}'|^2}] \times (\textbf{r} - \textbf{r}' )d^3\textbf{r}' , $$ where $t_r = t - |\textbf{r} - \textbf{r}'|/c$ is the retarded time. These equations seem to imply that neither of the fields "causes" the other. Instead, Jefimenko's equations imply that only the source charges and currents generate the fields (without the presence of external charges, currents, or fields). My question is related to this approach. Is it valid? What are the arguments for and against? Is the matter settled in the classical context of electromagnetism, or are there certain subtleties I've skipped over? As an extra question, is it instead better to consider $F_{\mu \nu}$, and treat it as one object arising solely from $J^\mu = (\rho, \textbf{J})$, instead of looking at the individual fields? Thanks in advance for any and all answers!


Does the term "dark matter" apply to nonluminescent bodies which still interact electromagnetically?

On the new Astronomy.SE site, I was having a short discussion on one of my answers. The basic discrepancy was; can MACHOs like black holes/brown dwarfs/neutron stars be termed "dark matter"? My reasoning is that these objects do not radiate EM radiation on their own but they do gravitate, and thus constitute a small part of the total dark matter in the universe. I agree that there is a lot of dark matter which doesn't In other words, can the term "dark matter" be applied to nonradiating (or faintly radiating) bodies which still participate in the electromagnetic interaction (baryonic or otherwise)? Or is it necessary for all dark matter to not interact electromagnetically?


Paper airplane physics

I am working on a game involving flying and steering a paper airplane for WP7. I want the plane to fly just like how normal paper airplanes fly (see this game for an example http://armorgames.com/play/7598/flight) but I can't seem to find an equation for how paper airplanes fly. Anyone have any experience with this? In my game now, it just follows the usual motion for an object in a vacuum, which makes for some flight, but it doesn't feel perfect, and traveling at a slight downward angle makes you lose speed, which isn't right. Thoughts?


Quantum computers: are they possible or impossible?

I know quantum computers are very complicated and my question is is there any way in "Principle" to create one? Are there already quantum computers being created?


How do magnets work?

I've read a classbook on the field theory (including EM): it perfectly describes quantitive patterns in EM-theory, but I have no luck understanding how and why it works. I mean, magnetic substances are described mainly by magnetic moments of electrons, but all explanations of the phenomenon I've found are rather of deep high-level focus on fields exclusively, than on explanation why it works and what underlying mechanisms bring all those ideas to life (including explanation of what field is, except that it is an abstraction). So, the question is: may anyone try to give (or point to) a popular and thorough explanation of magnets and on low-level mechanisms, which unifies and explains how this long-distance interaction really works (i.e., not only modelled and described mathematically)? p.s.: also, I'd like to see some papers on computing magnetic properties of bodies (iron ball, for example, finite plane, NeoCube's ball chains, etc). Thanks in advance!


A question about definition of Fermi energy

Wikipedia states the definition of Fermi energy as for "a system of non- interacting fermions". If we have to assume free electrons in a solid behave this way before we are able to calculate Fermi energy, how can Pauli exclusion be justified (because electrons are non-interacting)? Can Fermi energy be similarly defined for electrons confined to a single atom?


interior surface of furnaces

For industrial furnaces, in order to improve radiation heat transfer and save energy, some people say applying a high emissivity coating on to the interior surface of a furnace will do, while some people say applying a high reflectivity coating will do. we all know that at given wavelength and for an opaque object surface, emissivity + reflectivity =1, anyone can explain to me please? With best regards Sean


Do you use the magnitude equation to get speed from an accelerometer?

A guy suggested to me that getting speed from an accelerometer required the use of this equation: $\text{speed} = \sqrt{x^2 + y^2 + z^2}$ This does not make any sense to me, all that you would get from this equation would be the magnitude in $m/s^2$ of the acceleration in the $x, y$ and $z$ axes. Am I correct? Or is my reasoning flawed?


What is the meaning of inflection points in the dispersion relation inside the first Brillouin zone?

I have a question regarding the $E$ vs $k$ curve in the first Brillouin zone. Why does the curve have an inflection point at some value of $k$ in the curve? How does it physically support it? ![E vs k](http://i.stack.imgur.com/Y4DGl.jpg)


How do you explain Kepler's third law in general terms without complex math?

I understand the first law-elliptical orbits, and the second-equal area in same time, but I need help with the third one. Note that I am not in an AP course or taking calculus at the moment so simple quadratics/cubics/CP level explanations would suffice.


What's the meaning of the age of the universe?

I'm not asking about how we worked backward from an expanding universe to the age of the big bang, but rather what is the meaning of time in a near infinitely dense point in the context of general relativity? Wouldn't time flow infinitely slowly for a theoretical (though physically impossible) observer?


Are retrograde capture orbits "easier" than prograde capture orbits?

After reading up on irregular moons in the solar system - moons that are thought to be captured, most seem to be in retrograde orbit around their parent body. That led me to wonder if retrograde orbits are easier to capture objects than prograde orbits - say prograde orbits are more likely to gravitationally slingshot the object away from the parent body before capture, whereas retrograde orbits would be more likely to capture before flinging the object away. When viewing the capture from the perspective of the parent body, an object that is moving retrograde past the body appears to slow down as it interacts with the gravity well of the body, whereas an object moving prograde past the body appears to accelerate in the same frame of reference. Is there any validity to this, or is that just a flaw in reasoning?


How much detail can telescopes actually provide?

For example, could the numbers / letters on a postage stamp in a randomly specified location be clearly visible from space. This is to settle a discussion with a friend that piqued my curiosity.


What determines a progenitor's fate as a spiral or elliptical?

I was thinking about my answer to Are the inner planets on planar orbits because there was more dust in the inner solar system (early on in planetary accretion)? \- when it occurred to me that maybe I was reversing cause and effect. Specifically, that perhaps spiral galaxies simply arise from high- angular-momentum progenitors, while elliptical arise from low angular-momentum progenitors. In this scenario, the spiral wave-pattern would merely be how that excess angular momentum organizes itself, and the matter can't help but stir itself. For a warm-up question and consistency checker, do spiral galaxies in fact have substantially greater total angular momentum than elliptical?


Curve Fitting and Multiple Experiments

Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the average of all the experiments and then do fit a single curve to that?


Light Apparent Brightness

Let's say I have a triangular light source. From that light source I want to calculate a pyramidal frustum (tetrahedron with no apex). How would I calculate the maximum bottom area where light would be visible to the naked eye? And how would I calculate the apparent brightness?


How deep is the region near an event horizon where Hawking radiation is generated?

In other words, how strong does gravity have to be to cause Hawking radiation to occur?


Why does salty water heat up quicker than pure water?

> **Possible Duplicate:**   >  Why does adding solutes to pure water lower the the specific heat? Why do higher concentration salt solutions heat up more quickly?


Can the Hubble constant be measured locally?

The Hubble constant, which roughly gauges the extent to which space is being stretched, can be determined from astronomical measurements of galactic velocities (via redshifts) and positions (via standard candles) relative to us. Recently a value of 67.80 ± 0.77 (km/s)/Mpc was published. On the scale of 1 A.U. the value is small, but not infinitesimal by any means (I did the calculation a few months ago, and I think it came out to about 10 meters / year / A.U.). So, can you conceive of a measurement of the Hubble constant that does not rely on any extra-galactic observations? I ask because, whatever the nature of the expansion described by the Hubble constant, it seems to be completely absent from sub-galactic scales. It is as though the energy of gravitational binding (planets), or for that matter electromagnetic binding (atoms) makes matter completely immune from the expansion of space. The basis for this claim is that if space were also pulling atoms apart, I would naively assume we should be able to measure this effect through modern spectroscopy. Given that we are told the majority of the universe is dark energy, responsible for accelerating the expansion, I wonder, how does this expansion manifest itself locally? Any thoughts would be appreciated.


Terminologies for moment of inertia

Perhaps someone can suggest the right terms for the following mathematical objects related to moment of inertia?   1. A _inertia tensor_ $I$. $$I \equiv \begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} \\\ I_{2,1} & I_{2,2} & I_{2,3} \\\ I_{3,1} & I_{3,2} & I_{3,3} \\\\\end{bmatrix}$$   2. A _product of inertia_ is an off-diagonal entry in the tensor: $I_{1,2} = I_{2,1}$, $I_{1,3} = I_{3,1}$, or $I_{2,3} = I_{3,2}$.   3. A _principal moment of inertia_ is a diagonal entry in the tensor: $I_{1,1}$, $I_{2,2}$, or $I_{3,3}$. This is the semantic of moment of inertia discussed in elementary treatment of Physics.   4. What is the term for $I_2$ and $I_3$ in the last line below? $$\begin{align*} I_{1,1} &= \sum_{j} m_j\;\left(r^2_{j,2} + r^2_{j,3}\right) \\\ &= \sum_{j} m_j\,r^2_{j,2} + \sum_{j} m_j\,r^2_{j,3} \\\ I_{1,1} &= I_2 + I_3 \end{align*}$$


Speed of light as a universal speed limit

It follows from special relativity that nothing can travel faster than light. Einstein believed this would have to hold so generally that he assumed the Einstein-Podolsky-Rosen paradox to indicate a contradiction in quantum physics. Nowadays we know we have to be a bit more specific, and maybe it would be safe to say that nothing that can carry information can travel faster than the speed of light. If information can be transferred faster than light, it would be possible to change the past. If I understand correctly it is the contradictions arising from the hypothetical possibility to influence the past that lead to the assertion that nothing can travel faster than light. My questions:   1. Is this indeed the reason why it is said to follow from special relativity that nothing can travel faster than the speed of light?   2. Aren't the contradictions associated to influencing the past more of a philosophical nature than of a purely logical (or at least physical) nature? **EDIT** I realize now that instantaneous transfer (action at a distance) is actually physically in contradiction with special relativity: what is instantaneous/simultaneous in one inertial frame is not in the other, so the laws of physics would not be the same. I suspect that for finite velocities faster than that of light there must be a similar argument, I would appreciate if anyone could elaborate on that.


Can you clean the air in a small room using a vacuum cleaner?

You can use a vacuum cleaner to vacuum the floor but if you tried to vacuum the the air in a small room would this work or will the laws of physics stop this from working in any way? In what way would attempting to vacuum clean the air be different from using an air scrubber(in terms of physics). Thanks


Using Ampere's Law for a Solenoid

![solenoid](http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/imgmag/sol.gif) To calculate the magnetic field, a rectangle amperian loop was drawn, and since the sides of the rectangle are perpendicular to the magnetic field, and the top is too far away to have any field lines cross it, only BL makes a contribution. ![equation](http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/imgmag/sol2.gif) I don't understand why this magnetic field is only contained within the solenoid. Shouldn't the magnetic field at any point on the amperian loop equal μnI, according to Ampere's law?


Is this webcomic accurate?

I was considering this xkcd comic from 5/10/14, with the alt-text "Trains rotate the Earth around various axes while elevators shift its position in space." I'm wondering about its accuracy. ![http://xkcd.com/1366/](http://i.stack.imgur.com/EvUsH.png) I first considered the elevator scenario. As the elevator accelerates (to start or stop its motion), the occupants feel a fictitious force, demonstrating conclusively that it is the elevator and not the planet which is in motion. By this argument, an elevator is fundamentally different from a device which physically displaces the planet. Similarly, I suspect that a train doesn't "rotate the Earth," since such a rotation would be felt by occupants on Earth as an angular or centripetal acceleration. Any observer can conclude definitively that it is the train which is in motion around the Earth, not the Earth rotating under the train. For this reason, is the comic is not strictly correct? (Note that there is a minimal motion of the Earth resulting from the action- reaction force pair between the train and the planet, but it's safe to say that this motion is negligible relative to the distance traveled by the train, since $I_{earth} \gg I_{train}$.)


Is it possible to hear the past?

From this Stack Exchange Physics Post, I am certain that it is possible to view the past. But then this interesting question came to me. Is it possible to hear the past? Ok, you might say, "Well, we are hearing the past, aren't we?" and go onto immense details about the speed of sound against the difference and stuff. But what if I wanted to hear the sound of a place extremely close by days, weeks, months or even years back? Would that be possible?


Origin of Electromagnetic Interactions between Molecules

What is the origin of electromagnetic interaction between molecules? Anyway, it should have some relation with atoms. Also, These electromagnetic interactions are playing a major role in different properties of matter including the transition between solid-liquid-gas. Hence, what would be the **source** of these interactions..? If these interactions originate from atoms, then some other questions come into **focus** :   * Are they related to the transition of electrons between various energy levels and emission of photons from an atom..?   * Are they related to cohesive forces between molecules in solids & liquids?


Do really, two oppositely charged bodies (of equal charge in magnitude) attract each other?

It is well known fundamental behaviour that, _oppositely charged bodies attract each other_ (I don't know whether it applies also for charges of equal magnitude or not), and identical charges repel each other. It is also well known that, a system of oppositely charged bodies with equal charge in magnitude, has zero net charge. If a system has oppositely charged bodies, with equal charge in magnitude, it would imply that, there will be zero net charge, and the vector field's had got cancelled each other, like around a current carrying conductor, there is no electric field because, charge on current carrying conductor is zero, as one electron enters the conductor, the other will be leaving the conductor. Now, if there is no field, how could the oppositely charged bodies (of equal charge in magnitude) attract each other, they should not feel any force, isn't? Taking into account all the above statements, would it imply that, two oppositely charged bodies (of equal charge in magnitude) attract each other? _[All statements made, are up to my view. Any correction advisory is welcome]_ _LINKS_   * What are the fields produced around a current carrying conductor? (Provides an idea of whether, there exists a electric field around a neutral body or not, like the current carrying conductor)


equivalent spring constant for a drum floating in water

If I have a drum floating in water, how do I find the equivalent spring constant? I know that the water has density $\rho$, the drum has diameter $d$ and height $\ell$, and the positive direction is in the downward direction. Therefore, the buoyancy pushing up on the barrel is my spring. I have been reading the section of this topic in Mechanical Vibrations by Rao but there isn't any information that helps me figure this out.


> Quantum mechanics (QM; also known as quantum physics, or quantum theory) is > a branch of physics which deals with physical phenomena at nanoscopic scales > where the action is on the order of the Planck constant. -wiki Time is much less finite than matter. Matter can be split to the point where it is so small that it gains the classification of nanoscopic and subatomic. However the Planck constant is a measurement of mass and time so I find it difficult to separate the measurements. What measurement of time is so small that it qualifies as quantum? Could someone clear up the misconception?