Can black holes form in a finite amount of time?

One thing I know about black holes is that an object gets closer to the event horizon, gravitation time dilation make it move more slower from an outside perspective, so that it looks like it take an infinite amount of time for the object to reach the event horizon. It seems like a similar process should slow the formation of the black hole itself: As the star collapses, its gravitational time dilation make itself collapse more slowly. This make me wonder, are what astronomers claim to be black holes really black holes, or are they stars that progressively make themselves more similar to one without actually reaching the stage of having an event horizon? EDIT: Contemplating one answer, I realize the question is ambiguous. What does finite time mean in general relativity. Here is a less ambiguous question: Is there a connected solution of 3+1 dimensional general relativity with one space-like slice not have a singularity, and another space-like slice having one.


Specific energy and specific angular momentum of photon

In this PDF [1], is made reference to specific energy and angular momentum of a particle. If the particle has no mass, like a photon, how should I define these terms in the equations further down for the path of the particles? [1] Lecture XIX, Christopher M. Hirata, Caltech M/C 350-17


Relativistic corrections to quantum mechanics of Coloumb potential

Systems of charged particles (such as atomic nuclei and electrons) can be described by nonrelativistic quantum mechanics with the Coloumb interaction potential. A fully relativistic description is given by quantum electrodynamics which is much more complex. Is it possible to expand various quantities in QED as power series in 1/c i.e. around the nonrelativistic approximation? Examples of relevant quantities are:   * Ground state energy of a given set of charged particles   * Excited state energies   * Scattering cross sections of charged particles & their bound states (assuming we trace over the photons in the final state)


Using quantum entanglement to send messages back to the past

> **Possible Duplicate:**   >  Entanglement in time I heard that there is an experiment that uses quantum entanglement to try to send messages back to the past. I am having a hard time understanding how such experiments would work theoretically. Can anyone offer me some insights toward these types of experiment?


How do I combine three resistors which are in parallel with each other?

For this circuit (a and b are connected by a battery), ![circuit diagram](http://i.stack.imgur.com/CVtE3.png) Will I be able to find the total resistance of the circuit by adding resistors that are in series and combining resistors that are in parallel without using the method of Kirchhoff's Voltage and Current Law? I tried adding R3+R4 and this forms one resistor, which is in parallel with R2 and R1. And R2 is in parallel with R1 and R3, and R1 is in parallel with R2 and R3. This can be verified by Kirchhoff's Voltage laws. The problem is, I get stuck here. I know that if R1,R2 and R3+R4 can be combined into one resistor, then parallel-series would solve the problem neatly. But I can't see how these resistors be combined.


weak interaction coupling constant

in wikipedia, the weak interaction coupling constant is said to be 10^-13 times weaker than that of the strong interaction but hyperphysics (http://hyperphysics.phy-astr.gsu.edu/hbase/forces/couple.html) says it's 10^-6. It seems wikipedia is right but why many other resources also provide a value similar to that of hyperphysics?


Determine Charge With Electroscope?

You have three separate glass rods, and you know one is positive, one is negative and one is neutral. You also have an electroscope that is positively charged. How can you determine which rod is positive, negative and neutral? I think you will be able to determine which rod is negatively charged, as that rod will cause the electroscope leaf to rise. I'm not sure how to differentiate between the positive+neutral though, as both will cause the leaf to collapse. Could you place it next to the negative rod and see which repels? Essentially it would be great if somebody could explain the actual physics behind the situation. Why will the negative rod cause the leaf to rise and vice versa? Thanks!


What is being deprived when a photon is being watched in double slit experiment?

How are photons being watched in the double slit experiment? What exactly does being observed mean, as it is obviously changes the state of the photon somehow - it must be depriving the photon of something or emitting something that interacts with the photon.


Photons in a gravitational field

I have been really staring for a while in a MP-Beiser book and I totally disagree with a statement he does there. On a page 85 he states that photons act as they have a mass $m$. He derives this by stating that: $$ \begin{split} p &= m v\\\ \frac{h\nu}{c} &= m c\\\ m&= \frac{h \nu}{c^2} \end{split} $$ But I totally disagree with this. We have learned and derived that momentum of a particle is: $$ \begin{split} p &= m v \gamma (v)\\\ \frac{h \nu}{c} &= m c \gamma{(c)}\\\ m &= \frac{h \nu}{c^2 \underbrace{\gamma(c)}_{=0}}\\\ m &= 0 \end{split} $$ Something here is totally wrong, but what? How can an author state what he does? I know that on Harvard they did an experiment resulting in different $\nu$ of a photons falling in a gravitational field, but they must have been wrong or something... Please someone explain.


Using the covariant derivative to find force between 't Hooft-Polyakov magnetic monopoles

I am reading this research paper authored by NS Manton on the Force between 't Hooft-Polyakov monopoles. I have a doubt in equation 3.6 and 3.7. We assume the gauge field for a slowly accelerating monopole to be $A_0 = \epsilon^2 a_i t A_1$, where $\epsilon^2$ is an infinitesimal. Also, we write $\partial_0 \phi = -\epsilon^2 a_i t \partial_i \phi$. Using this he writes $D_0\phi=-\epsilon^2 a_i t D_i \phi$, where $D_i\phi=\partial_i \phi + [A_i,\phi]$. Isnt the sign of the second term wrong? Secondly, he says differentiation wrt t gives us, $D^0 D_0 \phi = \epsilon^2 a_i D_i \phi$. Shouldnt it be $\partial^0D_0 \phi$? Cause we are taking the actual derivative wrt t rather than the covariant derivative, WE should get some extra terms, do they cancel out? How does the minus sign disappear? Does the covariant derivative behave like a normal derivative in any case?


English translation of Helmholtz' paper: “On the Physical Significance of the Principle of Least Action”

I am asking about an English translation of a Helmholtz paper: > Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung. > _Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume > 100, Issue 2, 1887, Pages 137-166, and Volume 100, Issue 3, 1887, Pages > 213-222._ http://www.degruyter.com/view/j/crll.1887.issue-100/crll.1887.100.137/crll.1887.100.137.xml?format=INT. (Also see link and link.) The title in English: _On the Physical Significance of the Principle of Least Action._ Has it ever been translated (to English)?


For the long-term evolution of atmosphere/orbit, when is perihelion more important than mean distance?

When we want to figure out the long-term evolution of a planet's atmosphere/orbit, when is perihelion more important than mean distance? E.g. some processes (like Jeans Escape and escape of atmospheres) are disproportionately affected during perihelion (point of closest approach) rather than during aphelion.


Are the protons and neutrons in the nucleus arranged in any particular way?

I was wondering this: suppose you have two oxygen atoms. They will both have 8 protons and 8 neutrons in the nucleus (at least if they are the most common isotope). Now, will all those particles be arranged in the same way in both atoms? If they are, why would that be, and if not, does that affect the element's properties in any way? But then I also thought that maybe the uncertainty principle doesn't let us even ask this question. Maybe you can't tell the particles' positions so accurately, so all you can say is that you have 8 protons and 8 neutrons all together in a small space. So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?


Geometric structure of atomic nucleus?

The electrons in an atom have a certain configuration and we can predict in which space the electrons can be present of a certain energy level (s-orbitals have a sphere-shape,....). Recently I read that there are magic numbers for nuclei: special configurations of the nuclei that makes them more stable (the magic numbers are 2, 8, 20, 28, 34, 50, 82, and 126) - so there are definitely shells in a nucleus. I was wondering : do the neutrons and protons form certain geometric structures in these sub-shells of the nucleus? Or are they all stacked closely on each other?


The difference between the Wannier function and atomic orbit in a tight binding model

In a tight binding model, we usually start from the atomic orbits and linearly combine them to get the wave function of the crystal energy band. My questions are:   1. Since this kind of tight binding is an approximate method due to using atomic orbits, is it exact to use the Wannier function formalism? If so, how do I get the Wannier function systematically?   2. What is the use of maximally localized Wannier functions?   3. Why can't we get the maximum localized Wannier function when the Berry phase is not zero?   4. Also, in tight binding, formalism, taking atomic orbits or Wannier functions as the basis function, respectively, what does position operator (diagonal or not), velocity operator and angular momentum operator look like?


Does gravity affects temperature reading of a mercury thermometer?

I remember when I was in primary school, the science teacher put me in charge of a mercury thermometer. I do not quite understand the mechanics behind except that mercury expands when it is hot and contracts when it is cold, and that this could be read off a temperature scale along the stem of the thermometer. At the back of my mind, there is this doubt on how gravity affects the movement. Having seen most thermometers being hanged vertically on a wall, I did likewise, placing the thermometer on the floor, upright, leaning against the window sill beside my desk. This is fine until one day when a strong gust of wind blew and cause the thermometer to fall flat on the floor, smashing the glass casing and causing the mercury to leak out resulting in the evacuation of the whole class. After that incident, I learnt my lesson and place the new thermometer lying flat instead of upright. But, the lesson that I didn't learn is how gravity affects the behaviour of such thermometer. Hope to learn something here.


Solar sail area going to Proxima Centari

I have a physics question that I need some help with: "Proxima Centauri is a star in the Alpha Centauri solar system, it’s the nearest star to our sun (4.24light−years) http://en.wikipedia.org/wiki/Listofneareststars. How large of a solar sail would be needed to accelerate a solar sail of mass m to a velocity which will get the ship to Proxima Centauri in two lifetimes?" I'm assuming that the time is 160 years for 80 years each lifetime. Can anyone help?


Electric Flux Density - Ring Charge

A ring placed along $y^{2}$ + $z^{2}$ = 4, x = 0 carries a uniform charge of 5 $\mu$C/m. Find D at P(3,0,0) Should I be using Gauss's Law to solve this problem? I was considering using a spherical Gaussian Surface, and then using the formula D = $\epsilon_0 $E to find D, but I'm not sure how to set up my integral.


Some ambiguous points on Spontaneous Symmetry Breaking (SSB)?

Almost in every textbook of condensed matter physics, the standard description of SSB could be formulated as follows: Consider the lattice Heisenberg model in an external magnetic field $H=\sum_{ij}J_{ij}\mathbf{S}_i\cdot\mathbf{S}_j+hS_z$, where $h$ is the magnitude of magnetic field and $S_z=\sum_iS_i^z$. Now the average magnetization per site is a function of both magnetic field $h$ and number of lattice sites $N$, say $m\equiv \sum_i\left \langle S_i^z \right \rangle/N=m(N,h)$, where $\left \langle S_i^z \right \rangle\equiv tr(\hat{\rho }S_i^z)$ with $\hat{\rho }=e^{-\beta H}/tr(e^{-\beta H})$ the density operator. Then if $$\lim_{h\rightarrow 0}\lim_{N\rightarrow \infty }m(N,h)\neq 0$$, we say the system has SSB at temperature $T$. Now I get some questions: (1)We know at finite $N$ and zero $h$, $m(N,h=0)=0$ due to spin-rotation symmetry. But **there is no reason** for that $$\lim_{h\rightarrow 0}m(N,h)=m(N,h=0)—[1]$$, right? Since the function $m(N,h)$ may _not be continuous_ at $h=0$, from the math viewpoint. (2)If Eq.[1] is correct, and hence $\lim_{h\rightarrow 0}m(N,h)=0$, then $\lim_{N\rightarrow \infty }\lim_{h\rightarrow 0}m(N,h)=0$, right? (3)If Eq.[1] is wrong, say $\lim_{h\rightarrow 0}m(N,h)\neq m(N,h=0)$ and hence $\lim_{h\rightarrow 0}m(N,h)\neq0$, then what about $$\lim_{N\rightarrow \infty }\lim_{h\rightarrow 0}m(N,h)?$$ And why don't we use this identity to define SSB? Thank you very much.


How do alpha and beta particles ionise surrounding particles?

I've been wondering about this question for a while. If you have alpha and beta particles released from a radioactive core, how do they ionise surrounding particles?


Wikipedia's derivation of torque related to angular acceleration

Wikipedia derivation of the relationship between a torque and an angular acceleration is given here. Could someone help me to see how the following: $$\vec{\tau} = \left(-\sum^n_{i=1}m_i [\Delta r_i]^2\right) \vec{\alpha} + \vec{\omega} \times \left(-\sum^n_{i=1}m_i [\Delta r_i]^2\right) \vec{\omega}$$ is obtained from the following: $$\vec{\tau} = \sum^n_{i=n} (\vec{r}_i - \vec{R}) \times (m_i \vec{a}_i)$$ by using Jacobi identity, please? My attempt at the derivation is as follows: $$\vec{\tau} = \sum^n_{i=n} (\vec{r}_i - \vec{R}) \times (m_i \vec{a}_i) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} \vec{\Delta r_i} \times (m_i \vec{a}_i) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times \vec{a}_i) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i} + \vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})))\;\ldots\;\vec{R}\text{ is centroid} \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + (\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})))\;\ldots\text{ cross-product distributivity over addition} \\\ $$ Then, I try the Jacobi identity on the second term as follows: $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) \+ (\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\Delta r_i} \times \vec{\omega}) = \vec{0}$$ The last term on the LHS is $\vec{0}$ because $$(\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\Delta r_i} \times \vec{\omega}) \\\ = (\vec{\omega} \times \vec{\Delta r_i}) \times -(\vec{\omega} \times \vec{\Delta r_i}) \\\ = -[(\vec{\omega} \times \vec{\Delta r_i}) \times (\vec{\omega} \times \vec{\Delta r_i})] \\\ = \vec{0}$$ So, $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) = \vec{0}$$ That is different from the one stated in the Wikipedia. However, I try to continue with my own finding as follows: $$\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) + \vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) = \vec{0} \\\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = -[\vec{\omega} \times ((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i})] \\\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = \vec{\omega} \times -((\vec{\omega} \times \vec{\Delta r_i}) \times \vec{\Delta r_i}) \\\ \vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i})) = \vec{\omega} \times (\vec{\Delta r_i} \times (\vec{\omega} \times \vec{\Delta r_i}))$$ to have the following: $$\vec{\tau} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + (\vec{\Delta r_i} \times (\vec{\omega} \times (\vec{\omega} \times \vec{\Delta r_i}))) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i (\vec{\Delta r_i} \times (\vec{\alpha} \times \vec{\Delta r_i}) + \vec{\omega} \times (\vec{\Delta r_i} \times (\vec{\omega} \times \vec{\Delta r_i}))) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + \vec{\omega} \times ([\Delta r_i]^2 \vec{\omega} - (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i})) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + \vec{\omega} \times [\Delta r_i]^2 \vec{\omega} - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i}) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} + (0) - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i}) \\\ \hphantom{\vec{\tau}} = \sum^n_{i=n} m_i ([\Delta r_i]^2 \vec{\alpha} - (\vec{\Delta r_i} \cdot \vec{\alpha}) \vec{\Delta r_i} - \vec{\omega} \times (\vec{\Delta r_i} \cdot \vec{\omega}) \vec{\Delta r_i})$$ Then, I get stuck there. If you want to help, please show the complete derivation instead of just hinting here and there unless you have the complete derivation already in your mind.


Are Neutron stars transparent?

Neutrons have no charge so they would not, I think, interact with photons. Would a neutron star be transparent?


How do you actually define an orbit?

How do you actually define an orbit? I believe, Newtonian Mechanics describes an orbit as one object in free fall around another where projectile paths become elliptical. I think, Einstein describes an orbit as an object taking the shortest distance through curved space. And in Quantum Mechanics, orbits are quantized orbitals or states. Is there a definition for orbits, where all these characteristics are true?


Where does the "g" force that pilots experience come from?

I understand that it has to do with acceleration. Say a pilot does a quick maneuver and experiences a force of 5g. What exactly is happening here? And what is this force relative to? If someone can show an example with some calculations that would be really helpful. Thank you


Is the solar energy Infinite?

Is the solar energy coming from the sun infinite and will continue to be radiated to our earth forever? (discarding any outer factors) what's the sun's fuel?


How does time dilation work in this situation?

It is my understanding that when moving near the speed of light, time slows down relative to other things not moving so fast. Based on this principle, would it be theoretically possible to travel a thousand light years in a year, with the thousand years only having passed on the place you're moving relative to? If something is wrong with my premises or question, that knowledge is welcomed, too. (I am still a bit shaky on the whole subject of time-dilation because I don't fully understand how it's possible without a privileged reference frame.) Understanding is the prime goal, here.


Oil drop experiment and quantization of charge

How to systematically show that the resulting charges in oil drop experiment are integers multiplied by $e$ in other word how to extract $e$ from the data?


What are qubits made of in Wen's string-net theory?

In Prof. Xiaogang Wen's theory, photons and electrons are described as quasi- particles appeared as a result of the existence of the string-net liquid, which is the topological order of the qubits that form the space. This is a little confusing to me. Consider when we talk about other quasi-particles, for example, phonons, we say they are excitons appear from the oscillation of the atoms that form the solid, which means that 'atoms' are 'real' while phonons are 'quasi-particles'. But when we make an analogy between photon and phonon, (topological order with lattice model, qubit with atoms), what are those qubit in essence? Are they some kind of ultimate thing that build up our world?


Why Can We See Stars In The Sky

Why is it when we look up into the night sky we can see stars. but when you see pictures taken from the ISS you don't see any stars. Why is this?


Wick Order and Radial Ordering in CFT

I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case): $$\cal{R}j(z_1)j(z_2)~=~\frac{1}{(z_1-z_2)^2}~+~:j(z_1)j(z_2):$$ with $j$ defined as $$j(z)~=~\sum_k \alpha_k z^{-k-1}.$$ My question is should I start the calculation form the Wick ordered term and make the two others appear, because starting from the left side, I don't see how could I develop some calculus?


How good is current tsunami prediction?

We all know that predicting tsunami and earthquake is difficult, with too many variables involved. But with the advent in data collection and computing power and better models, one should be able to predict tsunami better than in the past. How accurate is current tsunami prediction?


Solar wind and the Earth's magnetic field

I have again an old question from a comprehensive exam I took a couple of months ago. Lucky for me one could pick 5 out of 8 questions, because on some of the problems I didn't even know how to start. Now that classes are over I've now the time to revisit those problems I was dumbfounded by, such as this one: (Abriged version) > Life on earth would be impossible if we were constantly exposed to charged > solar particles. Luckily, earth's magnetic field protects us from them. The > solar particles have a typical energy spectrum of $d\Phi / dE \propto > E^{-3}$ particles/$m^2/s/J$. What is the _minimal_ field strength of the > earth's magnetic field based on the anthropic principle, i.e., it couldn't > be weaker or else we wouldn't live to observe it. Well, this quantity $d\Phi/dE$ looks like a flux, so I guess the general setting is that of a scattering problem. But first, $d\Phi / dE$ isn't given completely, only a rough form of its energy dependence. And second, I'm not sure what a reasonably simple model for this entire process would be. Easiest in terms of calculation would be to assume some sort of homogeneous magnetic field aligned with the earth's magnetic axis, because I guess it's a pain to calculate the path of a particle in a dipole field... Maybe the idea of this is to calculate the total cross section of earth's magnetic field and then demand that it should "cover" the earth? Or they want me to solve an equation of motion for incoming solar particles and show that all of them are deflected? My problem right now is that I don't even know how to interpret the $d\Phi/dE$ quantity whose energy dependence I'm given. I guess it makes more sense to someone with a background in elementary particle physics? Right now I'm trying to write a vector potential $\vec{A} = \mu_o/(4\pi r^2) \vec{m} \cdot \vec{e}_r$ where $\vec{e}_r$ is the unit vector in $r$-direction in spherical coordinates, and then try to get equations of motion from the Hamiltonian $$H = \frac{(\vec{p} + q\vec{A})^2}{2m}$$ but I am not sure if I'll be able to solve whatever comes out of that, or if I'm completely on the wrong track with this. * * * EDIT * * * ![Schematic of Deflection of Solar Particles](http://i.stack.imgur.com/jbZDA.png) Image taken from here Maybe it helps trying to understand this schematic, but I cannot easily see how the Lorentz force would create such a trajectory. * * * ANOTHER EDIT * * * From further searching, I know suspect that this has something to do with how a plasma current (the charged particles) interact with a magnetic field. That would mean that I have to calculate the radius of the ensuing magnetosphere and then demand, via the anthropic principle, that it should be at least of the same size (or larger) as the radius of earth. So the Lorentz force would probably not directly have anything to do with it. But I also have no training in plasma physics. (Some of the problems in the exam were specifically geared towards Astronomy students, so I guess they'd find it a breeze).


What is your prefered toolkit / program for 3D visualisations of data?

If you recorded data which represents some physical value in space (e.g. electron density) and you need to explore this dynamically in 3d (say you have a isosurface and you change the value it represents) what toolkit/program would you use? Personally I used MayaVi and it has all the all the features I could need. Unfortunately it is sometimes unstable and often slow. So in principle I am looking for equally powerfull alternatives. Edit: I know this should probably not be in the physics SE but I think the answer to my question could be relevant for many physicists.


Changing the Half-Life of Radioactive Substances

Is there a way to extend or reduce the half-life of a radioactive object? Perhaps by subjecting it to more radiation or some other method.


What causes a gyroscope to eventually rotate/fall over?

Hey so I've just learned about angular velocity and momentum and how torque changes it. Looking at a wheel spinning around an axis, with one end being held up by a rope, what causes the wheel to rotate downwards over time, and eventually fall?


Why use lasers for intense, localized heat instead of some other light source?

Lasers are used in various industrial processes that need intense, localised, heat (3d printers and laser cutters come to mind). My question is: why use lasers? There are many other (cheaper, brighter) light sources. There are even other monochromatic and coherent light sources (LEDs and mercury vapour lamps respectively), and this video shows someone sintering desert sand using a Fresnel lens and sunlight, which is of course neither monochromatic nor coherent. So, what is it about lasers that make them so better than a conventional light source combined with appropriate focusing elements?


Neutral quantum particle in inhomegeneous magnetic field

I'm trying to understand Stern-Gerlach experiment on a computational level. Suppose we have a neutral particle with magnetic moment (e.g. a neutron), and apply an inhomogeneous magnetic field to it (let it change linearly with coordinate). As I understand, its Hamiltonian would look like: $$\hat H=-\frac{\hbar^2}{2m}\nabla^2+\left(\frac e{mc}\right)\hat{\vec s}\vec B$$ Now the spin operator is $$\hat s_i=\frac{\hbar}2\sigma_i,$$ where $\sigma_i$ is $i$th Pauli matrix. So, for magnetic field $\vec B=\vec e_x B_0 x$ we'd have Schrödinger 1D (Y and Z directions can be separated due to translation symmetry) equation: $$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+\left(\frac {\hbar e}{2mc}\right)\sigma_x B_0 x\psi=i\hbar \frac{\partial\psi}{\partial t}.$$ I now try to solve this equation numerically, taking initial wave function in the following form: $$\psi(x,t=0)=\begin{pmatrix}\psi_0(x)\\\ \psi_0(x)\end{pmatrix},$$ where $\psi_0(x)$ is a gaussian wave packet with zero average momentum. The problems start when I select $\sigma_x$ as is usually given: $$\sigma_x=\begin{pmatrix}0&1\\\1&0\end{pmatrix}.$$ The solution appears to look like showed below. I.e. both wave function components accelerate left! ![enter image description here](http://i.stack.imgur.com/Rrbsk.gif) I thought, what if I choose another axis as $x$, so I tried doing the same with $\sigma_y$: $$\sigma_y=\begin{pmatrix}0&-i\\\i&0\end{pmatrix}.$$ The result in the animation below. Now it's a bit better: the wavefunction at least splits into two parts, one going left, another right. But still, both parts are composed of a mix of spin-up and spin-down states, so not really what one would expect from Stern-Gerlach experiment. ![enter image description here](http://i.stack.imgur.com/8hyNA.gif) Finally, I tried the last option — using $\sigma_z$: $$\sigma_z=\begin{pmatrix}1&0\\\0&-1\end{pmatrix}.$$ The result is again showed below. Finally, I get the splitting into "independent" spin parts, i.e. one spin part goes left, another one goes right. ![enter image description here](http://i.stack.imgur.com/KPe1s.gif) **Now, the question** : how to interpret these results? Why does choice of active axis result in such drastic differences in results? How should I have done instead to get meaningful results? Shouldn't permutation of Pauli matrices not affect results?


Integrating the gauge covariant derivative by parts

I was watching a set of lectures on effective field theory and the lecturer said that you can always integrate the covariant derivative by parts due to gauge symmetry. For example, if I understand correctly, we can write: \begin{equation} \int {\cal D} \phi \exp \left\\{ i \int d ^4 x D _\mu \phi D ^\mu \phi + \, ...\right\\} = \int {\cal D} \phi \exp \left\\{ i \int d ^4 x - \phi D ^2 \phi + \,...\right\\} \end{equation} where $D_\mu = \partial_\mu + i g T ^a G_{a, \mu} $. This would be obvious if we didn't have the gauge boson contribution but why does integration by parts hold for the covariant derivative?


Can heat be transfered via magnetic field in a vacuum?

Say you want to store hot coffee in a container surrounded by a vacuum. To remove all sources of conductive energy loss the container is suspended in the vacuum by a magnetic field and does not have a physical connection to the sides of the vacuum chamber, My question is would the magnetic field be a path for energy to be conducted out of the suspended container? Another way to look at this question would be two magnets are suspended in a vacuum with their poles aligned. A heat source is attached to one of the magnets. Would the second magnet show a corresponding increase in temperature, excluding radiated heat transfer?


Relativistic Transverse Doppler Effect

In Minkowski spacetime, two observers, A and B, are moving at uniform speeds u and v, respectively, along different trajectories, each parallel to the y-axis of some inertial frame S. Observer A emits a photon with frequency $\nu_{A}$ that travels in the x-direction in S and is received by observer B with frequency $\nu_B$. Show that the Doppler shift $\nu_B/\nu_A$ in the photon frequency is independent of whether A and B are travelling in the same direction or opposite directions. Relevant equations: $$\lambda/\lambda' = \nu_B/\nu_A = \gamma(1-\beta\cos\theta)$$ Aberration formula: $\cos\theta' = (\cos\theta - \beta)/(1-\beta\cos\theta) = -\beta$   (for transverse case) The answer is apparently that the Doppler shift is independent of the relative direction of motion. I have tried to transform to the frame S' where B is stationary, finding the velocity of A using the addition of velocities formula - to then get gamma. I have used the abberation formula to insert $\cos\theta'$ into the Doppler shift formula above to get $\nu_B/\nu_A = \gamma(1+\beta^2)$. Plugging in the velocity of the emitter A in frame S' doesn't seem to get the required result.


Sequence of E and B field in radio waves and in single photons

In antenna technology we distinguish between nearfield and widefield. In the nearfield the electric and the magnetic fields are shifted by 90°. If you look closer you can see that there are two possibilities of this shift, 90° and -90°. To explain it you have to remember what this 90° means. Let (in vacuum) the coordinate systems X axis be parallel to the E field, let the Y axis be parallel to the B field and the Z axis is parallel to c * t . In Z equal zero let the E field be maximum and directed in the X direction. The B field is zero. 90° later (in terms of E = E(max) * cos α and B = B(max) * sin α) and this is a quarter of the wavelength the B field can be directed to the left or to the right. And this is natural because B = B(max) * - cos α is the second possible state of the nearfield radio waves. Regardless of the approach to see radio waves as one electromagnetic wave (statistical method) it is obvious that all radio waves are made from photons which are emitted during the electrons acceleration in the antenna rod. My question is, do these photons all have the same sequence of the E and B fields? The same question appears for quantum dots which produce single photons. Edit: There has to be a right or left hand rule because if half the photons have B-field with 90° to E and half have -90° then there wouldn't be magnetic field at all. Update: I get it. It's the right hand grip rule (conventional direction of current) because there is no principal difference to a straight wire.


Does this count as moving faster than light?

I'm not familiar with any complicated physics equation, however I do understand some basics. Suppose there is two objects, both of them are moving away from each other in a 3-dimensional space, which they both have the speed of half the speed of light ($c/2$). Relative to the reference frame of object A, object B would be moving away from itself equal to the speed of light.                  A          B     <-----      ----->       -c/2       c/2   to a 3rd observer        0          c    to A       -c          0    to B      This seems to break the law of "nothing can move faster than the speed of light". I believe that there is some other explanation for this case. I have done some research, and there are quite a few questions on this topic. Take this one and this one for example. However, they are about adding velocities together, which isn't quite the same as in the case I was describing. This question is also similar to my case, however the two objects described are both moving toward the same direction, and I'm not shooting any beam or laser toward the other object. Most probably in reality there are some extremely complex laws and equations which makes this question more complicated. However, the two objects are not interacting with each other and I'm not trying to "detect" the speed of the other object in the reference frame of A or B. Also none of the objects are moving at a significant fraction of the speed of light relative to the third observer, so special relativity doesn't apply(?) I just could not think of any reason why wouldn't B be moving away from A in the speed faster than light. A simple explanation would be great.


Would Einstein have accepted the presumptions that lead to the Bell inequality?

To check the correlation between Hidden Variable Theory and Quantum Mechanics, Bell calculated the expectation value $<\sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)> = \int d^n V \rho(\vec V) \sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)$ Here I am assuming that "Alice" is measuring the spin of an electron e along $\vec a$ and "Bob" is measuring the spin of the positron p along $\vec b$. Then $\sigma_{e}(\vec a,\vec V)$ and $\sigma_{p}(\vec b,\vec V)$ are the resulting spin values ($\pm \frac{1}{2}$) of the electron and positron, respectively. The vector $\vec V$ is an n-dimensional vector containing the hidden variables and $\rho(\vec V)$ is a probability distribution for the hidden variables. But does this not assume QM is probabilistic? I thought Einstein disagreed with the probabilistc nature of Quantum Mechanics (God does not throw dice).


Motion of charge in magnetic field with drag force

Say you have a charged particle in a region that contains a fluid that will produce a drag force that goes as $F=-kv$ where $v$ is the speed and $k$ is some constant. The region also contains a uniform magnetic field. Suppose you give the particle some initial velocity $v_0$ in the plane perpendicular to the magnetic field. What will be the particle's subsequent motion? Please provide semi-quantitative answers. Note that this is not a homework question.


How close can spaceship get to the Sun

If you want to fly a spaceship with human passengers as close to the Sun as possible, then what effects would the spaceship have to be designed to counteract in order to keep the passengers alive and how close to the Sun could you get before there would be no way to counteract the effects ?


If you run faster than speed of light what happens with your shadow?

If you are running faster than the speed of light, and behind you have an a object projecting a burst of light what Happens with your shadow? If you are running slower than the light, when the light touch you, its generate a shadow of the opposite side, right? So, when you are running faster than the light, your shadow runs counter the light "clashing" with her? **EDIT:** I know that is impossible to move's faster than the light in vacuum, it is logically a assumption. Sorry my bad English, please correct the errors.


Equation of a flying kite

My question is the following: _What is the shape of the rope which holds a kite flying?_ (Steady state.) I am not a physicist (I am a mathematician), so I can not work the physics part of the question. My guess is that it should be the same as catenary (hyperbolic cosine)


Computationally solving bodies that push and pull

Is there a way to find a solution for the positions (as a function of time) of multiple free bodies that push and pull on each other? Say for instance I have a collection of cells which can individually contract or expand, and are joined to one another. For simplicity, I'm assuming two dimensions; no gravity, friction, or other external forces; and the cells are circular, and therefore connect at a point. When a cell contracts or expands, it simply reduces or increases its radius, respectively. When it contracts, it pulls on any cells that are attached to it. When it expands, it pushes on any attached cells, and in both cases it does so with a particular force. The mass of each cell is known. I suspect this may be related to an N-body problem (perhaps even more difficult) and therefore cannot be feasibly solved exactly. If that's the case, can anyone suggest any approach for reasonably approximating this system?


Inward Pressure of Gravity

We are looking at White Dwarfs in Quantum and specifically how they are stabilized between gravitation pressure inwards balanced by the Fermi pressure due to electrons. > To start the problem off we are asked to calculate the inward pressure due > to gravity in terms of the total gravitational energy. We are examining a > white dwarf with the mass of our Sun (ignoring where this is > probable/possible). If we take the total gravitation energy to be $U_g$ then I figured $U_g=PV$ and that the inward pressure due to gravity would be $P=\frac{U_g}{V}$ where $V={4 \over 3}\pi r_{sun}^3$. However when I use this later on to calculate the radius of a white dwarf with mass equal to our sun I get an obscenely small number. While I understand the densities of white dwarfs are enormous I think I made a mistake in the inward pressure due to gravity part. Should it be that $V={4\over 3}\pi K f^3$ and then...? Not sure where to go or what I did wrong. Ideas?


Gravitational acceleration on the Moon and Mars

There are plenty of formulas that use gravity acceleration of Earth. This is represented with the symbol $g$. In my school work (I am a high school student) we usually take it as $g= 9,8 \,\text m/\text s^2$. This thing is obviously a number that is only usable on Earth. What I want to know is that, what if I want to make my calculations according to another planet? How the number is going to change?


At the instant of release of an object from rest. Is the only force that can act its weight?

Q3 from a mechanics exam past paper:![enter image description here](http://i.stack.imgur.com/95nD6.png) I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or torque, $I$ is the moment of inertia for the lamina about A and $\alpha$ is the angular acceleration. At the instant the object is released the only force acting on it is its own weight. Hence, $6g*0.8=9\alpha$ which yields $\alpha=5.227$. However the mark-scheme says the answer is $\alpha=1.65$, as they have taken into account the frictional couple. Here is the mark-scheme:![enter image description here](http://i.stack.imgur.com/l1s2i.png) **It was my understanding that at the moment of release no friction _(and hence no frictional couple)_ can act as there is no movement (yet).** So could someone please kindly explain what the mark-scheme is talking about?


Potential due to charge over infinite grounded plane conductor using the method of images

I am reading section 3.2.1 of Griffiths 3ed which explains how to calculate potential using first uniqueness theorem. Griffiths/3.2.1 ![](https://i.imgur.com/hTmCHFX.png) Griffiths/First Uniqeness theorem (Its corollary actually) ## ![](https://i.imgur.com/JlGMUEl.png)   1. As I understand we are taking volume $V = R^3$ , $\Phi (\vec{r})=0\ $on the boundary of $V$. But what does it mean by "boundary of $R^3$" ?   2. Since $\ \rho_1(\vec{r}) \ne \rho_2(\vec{r})$ which does not satisfy the prequisites of the theorem, we cannot use the theorem. Moreover no other distribution can be used which makes this theorem useless which it is not. So what am I missing here ?   3. Does not single charge distribution also satisfies the same boundary condititons ? So should not $\Phi$ also be $\frac{1}{4\pi \epsilon_0}\frac{q}{R}$ ?


Integration with Grassmann variables

How to show that $$ \int d\Psi d\bar {\Psi}e^{i \int d^{4}x\bar {\Psi} \hat {A} \Psi} = det (\hat {A})? $$ $\Psi , \bar {\Psi}$ refers to Dirac spinors (the second is $\bar {\Psi} = \Psi^{\dagger}\gamma^{0}$).


What do the latest FERMI results say about dark matter?

There was an announcement at a recent UCLA symposium on dark matter by the FERMI collaboration which hints at some evidence of dark matter. The results aren't yet published, but the symposium news release is here. Most of the news release is just general dark matter fluff, but it says that > One search technique involves using the vast amount of dark matter in our > galaxy. The NASA Fermi Satellite Telescope, an international collaboration > involving NASA, the Goddard Space Flight Center and the SLAC National > Accelerator Laboratory, searches for gamma rays — very high-energy light > particles — from this dark matter. > > There are models of dark matter that would allow a signal in the galactic > dark matter consistent with the claims at the meeting and provide a small > interaction consistent with the "null results" in the direct dark matter > searches all over the world. This obviously doesn't contain much information. Can anyone closer to the field expound on the findings?


Does the Photino have mass or is it mass-less like the photon

Does the photino in super-symmetry have a mass, Or is this different in different super symmetric models?


Realistic interpretations of quantum entanglement: Are there alternatives to the "Transactional Interpretation"?

Quantum entanglement bridges space and time: entangled particles show correlations independently of where and when they are measured. This is most evident in "delayed choice quantum eraser" experiments, where the outcome of single measurments _now_ depends on which measurements are done _in the future_ (though there is no retro-causality in the strict sense). I am looking for realistic interpretations which directly address this topic, i.e. discussions of underlying physical mechanisms or basic structures of the universe on which this correlations may be built. So far I found John Cramers "Transactional Interpretation" of 1986; Bohmian mechanics may also be considered. Are there other physical interpretations or philosophical papers on this topic of correlations over spacetime?


Solar panel maximum efficiency at specific frequency and temperature?

I talked to a professor about solar panels and their efficiency. It seemed that the main reason solar panels aren't that efficient is because it can only accept a single energy input size. Anything below is ignored and anything above only gives the energy of that single limit. That is, say the limit is 1W. A photon with 0.999W energy wouldn't give any voltage. A photon with 10W energy would give the same voltage as a 1W photon. Does this mean I can get very high efficiency from a light source with a specific frequency? For example a solar-powered calculator in a room with only sodium lights. How high? How does the efficiency change depending on the amount of light or the temperature of the solar cell? How about an extremely small amount of light, like 1nW, close to 0K? This could be terribly wrong as there was quite a language barrier between us. I want to learn, so please point out any misstakes.


scattering matrix superccurent density of states

while studying scattering formula for the supercurrent (Beenakker) I encountered that the density of states for discrete and continuous spectrum can be simultaneously described by taking $\epsilon\ ->\ \epsilon + i\delta $ in the following formula: $\rho(\epsilon) = -\frac{1}{\pi}Im\frac{d}{d\epsilon}\left\\{lnDet\left[1 - S_{A}(\epsilon)S_{N}(\epsilon) \right] - \frac{1}{2}lnDet\left[S_{A}(\epsilon)S_{N}(\epsilon)\right]\right\\}$. I am wondering if one could argument this by referring to scattering theory and Green's function - eg. Laszlo Szunyogh.


Moving wedge and pulley system

A wedge of mass $M$ rests on a rough horizontal surface with coefficient of static friction $\mu$. The face of the wedge is a smooth plane inclined at an angle $ \alpha$ to the horizontal. A mass $m_1$ hangs from a light string which passes over a smooth peg at the upper end of the wedge and attaches to a mass $m_2$ which slides without friction on the face of the wedge. A) Find the accelerations of $ m_1, m_2 $ and the tension in the string when $ \mu $ is very large. B)Find the smallest coefficient of friction such that the wedge will remain at rest. Attempts: No specification on whether or not any of the masses $m_i$ are bigger than the others, so let us assume that $m_1$ goes down and $m_2$ goes up the slope. Then A) is fine. (a minus error in the final answer for A) is introduced if I assume the other orientation) As for B) if we consider the wedge+masses as a system, then to conserve linear momentum if the masses move as above, then the wedge must move rightwards. So the frictional force acts leftwards until the wedge slips. The only other force acting on the wedge is via the normal contact force from $m_2$. A component of this force acts to accelerate the wedge horizontally. Let's write this relative to some fixed inertial frame of reference with $x$ rightwards. Then the above recast into symbols gives $$-\mu (M+m_1+m_2)g - m_2 g \cos \alpha \sin \alpha = 0 $$ at the point of slip of the wedge. However, solving this for $\mu$ gives a negative result. Where did I go wrong? ![enter image description here](http://i.stack.imgur.com/IQBDT.png)


What is charge?

I know this isn't the right place for asking this question, but in other places the answers are so awfull.. I'm studying eletricity, so, I start seeing things like "charges", "electrons has negative charges",etc. But I didn't quite understand what charge is. I search a little bit on the internet and found it related to electromagnetic fields, then I thought "negative and positive may be associeted with the behaviour of the particle in the field, great!", but the articles about e.m. fields already presuppose "negative" and "positive" charges. In other places, I see answers relating charges to the amount of electrons/protons in an atom, but if that's right, the "negative" electron is an atom without any protons? What about the neutron? So, my questions are (1) What are charges; and (2) How a particle can "be" electrically charged. What does that really mean? Thanks for your time.


Origin of electric charge

Baryons have charges that are the result of a polynomial calculation of their building blocks (quarks)'s fractional charges. But what gives these quarks electric charges? What interactions do they have with photons? And what about leptons, like muons and electrons? How do they get their -1 charge? And then, how do positrons and anti-muons get +1 charge? Same goes for quarks: how does up-antiquarks get -2/3 charge? Or I may want to formulate my question this way: **How does any elementary particle interact with the EM-Force to get an electric charge?**


What is charge actually? How to define it?

Is charge of something for (e.g.) an electron related to electromagnetic space if it exists due to energy, due to which it may have mass? I don't know about quantum mechanics or advanced particle models. Can anyone just simply give an intuitive idea? EDIT I want to mean what actually gives electron charge if it is not assumed fundamental but result of some other physical phenomenon or it is just the quantity defined to explain physical interactions?I think now it is clear


Particles and their charges

It is always known that electrons and protons have opposite charges but what gives electrons or protons the charges they have?


What exactly is charge?

If gravity is really the bending of space/time causing objects with mass to experience acceleration, is there a similar physical meaning to 'charge' besides 'a property of matter which causes it to experience a force when placed in an electric field.' What exactly is charge?


Is there any theory for origination of charge?

We have a theory of a Higgs field that describes how a particle gets mass. Since mass and charge both are intrinsic properties of a particle, is there any similar theory for how particles get electric charge?


What gives an elementary particle a charge?

We know that proton is positive, and electron is negative. But where does come notion of negativity and positivity? Does charge come from some specific particles, or they specific order?


Regularization of the Casimir effect

For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have on the subject is Wikipedia. Nevertheless I suspect this question is appropriate since I don't remember it being addressed in other textbooks Naively, computation of the Casimir pressure leads to infinite sums and therefore requires regularization. Several regulators can be used that yield the same answer: zeta-function, heat kernel, Gaussian, probably other too. The question is: > What is the mathematical reason all regulators yield the same answer? In physical terms it means the effect is insensitive to the detailed physics of the UV cutoff, which in realistic situation is related to the properties of the conductors used. The Wikipedia mentions that for some more complicated geometries the effect _is_ sensitive to the cutoff, so why for the classic parallel planes example it isn't? EDIT: Aaron provided a wonderful Terry Tao ref relevant to this issue. From this text is clear that the divergent sum for vacuum energy can be decomposed into a finite and an infinite part, and that the finite part doesn't depend on the choice of regulator. However, the _infinite_ part does depend on the choice of regulator (see eq 15 in Tao's text). Now, we have another parameter in the problem: the separation between the conductor planes L. What we need to show is that the infinite part doesn't depend on L. This still seams like a miracle since it should happen for all regulators. Moreover, unless I'm confused it doesn't work for the toy example of a massless scalar in 2D. For this example, all terms in the vacuum energy sum are proportional to 1/L hence the infinite part of the sum asymptotics is also proportional to 1/L. So we have a "miracle" that happens only for specific geometries and dimensions


Going through a ring of black holes

Mathematician here with a speculative physical question -- feel free to boot me if the level isn't right. Suppose one finds, or builds, a constellation of several black holes arranged in a circle. (To get a stable arrangement, presumably one could put a bunch of small positively charged black holes at equal angles around a central positively charged object, or alternatively suspend uncharged black holes gravitationally in a circle surrounded by a quickly rotating ring of heavy matter). Suppose moreover that the event horizons of neighboring holes overlap -- or if there's some reason this is theoretically impossible, suppose they're really, really close to being tangent, while leaving room for light and matter to pass through the middle of this ring. (Is this possible?) My question is whether a trajectory from point A to point B that goes through the middle of the ring is causally related to one that does not. For example say a space explorer measures the spin of an electron, writes down the result and puts it in a box which he leaves at some fixed point in space (we've chosen a reference frame), then goes in a loop through the middle of the ring of black holes and comes back. When he comes back and opens the box again, will the recorded result be the same? I got this question after someone told me about quantum decoherence, and I was curious whether the world can change significantly if you go around a loop that either cannot be contracted or cannot be contracted without at some point becoming ridiculously long (where it's up to you to decide the value of "ridiculously").


Is the expansion accelerating or decelerating?

I have already asked this on Astronomy.SE but I couldn't understand the answer there. > According to Hubble's Law, the farther a galaxy is, the farther it is moving > away. But do we take into account the fact that we are actually looking in > the past? > > For example, there are two galaxies A and B at distance of 5 and 10 billion > light years respectively. Now, when we observe A we are looking at how it > was moving 5 billion years ago. The same applies for B. So, now we conclude > that 5 billion years ago space was expanding at a slower rate while it was > expanding comparatively faster 10 billion years ago. What's wrong with this > conclusion?


What is the name and value of the constant that relates to electrons and that coincidentally has the same exact value as the speed of light?

There's some constant relating to electrons that also has the same value as the speed of light. What is it, what is the value, and how are they related? EDIT: Is it the fine-structure constant?? Are there any other similar constants? If you posted that answer before, you shoulda left it (to the person who deleted their answer)!


Is a vacuum needed in photoelectric effect?

This question was asked to me. My first thought was that electrons may ionise the air and potential difference that was applied may increase or decrease the current which should have been observed. I'd like to know whether this is the right reason or if there is any other reason which I should take into account. $$hv-w.f=hv'=eV$$ where $V$ is stopping potential , my second question is: if there isn't a vacuum will that stop potential change?


What is the ram-facing side of spacecraft?

what is the ram-facing side of a satellite? What does it mean and why is it called "ram"-facing? Thanks.


How do we know that some radioactive materials have a half life of millions or even billions of years?

If a radioactive material takes a very long time to decay, how is its half life measured or calculated? Do we have to actually observe the radioactive material for a very long time to extrapolate its half life?


Poisson brackets: prove that they are canonical invariants

**EDIT** : I haven't forgotten to accept answer, the question is still open.. * * * I need a clarification about Poisson brackets. I'm studying on Goldstein's Classical Mechanics (1 ed.). Goldstein proves that Poisson brackets are canonical invariants for any functions F and G. But there is a step that I can't understand. After some steps, he says that: $$ \tag{1} [F, G]_q,_p = \sum_k ( \frac { \partial G}{\partial Q_k} [F,Q_k]_q, _p +\frac {\partial G}{\partial P_k}[F, P_k]_q, _p)$$ After other steps, he writes: $$ \tag{2}[F,Q_k]= - \frac {\partial F}{\partial P_k}$$. and $$ \tag{3}[P_k, F]_q, _p = \sum_j \frac {\partial F}{\partial Q_j} [P_k, Q_j] + \sum_j \frac {\partial F}{\partial P_j}[P_k, P_j]$$ -> $$ \tag {4} [F,P_k]=\frac {\partial F}{\partial Q_k}$$ and now he replaces these relations in the first expression I have written, obtaining: $$\tag {5}[F, G]_q, _p=[F, G]_Q, _P$$ Why does he obtain in the second last step $\frac {\partial F}{\partial Q_k}$ and not $-\frac {\partial F}{\partial Q_k}$? $[P_k, F]=-[F, P_k]$ isn't it? **EDIT** : Golstein starts from (1) and substituites $Q_k$ to $F$ and $F$ to $G$ and so he obtains (2). Then he substitutes $P_k$ to $F$ and $F$ to $G$ and obtains (3). Immediately after he writes (4), that according to me is opposite to (3). And so I have thought to a printing error. I have tried to substitute $-P_k$ to $F$ and I have obtained $$[-P_k, F]=\frac {\partial F}{\partial Q_k}$$ Then, as $[-P_k, F]=[F,P_k]$, I can say that $$ [F,P_k]=\frac {\partial F}{\partial Q_k}$$. And so I can obtain (5). Could you confirm that my argumentation is correct? Many thanks


Thermal Velocity

What is thermal velocity? What is it's physical significance? Wikipedia says: > The thermal velocity or thermal speed is a typical velocity of the thermal > motion of particles which make up a gas, liquid, etc. Thus, indirectly, > thermal velocity is a measure of temperature. But I cannot understand and comprehend it.


Can exhaust gases be diverted to other cylinders during engine operation?

Here's a description of new combustion engine improvement by Mazda, called SkyActive-G. They claim that in a "generic" engine... > when the exhaust manifold is short, the high pressure wave from the gas > emerging immediately after cylinder No. 3’s exhaust valves open, for > example, arrives at cylinder No.1 as it finishes its exhaust stroke and > enters its intake stroke. As a result, exhaust gas which has just moved out > of the cylinder is forced back inside the combustion chamber, increasing the > amount of hot residual gas I always though, that exhaust manifold and intake manifold are separated, so exhaust gases just can't possibly enter other cylinders combustion chambers as described in the above quoted text. Can exhaust gases be diverted into other cylinders as claimed in that text?


Is the structural similarity between atoms ( smallest) and universe (biggest) a conincidence. Or there can a reason for this beyond imaginations

Is the structural similarity between atoms ( smallest) and universe (biggest) a coincidence? Or there can a reason for this beyond imaginations? It seems like, if one starts travelling out from atoms... and grows bigger and bigger, one ends in a similar structure somewhere in the universe. Kind of a circular ring. PS: By structural similarity I mean : single nucleus and electrons revolving around in an atom, is structurally similar to planets revolving around the sun.


Computational Science involve programming?

I read what is computational science in Wikipedia but the explanation and understanding are not very clear. So, I could you please give a simple example computational science project and what all basic skills a person should have? Also,   1. Does computational science involves programming?   2. How different are computational science and computational materials science?   3. I am from Electrical and Computer Science (basically programming) background. I was assigned a computational materials science project. So, is it in my scope? Probably, the prof assigned based on what individual subjects I studied (Engineering Math, Engineering Physics, Engineering Chemistry, Probability, Programming).


Acceleration of table tennis racket

I can't find any numbers about table tennis. What accelerations occur during a forehand smash for example? Thanks in advance & kind regards, Hans


picture of newtonian gravity and black hole

![Picture of gravity](http://i.stack.imgur.com/Vl3a6.jpg) We assume that the earth planet is a black hole. When a light beam is fired. The only way that even light can move is A and B which means this is impossible that light move on other ways C, D, E Isn't it true?


Force-Weight Pinewood Derby Car

Force-Weight Pinewood Derby Car. I'm trying to make a fast derby car. I was wondering if I could make a car that can drop/eject weight once it comes off of the incline. Would that increase speed or would it make it slower?


Nonlinear waves superposition

Non-linear waves do not superimpose to each other, but why? What characteristics give this property?


2 Nucleon Potential

I am looking at a 2 Nucleon potential of the form $$V(r)=V_0(r)[a+bI_1\cdot I_2]$$ Where a and b are constants. $I_1,I_2$ are isospins. $V_0(r)$ is of the square well form. My goal is to find an equality for a and b, given that deuteron exists, and that diproton and dineutrons do not. My approach has been the following: 2 nucleons can either be in an isosinglet or an isotriplet. The Isosinglet has I=0 and is the following: $$|00\rangle=\frac{1}{\sqrt2}(pn-np),I_1\cdot I_2=-\frac34$$ The isotriplet has I=1 and $$|11\rangle=pp,I_1\cdot I_2=\frac12$$ $$|10\rangle=\frac{1}{\sqrt2}(pn+np),I_1\cdot I_2=\frac12$$ $$|1-1\rangle=nn,I_1\cdot I_2=\frac12$$ So I get the following equations based on V $$\text{Isosinglet: }\frac{V(r)}{V_0(r)}=a-\frac34 b$$ $$\text{Isotriplet: }\frac{V(r)}{V_0(r)}=a+\frac12 b$$ Now I am supposed to use the fact that $V_0(r)$ is of the square well form to create inequalities, but I am unsure of where the inequalities come from.


how to determine direction of gyroscopic couple on car?

How can I determine the direction of the force acting due to gyroscopic couple on a car's wheels when it is taking a turn to it's left side?


Temperature of rod if end points are different temperature

How does the temperature vary along the length of the rod if its both ends are at different temperature. As an example, consider the problem: > 20 cm long rod has rod at one end 100 ºC and another end at 0 ºC. Find the > temperature at the center of the rod when it's in thermal steady state.


What causes light to move through a vacuum

I was looking at rockets and stuff and thought about how they move through a vacuum using newtons 3rd law, and then I started thinking of any other ways you could move through a vacuum without using this and then I thought about the photon. I then thought how does the photon move through a vacuum? So I searched it online and couldn't find an answer. So my question is simply how does a photon move through a vacuum? Is it because its massless? Is it because it is thrown from an electron or whatever like a rock from a slingshot? Or is it something completely different? I'm very curious!


non-exotic traversable wormholes and string theory

In this article it is claimed that certain type of string theory called EGBd allows for traversable wormhole solutions that do not require exotic matter. What is this EGBd model and how it fits in the grand scheme of things of string theory? is this fringe science or is serious? (i know that there might be fringe authors in ArXiv). But my main concern is how general or realistic is this result?


Is the spring constant k changed when you divide a spring into parts?

I've always been taught that the spring constant $k$ is a _constant_ -- that is, for a given spring, $k$ will always be the same, regardless of what you do to the spring. My friend's physics professor gave a practice problem in which a spring of length $L$ was cut into four parts of length $L/4$. He claimed that the spring constant in each of the new springs cut from the old spring ($k_\text{new}$) was therefore equal to $k_\text{orig}/4$. Is this true? Every person I've asked seems to think that this is false, and that $k$ will be the same even if you cut the spring into parts. Is there a good explanation of whether $k$ will be the same after cutting the spring or not? It seems like if it's an inherent property of the spring it shouldn't change, so if it does, why?


Question on ladder operators

Suppose we have a finite , discrete set of orthonormal states $|k\rangle $ We can construct raising and lowering operators intuitively, for example $$a_+ =\sum_{k=1}^nC_{k+1}|k+1\rangle \langle k|$$ However most textbooks begin by defining ladders in terms of the linear combinations of hermitian operators. How do we get from the above construction to showing that such ladder operators have the form $$a_\pm = \frac{\alpha\pm i\beta}{\sqrt{2}}$$ where $\alpha,\beta$ are hermitian?


How does cooling scale with volume?

What equation would give me the answer to the question, "If i have a cup of water at a tempature of say boiling, how long would that cup of water take to cool off compared to say half that size of a cup of water." So the volume is in half. Its a general question I am just looking for where to start.


What's the amount of deviation of cellestial orbits from perfect ellipses

It's well known that the planets don't orbit the sun in perfect circles and the characteristics of the elliptical orbits which serve as better approximations to their motion have been calculated fairly accurately. How accurately do elliptical orbits model their actual paths?


Calculating the Uncertainty for an Average Value

How would I calculate the uncertainty for the average of this set? $32.5 \pm 0.1$ $32.0 \pm 0.1$ $32.3 \pm 0.1$


Would time freeze if you can travel at the speed of light?

I read with interest about Einstein's `Theory of Relativity` and his proposition about the speed of light being the speed limit for anything with mass. So, if I were to travel in a spacecraft at the speed of light, would I freeze and stop moving? Would the universe around me freeze and stop moving? Who would the time stop for?


How does a particle of light reach the max speed of light?

> **Possible Duplicate:**   >  How can a photon have no mass and still travel at the speed of light? First of all I am not a professional physicist. I was curious as to how a particle of light can reach the maximum speed of light c given that particles experience an increase in mass as they approach c ?


If photons have no mass, how can they have momentum?

As an explanation of why a large gravitational field (such as a black hole) can bend light, I have heard that light has momentum. This is given as a solution to the problem of only massive objects being affected by gravity. However, momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum. How can photons have momentum? How is this momentum defined (equations)?


How can a photon have no mass and still travel at the speed of light?

I've read a number of the helpful Q&As on photons that mention the mass/mass- less issue. Do I understand correctly that the idea of mass-less (a rest mass of 0) may be just a convention to make the equations work? From a layperson's view, it's difficult to understand how a particle of light (photon) can be mass-less. A physical object (everyday world-large or quantum- small) must have a mass. Yet, if my understanding is correct, the mass of a moving object/particle increases in proportion to its speed/velocity...so that at the speed of light, its mass would be infinite. A photon travels at the speed of light, but it obviously doesn't have infinite mass, right? Can someone formulate a practical explanation that can be understood by middle- school to high school kids? Much thanks for the help. * * * Wow--your answers to my original Q below clear up much of my confusion. I now have the daunting task of going over these nuggets and working up an equation- less (hopefully) explanation of the mass-less photon for non-physicist types. Yes, _from a layperson's view_ , it does seem remarkable that an existing piece of _matter_ \-- which has to be made of _physical substance_ \--could have zero mass at rest (though a photon is never at rest). It would be almost understandable if a piece of matter made of _nothing_ had zero mass, but that seems to be an oxymoron, and "nothing" would equate to nonexistent, right? In case you might find it interesting: I'm working on a writing project that posits we inhabit a universe that consists of matter (physical stuff) only, and that the NON-physical (aka supernatural) does not (and cannot) exist. For instance, if a purported supernatural phenomenon is found to actually exist, then by definition, its existence is proof that it is mundane/natural. All it would take to disprove this premise is reliable proof that ONE supernatural event has occurred. Despite thousands of such claims, that's never yet happened. Who else better than physicists to confirm my premise? However, I do wish the TV physicists would explain the terms they throw about, some of which mislead/confuse their lay viewers. Case in point: "The universe is made up of matter and energy" (without properly defining the term "energy" as a property of matter). The result is that laypersons are left with the impression that energy must therefore be something apart from or independent of matter (ie, nonphysical). Their use of the term "pure energy" without specifying exactly what that means adds to the confusion. (Thanks to your replies on this forum, I now understand that "pure energy" refers to photon particles.) However, "psychics" and other charlatans take advantage of such confusion by hijacking terms like energy (as in "psychic energy"), frequencies, vibrations, etc to give perceived scientific legitimacy to their claims that a supernatural spirit world, etc., exists. As you may realize, the majority of people in the US (per 2009 Harris Poll) and around the world believe in the existence of nonphysical/supernatural stuff such as ghosts and spirits. My purpose is to give laypersons the information they need to distinguish what's real from what's not. Thanks so much for help...And, PLEASE, add any further comments you think might be helpful/insightful to better inform laypersons.


How does the solar sailing concept work?

Wikipedia describes solar sailing as > a form of spacecraft propulsion using a combination of light and high speed > ejected gasses from a star to push large ultra-thin mirrors to high speeds. I understand the part where ejected gasses bump into the sail pushing the spacecraft. On the other hand, I don't understand how light can do this, since light has no mass. How does that work? Does this mean that if I have a mirror balancing on a needle I would be able to push it over with my flashlight?


Clocks tick steadily, so why is there no photon time?

Consider a photon bouncing left and right between two mirrors in a photon clock. Seen from inside the clock, the photon bounces at a constant frequency. Time ticks regularly. No matter whether the clock begins moving relative to some outside observer, the clock continues to tick regularly. Light always moves at the same speed relative to the observer in the clock. If the clock is moving forwards relative to an outside observer, we know that _to that outside observer_ , the photon clock appears to tick more slowly. This happens because the speed of light relative to that observer is constant, and, since the photon is moving diagonally relative to the outside observer, the distance traveled by the photon in the clock between ticks is greater, so the photon must take longer to pass between the mirrors and the clock must tick more slowly. But notice how, if you are inside the clock, it continues to tick at the normal speed. This brings me to my question. It is widely reported (including on this very QA site) that a photon traveling uninhibited in a vacuum experiences no time. This seems to be because all the speed of the photon in the clock is now taken up keeping up with the forwards movement of the clock, so there is no speed left to deal with left/right bouncing and ticking. The clock stops ticking. Well, this is the thing. Surely the clock only stops ticking _from the perspective of the outside observer_. If you're inside the light-speed clock, time continues to tick normally. The photon continues to bounce. So why do we say a photon experiences no time? Surely we mean it _appears to us_ as stationary observers to experience no time? If you're a photon, time is just the same as it is to the rest of us, right? Fair enough, your average photon might well feel that it got from A to B in no time at all, because distance was compressed to zero and all the other clocks appeared to have zipped along beyond the end of days in less than an instant, but its own clock, if the photon could observe it, is just ticking normally. This leaves my brain mushed, because now the photon has seen the entirety of an infinity of time in no time at all, yet its clock is still ticking. So what does that mean? What happens the moment after the infinity of external time has completely passed? Surely there is nothing left to happen?


Does distance really exist at least for observer which moves at the speed of light?

To our time a light year is the distance that light travels in a vacuum but if we consider time of observer which moves at speed of light we found for light time and distance doesn't really exist. Does distance really exist? ($d\tau=dt\gamma^{-1}$ where the at speed of light $\gamma^{-1}=0$) isn't universe just a point for observer at light speed? * * * it is not duplicate guys!.


Frame of reference of the photon?

In the frame of photon does time stop in the meaning that past future and present all happen together? If we have something with multiple outcomes which is realized viewed from such frame? Are all happening together or just one is possible? How the communication between two such frame s work meaning is there time delay for the information as $c$ is limited? If there is time delay does it mean that time does not stop? My question does not concern matter at that speed rather how it looks viewed from the photon reference. Thanks Alfred! I think I understand it now.


Speed of light and lorentzian factors

> **Possible Duplicate:**   >  How can a photon have no mass and still travel at the speed of light? If light travels at the speed of light, and anything with rest mass will experience relativistic effects based on the Lorentzian equations, why doesn't light experience these kinds of effects? For example, relativistic mass and rest mass are related via $$m = \frac{m_0}{\sqrt{1-\dfrac{v^2}{c^2}}}$$ Shouldn't light therefore have no rest mass (since the Lorentzian is $0$)?


Why does Lorentz factor not hold for relativistic mass when we apply it to photons?

We know that the photon itself is massless particle $m_0=0$. But we also know, that the mass of the objects does increase with their energy. And we know that under certain circumstances (gravity, collision with objects) the light does behave like a beam of particles that do have a mass. Now, this is the equation to get the (relativistic) mass of the object with certain speed: $$m=\frac{m_0}{\sqrt{1-v^2/c^2}}$$ We already do know some things: $m_0 = 0$, $v = c$. $$m=\frac{0}{\sqrt{1-c^2/c^2}}$$ $$m=\frac{0}{0}$$ Expression $\frac{0}{0}$ is **not** equal to 0. So what is this?   I'll remind you of those weird clock that are run by light - I failed to find any images, but the clock principle was similar to windmill. And it works.   Also there was an idea to make long-distance autonomic spaceships that are pushed by light rays from stars.


how photons are said to be in momentum if they are considered to have no mass and no charge?

According to quantum theory,Einstein said that light waves have small packets called photons and they are said to have no mass and charge but having energy E=hv and they behave has both particle and wave character. And they travel with certain velocity. How photons have momentum and energy if they said to have no mass? Momentum = p=mv ,where m= mass v=velocity.


How can light travel at speed of light?

I am well aware that the speed of light is a universal constant at which nothing but light can travel. But why? Why can even light travel at such a speed? Our maths tell us that anything with rest mass takes infinite energy to reach speed of light, so do we just mean that light is the lightest of all things and it can reach speed of light with negligible energy? Why isn't the speed constant which appears not only as speed of light, but various other places like GR etc something even more than speed of light? Why can light reach the speed of light? I think my question was highly misinterpreted, my question has nothing to do with mass of light. My question is why is the speed limit that appears universally equal to speed of light, why isn't even light allowed to reach the universal speed limit? Yes after that I did ask, the reason for light reaching the highest possible speed. Whether it is it's unusually less mass(mass-less) or something else. So to summarize my question is : 1\. Why anything at all can reach the top speed limit, why is even light permitted to reach the top speed limit? 2\. Why light can reach the top speed? (Only this part is related to the question whose duplicate my question is marked)


Does $p=mc$ hold for photons?

Known that $E=hf$, $p=hf/c=h/\lambda$, then if $p=mc$, where $m$ is the (relativistic) mass, then $E=mc^2$ follows directly as an algebraic fact. Is this the case?


Masses of all the particles in the Standard Theory

> **Possible Duplicate:**   >  If photons have no mass, how can they have momentum? I'm sure this question has been asked here before but I wasn't able to find it clearly answered in one q/a session. I'm a newb, yes. In the standard model (which I understand is well experimentally supported except for the absent Higgs Boson), all the elementary particles must carry some mass, correct? For E=MC2 to be true, you can't have energy without mass. And energy has been carried away in collisions of sub-atomic particles that can't be accounted for (the Higgs). So EVERY elementary particle in the Standard Model must have mass, correct?


How does light travel at the speed of light?

If it is impossible for matter to accelerate to $c$ (because doing so would take infinite energy), and if light can be deemed matter (because of wave- particle duality, photons are matter, right? And photons clearly have mass per solar radiation pressure), how does light manage to travel at $c$?


What happens with time at the speed of light?

Say that by some means you could become massless, and therefore travel at the speed of light. I understand that the closer you get to the speed of light, the slower time appears to the person traveling. Like say you were traveling a 99.999 % of the speed of light, ever year for you would be 200 or so years for people not traveling. So I suppose at the speed of light, time stops for the traveler. Would that mean you'd be frozen in time and an infinite amount of time would have passed for the rest of the world while you were travelling at the speed of light?


If photons don't have mass, how can they accelerate objects?

As far as I know photons don't have mass but they do have momentum ($p=mv$). Scientists say that if we put a shiny (reflective) shield of large radius in the vacuum of space, then light from sun will will start moving the shield. How can these two facts be commensurate?


attaining the speed of light??

WHY our time would run slow if we move with the speed close or equal to the speed of light, and is that time dilation restricted to only just '' time' ' or it affects our biological time also (does it stops or slows down our ageing process) ??


Are photons travelling in future? Are they lifespan 0 (or close to 0)?

We know that if we are going closer to speed of light, we are travelling in future. Since the photons are travelling at the speed of light, that means that they are travelling into future? Also, if we could possible attach a clock to a photon and measure how much time passed for 1 earth year(we put the clock to be 0, then we wait 1 year, then we look at the clock), would that clock show 0?


Do photons have mass?

As a student in a highschool physics class, my teacher has repeatedly told me that photons are massless. Yet, I have also heard from other sources that photons have momentum. If photons were to have momementum, that would mean that they have mass as according to p = mv. Do photons really have mass? Also, how would this mass be calculated?


> **Possible Duplicate:**   >  Would travelling at relativistic speeds have any impact on human biology? I am asking myself this question for a few day. What is the answer on: Does the increase of (relativistic) mass, while flying near speed of light, has any impact on astronauts? I mean let's pretend a spaceship will be able to travel in space with nearly the speed of light. Does the astronaut on this ship, while its (relativistic) mass increases, feel any difference on gravity or anything else? Or is the increase simultaneous to his own?


Why doesn't light travel instantly?

I've read that the faster you travel in space, the slower you travel in time. And when you reach the speed of light (which we won't be able to) time will stand still. So when light travels at the speed of light, it doesn't move in time, so no time will have past when the light arrives at it's "destination". Right?


1. What is the mass of a photon moving at the speed of light?    2. And if it does not have mass, how is it affected by gravity?   3. Also why does Einstein's general relativity support that a gravitational wave must travel at the speed of light? I'm just an A-level student. So, I would appreciate it if you could explain it as idiotically as possible.


Photons have no mass. So, why does $E = pc$ hold?

It's a somewhat theoretical question. In special relativity, The energy of a photon is given by $E = pc$. But, my argument is that, since photons have no mass, how can they have a momentum $p$? The energy $E$ turns out to be 0 always. So, why does this equation hold?


How to get the accurate relativistic momentum form for photons?

I have studied from Griffiths, the relativistic form of momentum is $$p = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} m_0v$$ Now when I evaluate the momentum for photon, I just insert $v=c$ and $m_0=0$ and I get $p= 0/0$. How does it make sense? Can you tell me that where I am wrong?


Would travelling at relativistic speeds have any impact on human biology?

If a person was sitting on a craft that has accelerated to near light speed speed from Earth (e.g. 99.999% of light speed) would there be any impact on his or her human biology due to relativistic effects on mass, time and distance? For example, would the heart still be able to pump blood effectively if the relativistic mass of the blood has increased? Or would accelerating the blood a bit further towards light speed requires more energy than when stationary on earth?


Why does the cart move?

A while ago someone proposed the following thought experiment to me: A horse attached to a cart is resting on a horizontal road. If the horse attempts to move by pulling the cart, according to the 3rd Newton's Law, the cart will exert a force equal in magnitude and opposite in direction, cancelling each other out and thus the horse and the cart _should_ not move. And yet it moves (pun intended ;) Why? I never got a satisfactory answer, my guess is that the answer lies in the frames of reference involved: horse-cart and horse-road. Any ideas?


Topolgical insulators order parameter

_For topological insulators_ **Is there any way to defineorder parameter for topological phase transitions?**


Why is this not a violation of parity invarance for EM

I read that Wu's experiment illustrates that parity violation is possible for weak processes. In that experiment, when Co-60 undergoes beta decay, the emitted electrons come out opposite to the direction of nuclear spin. The mirror image process does not occur. I also read that parity is conserved for EM and strong processes. But consider the following from EM. Say I have a positively-charged particle moving to the right, and the magnetic field is into the page. Then force on the particle is up (i.e. the particle is accelerated upwards initially). And here, the mirror image also does not occur! The mirror image (the mirror I am thinking of lies in the plane that cuts your head between the eyes) would be moving left, magnetic field into page, acceleration still pwards - but EM would predict that the acceleration is supposed to be downwards!! Why isn't this an example of parity violation?


How to convert physical ligth quantities (frequency, intencity, spectrum) to RGB and back?

Suppose I would like to have maximally physical image format. Which quantities should I store in the pixel? Obviously, it should be frequency spectrum, i.e. table with frequency vs. WHAT? How to convert RGB values to any sort of physical quantities and vise versa?


Is there Phase difference between voltages at end points of a long AC Power line

I found this explanation somewhere: Since wavelength=c/freq so in a AC power line of 50 Hz, wavelength=(3*10^8)/50 = 6000 Km, so voltage phase reverses after 6000 Km in a ac Power line. Now this would have been correct if it was a 50 Hz E.M. Transmission line, but does the same occur in metal wire with A.C. ? Clearly the power transmission through the wire would not be through E.M. waves. I am also confused if a wave phenomenon does occur in a AC wire or not. If it occurs then how (in terms of flow of electrons) and at what speed/wavelength ?


Energy stored in a capacitor in an RC circuit

Suppose we have a capacitor of capacitance $C$ and a cell of emf $E$, why is the maximum energy stored in a capacitor equal to $\tfrac{1}{2}CE^2$? I feel confused because, the potential difference against the capacitor will be less than the emf of the cell because of the potential drop across the resistor. What is the flaw in my thinking?


What is the observable Earth we can see?

Yes I know we can see the whole earth, but how far can we see left to right waving out the limitations of sight. Because it's impossible to see the whole earth right? because it's spherical? So is there some kind of equation to work this out?


What is the invariant associated with the symmetry of boosts?

Noether's Theorem states that if a Lagrangian is symmetric for a certain transformation, this leads to an invariant: Symmetry of translation gives momentum conservation, Symmetry of time gives Energy conservation etc. The Galilean principle stating that all reference frames that move with constant speed relative to each other are equivalent is also a symmetry principle: Setting up a physical system that is identical to the original except for a constant velocity (boost) added will have the same behaviour. Shouldn't there be an invariant associated with this symmetry? If yes, what is that invariant?


Noether's charge due to lorentz transformation

> **Possible Duplicate:**   >  What conservation law corresponds to Lorentz boosts? For a relativistic free particle, What is the Noether charge generated due to Lorentz transformations? What is conserved quantity associated with boosts?


What is a correct and simple definition of quantum physics?

Is it correct to define Quantum Physics as the study of Physics in sub-atomic scale? Does Quantum Physics studies something else other than sub-atomic phenomena? This may be a very stupid question but realize that I am **!!!VERY NOOB!!!** at Physics.


Does (it make sense to say that ) the universe has a center?

I was reading this page: http://www.guardian.co.uk/science/2011/oct/23/brian-cox-jeff-forshaw-answers and I found this sentence by Brian Cox: > That seems to imply that everything is flying away from us and we're > therefore somehow in a privileged position; that isn't true. The way it's > often described is if you imagine some bread with raisins in it that you're > baking in the oven and as you heat it, it expands. On any particular raisin, > if you look, you can see all the other raisins receding from it. So it's > space that stretching, it's not that everything's flying away. I already heard this raisins analogy, but it never persuaded me: I understand that the "big bang" is more like a "big stretch", and I see how every 2 observers in the universe are being distanced farther and farther away (regardless of their position) Yet one of the Big Bang ideas is that the universe isn't anymore considered infinite and completely homogeneous But the fact that the universe is finite, while inflating to me implicates that it should have some kind of bounds (not that we can reach these "bounds", since our distance to them is getting bigger, but they should still exist) (And the fact that it's spreading inhomogeneous mass and energy over big distances, is thus making it more homogeneous, but this doesn't probably matter) So: the very idea of a big bang seems to me in contradiction to the assertion that there's no such thing as a "center of the universe": If it has a finite mass and some kind of bounds, then it should also have a barycenter. And if we consider the bread with raisins analogy: the bread has a center from which it's expanding Surely, the universe isn't homogeneous (like the distribution of the raisins), and so, in its hypothetical center, there may not be actually anything... but I think (even if it's really unlikely) it should still be theoretically possible to have a raisin in the exact centre of the bread


Finding possible values of $L_x$ given $L^2$

Here's a homework problem I'm working on. I am not asking for the answer, but any guidance or comments on the approach are appreciated. > Given that a measurement of $L^2$ for a free particle has resulted in the > value $6\hbar^2$, what are the possible results for a measurement of $L_x$? I've been following the approach given on pp. 378-380 in Liboff (4th ed.). I've made it as far as expressing the eigenfunctions of $L_x$ as sums of the eigenfunctions of $L^2$... Here's what I've got: $$\begin{align*} X_1 &= \frac{1}{2}\left(|2,2\rangle - |2,1\rangle + |2,-1\rangle - |2,-2\rangle\right)\\\ X_2 = -X1 &= -\frac{1}{2}\left(|2,2\rangle - |2,1\rangle + |2,-1\rangle - |2,-2\rangle\right)\\\ X_3 &= \frac{1}{2}\left(\sqrt{\frac{3}{2}}|2,2\rangle - |2,0\rangle + \sqrt{\frac{3}{2}}|2,-2\rangle\right)\\\ X_4 = -X3 &= -\frac{1}{2}\left(\sqrt{\frac{3}{2}}|2,2\rangle - |2,0\rangle + \sqrt{\frac{3}{2}}|2,-2\rangle\right)\\\ X_5 &= \frac{1}{2}(|2,2\rangle + |2,1\rangle - |2,-1\rangle - |2,-2\rangle)\\\ X_6 = -X_5 &= -\frac{1}{2}\left(|2,2\rangle + |2,1\rangle - |2,-1\rangle - |2,-2\rangle\right) \end{align*}$$ The next step appears to be to express $L^2$ as sums of the eigenfunctions of $L_x$, which does not seem possible. It was a lot of algebra, matrices, determinants, etc. to get to this point so it is certainly possible that I've miscalculated something accidentally, but does anyone see anything fundamentally wrong here with this approach? And/or is there a simpler way?


Mathematically possible vs physically probable outcomes

A good buddy of mine and I have had a friendly debate about the origins of the current state of our universe (namely; Earth and life on Earth) and have fundamentally disagreed in our stances with respect to probabability, infinity in time and space and possible/probable event outcomes. He maintains the position that given a set of possibilities and enough trials, each outcome _must_ have occured; which is his reasoning for why life must exist. I do not necessarily take issue with this particular concept, as given an infinite amount of time and states of matter life is bound to come from one of those states. In our discussions, however, we have been using a specific example in which I disagree vehemently with his stance. The example: > If you throw a handfull of sand in the air an infinite number of times, and > that sand lands on a flat surface, every configuration _will_ happen > (according to his position). For instance, the sand landing in a pattern > which spells your name out is a mathematically possible outcome, and will > therefore happen given enough trials. To counter his stance on this example, I took the position that there is a mathematical (but not physical) possibility that every grain of sand lands in the same one inch square of the surface; but I maintain that even though it is a mathematically possible outcome it will never happen because of the way the physical world works - that sand will be roughly evenly distributed for each throw, even if over an infinite number of trials, assuming consistent and fair trials (ie, no God or other being moving grains of sand). I submit that even though it is a mathematical possibility, you'll never see your name spelled out in block letter English anywhere in the universe without the influence of intelligence, even if you were able to attempt a verification for this - he disagrees. I held him liable for mathematical/physical proof of reasoning for his stance and he has taken to dismissing me as ignorant of probability and infinity. Can anyone provide some good reasoning for either side of this argument? I realize that either is an impossible stance to prove, since we can't verify our positions, but any well-reasoned insight will be appreciated. A similar question, with an answer I found to be relatively useful: Infinite universe - Jumping to pointless conclusions **EDIT:** After reading some of the comments and answers here it has become apparent that I may have misrepresented my ultimate question. I realize that given a non-zero probability and an infinite number of trials, the mathematical probability of encountering the event described by said probability converges to 1. Some have taken the position that there is no disconnect between a mathematical probability and the likelihood (read: possibility) of a physical event happening. To simplify the argument, the surface can be thought of as a grid - in which case every single configuration has some mathematical probability associated with it. My stance regards certain configurations as physically impossible, however, which is the reasoning behind my one-inch-square analogy. Can anyone show clear reasoning (and sources!) for their belief that it is possible to toss a handful of sand into a one inch square?


Losing mass in space

So I came across a question while studying laws of motion. Roughly, this is how it goes: There are two astronauts in a space shuttle, who together have mass 200 kg. If by doing exercise, they manage to lose 80 kg, what will be the percentage increase in speed of the shuttle. The question is pretty straight forward, if thought about directly. However, my instant reaction was that by conservation of mass, the mass that the astronauts lose will still be contained within the space ship in the form of water, CO2, etc. So technically there won't be any change in mass, thus no change in speed. I would like to know if this assumption is correct and in what forms is the mass we lose released.


insufficient electrons in a conductor

We all know that if an electric field is applied across an isolated conductor, then charges are induced at either surface such that the net electric field inside becomes zero. Now if the applied electric field is so strong that there are not sufficient free electrons in the metal to move and cause the required surface charge seperation. Then will the conductor now behave as a non- conductor(i suppose so) or will it ionise nearby gas and extract electrons from it .


Including air resistance, what is the escape velocity from Earth?

Including air resistance, what is the escape velocity from the surface of the earth for a free-flying trajectile?


Conservation of energy (or lack thereof) in Doppler cooling

I did not find my question answered elsewhere, so here it is. I'm doing a project in my first optics course, and we are reading a bit about Doppler cooling. I understand that a laser is tuned to a frequency where the photon energy $E_p = h \nu$ corresponds to an energy slightly below the energy $E_{abs}$ required to excite the gas atoms, and that an atom moving towards the laser at a high speed (relative to rest of the gas) will experience a blueshift that allows the atom to absorb the photon, thus slowing down the atom. I also understand that the photon is re-emitted in a random direction, which of course means that the atom's velocity in the direction of the laser will be decreased. What I'm not sure I understand is how the re-emission of the photon doesn't just bring the atom back to the same kinetic energy (but with a different direction of motion). I have not been able to find a source explaining this, but here's my own attempt at an explanation (which I am not sure is correct, which is why I am asking here): The absorbed photon has an energy (in the frame of the laboratory) of $E_1 = h \nu_1$ which is slightly less than the absorption energy $E_{abs}$. But after the absorption the atom's speed is reduced, thus the photon will be emitted with a different frequency $\nu_2$ (in the frame of the laboratory). Since the speed of the atom (relative to the frame of the laboratory) has been reduced by the absorption, the energy $E_2$ of the emitted photon will be closer to $E_{abs}$, thus $|E_{abs} - E_2| > |E_{abs} - E_1|$ (the last step of the explanation only works if the atom's speed perpendicular to the laser is not greater than the speed parallel to the laser for most atoms, but since the laser is tuned to only hit the fast moving atoms, this seems improbable enough to not invalidate the explanation). Is this explanation correct or have I made mistake? I have taken a special relativity course but no quantum mechanics yet, so please keep the quantum mechanics to a minimum if possible.


Modeling of exact stick-slip-friction using Linear complementarity problems (LCP)

Within my master's thesis, I try to model a machine elements and have to model the exact slipping and exact sticking between the bodies. So I noticed that such problems can be formulated as a LCP. However I have difficulties at finding any literature where the usage is described well with examples.. I would like to have something like a 1-DOF model such a box on a plane pulled with a spring and friction modeled with coulomb or so. I would be very grateful of mentioning any literature/sites which can bring me further. I am really desperate at implementing the coulomb friction..


Dalitz plot analysis

I have seen a few Dalitz plots so far and tried to understand how they are useful. So one of the advantages of these plot is that the non-uniformity in the plots can tell something about the intermediate states that we cannot detect. My question is how do you extract the mass of these resonant particles from such plots? What additional information would you need?