Transformer and magnetic field/flux through the iron core?

I am familiar with how a transformer works. This is the diagram that I found on the wikipedia page:![Mangetic flux in a transformer](http://i.stack.imgur.com/9ccpO.png) The current through the red solenoid varies with time, which creates a time- dependent magnetic flux (green) that also goes through the blue solenoid therefore inducing an electromotive force. Both solenoids are wrapped around a magnetic object so that this "carries" the magnetic flux created _inside_ the solenoid (which is much bigger than the one outside of it) around. I am not quite sure why the magnetic flux bends in the way displayed in the diagram. The magnetic material is ferromagnetic right? So shouldn't the magnetic field point produced inside the red solenoid force all the other domains to align with it? In other words shouldn't the magnetic field point in the same direction everywhere in the magnetic material (as I have tried to illustrate below) ?![enter image description here](http://i.stack.imgur.com/vNeEM.png)


Are there any musical instruments which use resonance tubes with two closed ends?

Many musical instruments use resonance tubes with one closed end - all brass instruments (I think), clarinet, etc. There are also instruments where both ends are open (flute, pipes) Are there any which use two closed ends, that is, a resonance tube with nodes at both ends? I am aware of the problems which would exist for such an instrument - essentially not be very loud - but I am just curious if anyone knows of any.


Excited Energy levels of Hydrogen vs Solids

My question has to do with Excited Energy Levels. I keep reading and learning that all objects/most solids emit infrared radiation. But to emit radiation, an object's particles must first enter a higher energy orbital (electrons) which will then immediately lead to the atom returning to its ground state, releasing the energy of the level jump from ground state to level 2 in eV in the form of a photon wave. BUT! When I looked for information about energy levels of elements, I could only find hydrogen, and when I plugged in the energy for: l=hc/E where l is wavelength, h is Plancks constant, c is the speed of light, and E is energy of the photon. The wavelength came up to be 1.121x10^-7! That is in the ultraviolet range! That means that it takes a ton of EM Energy to emit a photon wave! This doesn't make sense unless solids require much less energy to move up orbitals. Is this true? What happened to Infrared or Radio? And how do I find out about energy levels of solid elements and understand the data?


2-body interaction energy of 3 particles

If only two-body interaction is considered then what's the energy if I put three particles on one site? Assume delta-interaction and interaction strength is proportional to scattering length $a_F$ which depends on the total spin of the two-particle state. For two particle case, the interaction is simply $\hat U |F,M\rangle = U_F |F,M\rangle $ Suppose the three particles are fermions, then I will take a completely anti- symmetrized spin wave function. But now, how to express the interaction energy in terms of two-body scattering lengths?


Young experiment: square of classical real wave function

I can't understand why the sum of two real waves result in a time dependent wave, but not so for the complex waves. In details, I can't get this passage on p.38-39 in A.C. Phillips, _Introduction to Quantum Mechanics:_ $$\tag{3.9}\Psi ~=~ A\cos (kR_1 - \omega t) + A\cos (kR_2 - \omega t)$$ $$\Downarrow$$ $$\tag{3.10}\Psi^2 ~=~ 2A^2\cos^2\left(\frac{k(R_1-R_2)}{2}\right)\cos^2\omega t.$$ This is driving me crazy!


In what way do Cooper pairs of electrons bond and stay bonded in superconductors?

I understand how electrons initially move into another's vicinity, but nowhere can I find a fathomable answer to this. Also, does the pairs forming 'a condensate' mean a Bose-Einstein condensate?


What electric potential is found over individual resistors connected in series to an AC power supply?

Given a set of resistors connected in series to an AC power supply:   1. What formula governs the peak-to-peak voltage which will be measured when voltage is probed over individual resistors and sets of resistors in that circuit?   2. What is a good way to think about this construction so the answer to question #1 is obvious and intuitive? Intuitively, I would think that the measured voltages would be the same as in a DC circuit such that:               V_out = (R_partial/R_total) V_in      Where V_out is the voltage which is probed, R_partial is the total resistance of the sub-set of resistors which are being probed and R_total is the total resistance in the circuit. This formula makes sense intuitively because the current throughout a circuit of series components is everywhere constant and according to Ohm's law for a constant current, the higher a given resistance over an area the higher the voltage potential measured there. Is this correct for AC circuits as well?


Why Are Even and Odd Regge Trajectories Degenerate?

The Gribov-Froissart projection treats even angular momentum differently from odd angular momentum. But in QCD, I believe that the odd trajectories interpolate the even trajectories--- the two trajectories are degenerate. Is there a way of understanding why the even-odd trajectories are degenerate? Is is a symmetry argument? Can you find a natural system where they are not degenerate? A quick google search revealed this reference, which doesn't answer the question, but gives an experimental signature for the degeneracy: http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-0576.pdf , There is probably excellent data by now on this.


Parity and Helicity of the Higgs Boson

I have been studying how the spin and parity of the new boson discovered at the LHC will be studied and have run into some confusion. The Standard Model Higgs is expected to be a scalar (i.e. have even parity with spin 0). My question is how is the intrinsic parity of a particle experimentally determined? Is it related to the helicity amplitudes? Why is parity even an important quantum number to determine?


Relationship between boiling point and pressure

In the bubble chamber, liquid hydrogen is kept at high pressure. When a stream of charged particles are injected into the bubble chamber, pressure on the liquid hydrogen is released and bubbles of gas are produced around the charged particles. However my question is that:                  why would the boiling point of hydrogen decrease when pressure decreases?


Torque due to a magnetic field

We have two infinite perpendicular wires, each carrying a current "i" and the distance between them is l. If both the wires are free to move, how will they arrange themselves because of the torque exerted by the magnetic field produced by the two wires?


If a photon has no mass, how can it be attracted by the Sun?

I read that the photon doesn't have mass, but my teacher says that the photon has mass because the sun can attract it (like in the experiments to prove the theory of relativity). I think that there is another reason to explain that. How can I explain that the photon doesn't have mass and the sun attracts photons?


How thick (in molecules) would a piece of paper be if it was folded to the moon?

Apparently you can fold a piece of paper 42 times and reach the moon! 42 is the answer the everything after all! Oh, Douglas Adams, you clever son of a bitch. The reasoning behind this is that as you fold a paper, it becomes twice as thick. It's exponential and can be modeled by the following formula for a paper that is say- 0.001 cm thick. 0.001(2)^42 = space travel I want to test this. You have to figure that as you fold a paper and it becomes twice as thick, the area is also halfed. So, for a standard piece of A4 paper, being 8.5x11, would have an area of 92 in^2 or about 603 cm^2. The equivalent decrease in area can be modeled by this formula: 603(0.5)^42 Now, for the number of molecules in a sheet of paper. The number of molecules for a piece of paper consisting solely of cellulose (C6H10O5) strains can be found by using the molar mass of cellulose and the mass of the paper in question. So the calculations for a standard piece of A4 paper with a mass of 5 grams should look something like this: 5g × (1 mol C_6 H_10 O_5)/162g =3.09 × 〖10〗^(-2) mol damn stackexchange formatting And since a mole of cellulose contains 6.022 x 10^23 molecules, you would have about: 1.8586 x 10^22 molecules in that piece of paper This is where that math breaks down, I've tried to divide the number of molecules by the resulting area, but the answer doesn't seems quite right. :/


electron flow in capacitor with dielectric

if I make a circuit with a battery and a capacitor (with a dielectric inside), how it is possible to get a current in the circuit? If electrons go from one pole of the battery and they arrive to one plate of the capacitor they cannot flow to the other plate of the capacitor because the plates are separated by a dielectric and this dielectric doesn't have free charges, they are bound charges. How can I explain a net flow of electrons in this circuit? Thanks.


estimating ion collisions from Vlasov equations

Suppose you have a distribution function $ f_{\alpha}( \vec{r} , \vec{p} , t)$ obtained from Vlasov equations for a certain $\alpha$ species, say some ions. Is there a rigorous way (in the domain of validity of the model, that is) to estimate a collision density function $ p^{collisions}_{\alpha}( \vec{r} , E_{CM} , t)$ , where $E_{CM}$ is a variable describing collision energy parameters? A link or reference would be great


Why didn't the control rods in Fukushima shut down the reactor?

With all the hubbub over the reactors in Japan and nuclear safety, I was wondering why the insertion of the control rods in Fukushima did not shut down the reactor? Shouldn't they slow the fission reaction so far that cooling is not needed? Why do they need to continue cooling the core when I would expect no more exothermic reaction to take place?


What mechanism produced the baryon excess in the universe?

A baryon is a composite subatomic particle made up of three quarks (as distinct from mesons, which comprise one quark and one antiquark). Baryons and mesons belong to the hadron family, which are the quark-based particles


decay time for a blast wave

The blast wave that is formed as a result of an explosion has some import parameters. One of them is the peak pressure which is basically the pressure measured when the blast wave arrives at a given location. It is the case that this peak pressure decreases as one moves further and further from the center of the explosion. Ignoring the actual mathematical form, this is plausible. Now the other parameter of interest is the decay time. It is the time it takes for the peak overpressure to decay to the atmospheric pressure at a fixed point in space. One could imagine an experimentalist putting a pressure gauge at some location. When the blast wave arrives, the pressure rises to the peak pressure and after some time $t_{d}$, it decays to atmospheric pressure. This $t_{d}$ is the decay time. From my readings so far, I don't seem to have an intuitive grasp of the decay time. The books I read suggest that the decay time increases as one moves away from the source. Could any body provide me with some simple explanation as to why this might happen?


Einstein's special relativity beyond the speed of light

Could someone with access to this paper which claims to have new transformations between frames with relative motion faster than light which are supposedly consistent with special-relativity, say what the new transformations are and why they might make sense ?


Mass-Energy Equivalence

I've asked before, but I'm still confused as to what the mass energy equivalence implies. I've taken an introductory course in relativity, so I only covered special relativity. From what I gather, all things have a rest mass. As you increase your energy, your mass increases as well, and the rest mass behaves as some potential. But then when we talk about fusion and fission, we are splitting the atoms to generate energy-breaking apart the potential stored in forces between atomic components. This kind of makes sense to me. But then are _all_ energies and masses equivalent? An electron has a rest mass, is this considered energy as well? Is the mass stored in objects I hold due to the gravitational field simply too small for me to notice? Furthermore, if mass is energy, then why do nuclei weigh less? Is some of their mass going into the bonds? The fission and fusion also confuse me, both are building towards a more stable atom (32?) but somehow both paths generate energy.


Why are most metals gray/silver?

Why do most metals (iron, tin, aluminum, lead, zinc, tungsten, nickel, etc.) appear silver or gray in color? (What atomic characteristics determine the color?) What makes copper and gold have different colors?


How to show that value is conserved along geodesics?

Let's have the motion of charged particle in a field of Reissner-type black hole. The equation of motion looks like $$ \frac{d^{2}x^{\mu}}{d \tau^{2}} + \Gamma^{\mu}_{\nu \lambda}\frac{dx^{\nu}}{d \tau}\frac{dx^{\lambda}}{d \tau} = \frac{q}{m}F^{\mu}_{\quad \nu} \frac{dx^{\nu}}{d\tau}. $$ How to show that value $$ E = m\left( 1 - \frac{2M}{r} + \frac{Q^{2}}{r^{2}}\right)\frac{dt}{d\tau} + \frac{qQ}{r} $$ is conserved along geodesics, $u^{\mu}D_{\mu} E = 0$? The first step, maybe, must be the attention to find the $\frac{dt}{d\tau}$. But what to do after that? It doesn't seems like convolution $\varepsilon_{\mu}u^{\mu}$ or to $u_{\mu}u^{\mu}$, where $u^{\mu}$ is the 4-speed. So what to do next?


Can we create a magnet with only one Pole?

> **Possible Duplicate:**   >  How to make a monopole magnet? No matter how many times you cut a magnet, we always end up with 2 poles. Is there any possibility of creating a monopole magnet?


Observables still commute even if fields only anti-commute

In Peskin & Schroeder page 56, after introducing _anti_ commutation relations for the fields instead of commutation relations (in order to fix the negative energy problem as well as to have proper causality conditions stemming from particles propagating from y->x and anti-particles propagating from x->y), it is stated that even though now we have anti-commutation relations, any two observables evaluated outside of the lightcone should still commute (and not _anti_ commute) and thus our theory still preserves causality, in the sense that if two observables commute then measuring one does not affect the measurement of the other. The reason they give for this is that any observable is ultimately going to be composed out of an even number of fields. I have tried to put this claim into mathematical sense, by doing the following:   1. Assume $g(x-y,x-y)<0$ where $g$ is our metric.   2. Thus we know that $<0|\\{\Psi(x),\,\bar{\Psi}(y)\\}|0>=0$ from our previous calculations.   3. Define $O_1(x):=<0|\bar{\Psi}(x)A_1(x)\Psi(x)|0>$ and $O_2(x):=<0|\bar{\Psi}(x)A_2(x)\Psi(x)|0>$. $A_1$ and $A_2$ are two differential / spinor operators which are thus far arbitrary.   4. Try to show that $[O_1(x),\,O_2(y)]=0$, because $g(x-y,x-y)<0$. I have failed at showing 4. assuming 2. (I have tried to somehow make linear combinations of the commutator as anti-commutators of the fields and it didn't work out). My question is:   * Is this the most general formulation of the claim that one could make? If not, how would you better formulate it?   * Once we have the formulation we want, how to prove it assuming 1. thus 2.?


Otto cycle efficiency described in pressure ratio (instead of compression ratio)

The Otto cycle's thermodynamic efficiency is often described using the compression ratio: $$\eta_{th} = 1 - \left( \tfrac{1}{CR} \right ) ^ {k-1},$$ (where CR is the compression ratio, and k the ratios of specific heat). Now I tried describing this using a pressure ratio (So instead of $\tfrac{V_1}{V_2}$ using $\tfrac{p_1}{p_2}$. Taking into account the compression in a cycle is (in the ideal case) an isentropic process, one could use $$p_1V_1^k = p_2V_2^k$$ $$\tfrac{P_2}{P_1} = \left ( \tfrac{V_1}{V_2} \right ) ^ {k} = \Pi$$ Combining these I would get: $$\eta_{th} = 1 - \left( \tfrac{1}{\Pi ^ {\tfrac{k-1}{k}}} \right ) $$ However this is **exactly** the same as the brayton efficiency.. While the cycles are different (otto uses isochoric combustion, while brayton uses an isobahric expansion during combustion). I wanted to compare those two, what did go wrong? EDIT: PS why is in the mathjax "\frac" not working, yet \tfrac does?


Physics problems requiring optimization?

I would like to know if there are simple benchmark physics problems to test a Genetic Algorithm C++ library I developed. The library supports single and multi-objective constrained optimization and I would like to publish the code showing its performance in the optimization of some physics examples.


A curious case of Relativistic Velocity Addition

The relativistic velocity addition formula is $$u = \frac{v+u'}{1+ \frac{vu'}{c^2}}$$ Where $u$ = velocity of projectile seen by rest observer "A" $v$ = velocity of moving observer "B" as seen by rest observer "A" $u'$ = velocity of projectile seen by B Now the question is this: If $v=c,$ and $u'=-c $ I get an undefined answer. i.e. the relavistic velocity addition formula is undefined. Whats wrong with setting $v=c,$ and $u'=-c $?


Is there no radioactive decay between nuclear fusion and solid material formation?

I'm aware my question might be considered a duplicate of this one: Radio- dating and the age of the earth I read that one and I looked everywhere and I still can't find my piece of mind. I would really like to understand the following points:   * does one uranium nucleus start the radioactive decaying process the instant it is born inside a supernova ? (I understand that for a single nucleus the chance of decaying which is an instantaneous event, grows from 0% to 100% in the time from _the beginning_ to the half-life)   * while talking about a collection of atoms, Wikipedia states here that " _This predictability allows the relative abundances of related nuclides to be used as a clock to measure the time from the incorporation of the original nuclides into a material to the present._ "   * @JohnRennie helped me understand that chemically speaking zircon and lead do not mix whereas zircon and uranium do. So basically macroscopic contamination via accretion with flying lead is an event too rare to take into consideration right ? Also... I understand that the uranium-lead decay series has very short half-lives for every nuclide except for the beginning and the end (uranium and lead) and so, from a cosmological perspective it's quite improbable that zircon would mix with anything other than uranium, should a chemical compatibility exist with another nuclide. Please bare with me and excuse my "highly scientific" vocabulary. I have an experiment I would like to imagine in order to discover what I'm doing wrong:   1. A star goes supernova   2. We concentrate our attention on one uranium nucleus which is fused together during the violent event   3. The uranium nucleus travels for millenia towards the an accretion cloud surrounding a newborn star   4. The nucleus collides with _brand new molten zircon crystal_ and becomes part of it. Let's say that event 4. was happening after 0.99 uranium half-lives and the uranium nucleus still hadn't decayed. Isn't it imminent that the nucleus will decay pretty soon after event 4. ? I mean, I understand that if the nucleus would've become **thorium** or **protactirium** or **lead** it wouldn't've gotten stuck in the crystal for chemical reasons, but if it was still uranium what does that say about the internal clock of the nucleus which **had** been ticking for 99% of the half- life ? Is that clock reset because of external electromagnetic reasons the moment the nucleus becomes part of the crystal ? Thank you for your indulgence


A Variation on Laplace's equation (context: Yang-Mills N-Instantons, Rajaraman's book)

**Statement of the problem** I need to solve the equation \begin{align} 0 = \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi \hspace{20mm} (1) \end{align} where $\phi$ is a scalar field and we're working in Euclidean 4-space. More specific questions are asked below. Basic advice on this would also be appreciated. **Context** I'm trying to work through 't Hooft's Euclidean Yang-Mills N-instanton solution. The idea is that we make the ansatz for the gauge potentials \begin{align} A_{\mu} = i \overline{\sigma}_{\mu \nu} \partial^{\nu}(\ln \phi(x)) \end{align} where $\overline{\sigma}_{\mu \nu}$ can be thought of as a constant 4x4 matrix that is unimportant to my question. We use the self-duality of the field strength tensor that arises from this ansatz to obtain the equation (1) for $\phi$ **The Difficulties** One generally good source has been Rajaraman's "Solitons and Instantons." The relevant section is displayed ![Relevant section in Rajaraman](http://i.stack.imgur.com/mdLWX.png) My difficulties are:   1. I am assuming by "singular" Rajaraman means that the function contains singularities. Why, explicitly, does $\phi$ being non-singular imply that we we need only solve $\Box \phi = 0$? Surely if $\phi$ contains any zeros then $1/\phi$ raises issues?   2. Why does $\Box \phi = 0$ only permit the constant solution and not say $\phi(x) = a_{\mu} x^{\mu} + b$ where $a_{\mu}$ and $b$ are constants? Is this because of the form of the ansatz, or is it something I'm missing?   3. In the case where $\phi$ is allowed to be singular, why does $\phi = 1/|x|^2$ solve the equation (1) for $x \neq 0$? Surely we have \begin{align} \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi &= |x|^2 \partial_{\sigma}\partial_{\sigma} \frac{1}{|x|^2}\\\ &= |x|^2 \partial_{\sigma}\bigg(-\frac{2 x_{\sigma}}{|x|^4}\bigg)\\\ &=|x|^2\partial_{\sigma} (\frac{-2 x_{\sigma}}{|x|^4})\\\ &= |x|^2\partial_{\sigma} (\frac{-2 x_{\sigma}}{(\sum_{i} x_i^2)^2})\\\ &= |x|^2\frac{-2}{(\sum_i x_i^2)^2} + |x|^2\frac{8 x_{\sigma}^2}{(\sum_i x_i^2)^3}\\\ &=\frac{-2}{|x|^2} + \frac{8 x_{\sigma}^2}{|x|^4}\\\ &\neq 0 \end{align} ( _edit_ ) I see where I went wrong above, the last three lines should be \begin{align} &= \sum_{\sigma}(|x|^2\frac{-2}{(\sum_i x_i^2)^2}) + |x|^2\frac{8 x_{\sigma}^2}{(\sum_i x_i^2)^3}\\\ &=\frac{-8}{|x|^2} + \frac{8 |x|^2}{|x|^4}\\\ &= 0 \end{align}


Linear independence of the Covariant Derivative

What's the easiest way to show that the covariant derivative $\nabla U^{\mu}$ is linearly independent to $U^{\mu}$, which is a vector? I mean I'm assuming they are since I'm proving the second Bianchi identity and my proof requires this to be the case; and I'm quite confident that the Bianchi identity is true! So I start with the Jacobi identity $$[\nabla_a,[\nabla_b,\nabla_c]]+\text{cyclic}=0$$ I then substitute in the definition of the Riemann tensor, giving me $$\nabla_a(R^d_{\space\space\space ebc}U^e)+\text{cyclic}=0$$ which gives $$\nabla_a(R^d_{\space\space\space ebc})U^e+R^d_{\space\space\space ebc} \nabla_aU^e+(\text{cyclic in }a,b,c)\space\space=0.$$ So we have 6 expressions, from which if the second Bianchi identity is true, the vector and the covariant derivate of the vector must be linearly independent. Or am I just OK to say that as the covariant derivative is (1,1) tensor and a vector is a (1,0) tensor, the result must follow straight away?


normal ordering from anti normal ordering of creation and anhilation operators

I am working on entanglement of systems in beam splitters. I constantly come across equations of creation and annihilation operator (ladder operators) in anti normal ordering. I want those equations in normal ordered form. Does anyone know the the relation between normal ordered and anti normal ordered. ( a^m * a' ^n)= ?


derivation of noise spectral density for Poisson distribution

I am trying to follow this article online to derive the noise spectral density of a Poisson distribution. I was able to follow it until equation (18). I cannot understand how he applied Wiener-Khintchine theorem to get that. It doesn't even look like the WK theorem that I know of (link to wiki). Does anyone know how he got that? The full article is from here. ![enter image description here](http://i.stack.imgur.com/YlVF5.png)


Quantum frequency vs classical frequency and Energy dependence

I recently realized what seems to be a contradiction in what is called "frequency" for light, and would enjoy some clarification of some more educated mind on the matter, thanking you in advance for your time. At the classical level, light is, in an ideal case, a plane EM wave. The electric field $\mathbf E$ for some light frequency $\nu$ is $$ \mathbf E(t,\mathbf x) = \mathbf E_\nu \sin(2 \pi\nu t-\mathbf k \mathbf x ) $$ with $\nu=c|\mathbf k|$ and $c$ is the speed of light. The energy density of the said wave is readily found to be proportional to $|E_\nu|^2$ (magnetic field is omitted for clarity). Light is described as a wave packet of such waves. The light spectrum : Radio, visible light, UV , $\gamma$ and all others in between is classified by the value of $\nu$. In quantum mechanics (or rather QED) one readily defines the energy of a light quantum, a photon, as (having usually put all constants to 1) a frequency $\nu'$. $$ E= \nu'. $$ Now to my question. For sure jumping from photons to classical wave is no light matter (!), but at a first glance, one can reasonably assume that $\nu'$ is linked to $|\mathbf E_\nu|$ due to the energy relations. One would get something like $$ |\mathbf E_\nu|^2 = \mbox{density of photons} \times \nu'. $$ However, while discussing radioactive decay for example, photons of fixed energy i.e. $\nu'$ are often emitted. At the same time the same terminology is used. That is we say some $\gamma$ ray is emitted. In what sense is that to be understood, as it refers to $\nu$ and not $\nu'$ ? It seems to me that by definition there are NO links between $\nu$ and $\nu'$, even if one introduces wave-packets to do things more rigorously. Wishing you a good day, Frédéric Edit: Thank you for your comments. The reasoning from QED, i.e. on some state clarifies the identification of $\nu$ and $\nu'$. I stupidly only recalled the classical average energy over time (which is $\nu$ independent and thus mislead me), where one should really consider the Poynting vector. Thanks again for your quick comments.


Could the measurement problem be solved by string theory / other ToE?

A quick wikipedia search will reveal the many different approaches to the measurement problem in quantum theory. What I find strange is that they even make up all these weird hypothesis like true randomness, many worlds, bohm and other interpretations when there isn't any ToE they're working it out from. I'm just a layman so maybe my assumption is wrong, but wont a fundamental theory inevitably change quantum theory? Is really a realistic model where superpositions are nothing but lack of information on our part not likely to come from the ToE? To my layman mind it seems premature to even begin to try to answer these questions if a theory of everything isn't in place. Or is there actually a reason that people are trying to solve this now? Such as: no matter what the ToE is, quantum theory wont change, superpositon will always exist etc.?


Given temperature, composition, column density, and radial velocity, can I find the bulk flow of a gas cloud?

We've got a hot star in the middle of a gas cloud. We point a spectrometer at the star, calculating the following attributes of our line of sight at the star through the cloud:   * Total number of various elements and molecules (H, H2, C, O, N, CO, lots of ions of all of the above) along the line of sight (units are number/cm^2)   * Average temperature of the atoms in our line of sight   * Average radial velocity of the atoms in our line of sight All the radial velocities are roughly the same, so we infer that the line of sight is more or less a single coherent blob of gas. Good! Now, the star itself which illuminated the line of sight is moving about 40 km/s towards us, while the gas cloud chunk we observed is moving at around 20 km/s away. Also, from comparing the luminosity of the star to the temperature of the gas, we conclude that the gas cloud is a light year or two away from the star. Given all this information, here is your task: If we take another spectrum of the _same line of sight_ ten years from now, will it be the same blob of gas? I will tell you right away that the radial velocity will only make the gas 0.002LY closer to the star, which is way less than what we need. So, in other words, given all the above, **can you conclude anything about the transverse velocity of the cloud?** Can you tell anything about the bulk flow and determine whether it is probable that the gas in our line of sight has moved with respect to the star? My physical intuition says that you'd expect yes, since the radial velocities are so different, but I'm not totally sure - the radial velocity doesn't necessarily imply anything about the transverse velocity. Or does it?


How to compute the speed of sound in relativistic hydrodynamic?

In Weinberg: Gravitation and Cosmology chapter 2.10 (Relativistic Hydrodynamics) the speed of sound is derived as $v_s^2 = \left(\frac{\partial p}{\partial \rho}\right)$ and the equation of state is given as $p=(\gamma - 1) (\rho - nm)$ with $\gamma = 4/3$ for the case of a relativistic gas and $\gamma=5/3$ for the case of a nonrelativistic monoatomic gas ($nm$ is the rest mass density and $\rho$ the energy density). Then Weinberg computes the sound speed for the relativistic case as $v_s= \sqrt{\frac{1}{3}}$ in natural units and states that this is still safely less than unity (the speed of light). However what he doesn't show is the same calculation for the nonrelativistic case. This would yield $v_s^2 = \frac{\partial ((5/3 - 1) (\rho - nm))}{\partial \rho} = 2/3$ and so the speed of sound in the non-relativistic case would actually be $v_s =\sqrt{\frac{2}{3}}$. So the speed of sound for a non-relativistic gas is actually higher than the speed of sound for an relativistic gas!? However one usually computes the non-relativistic speed of sound with another equation of state $p=K\rho^{\gamma}$ which yields $v_s^2=\frac{p}{\rho} = K \gamma \rho^{\gamma-1} = \gamma \frac{p}{\rho}$ This is a completely different result compared to the one we get from the equation of state given by Weinberg. So is the equation of state given by Weinberg wrong? If so, what is the correct equation of state for a relativistic gas and what is the real maximum speed of sound for a relativistic gas? If not, what is wrong with my calculation of the non- relativistic sound speed based on the equation of state given by Weinberg?


Electrons in Rutherford's gold foil experiment

I have been reading a chapter in my physics text. In it there is Rutherford's gold foil experiment. There it is written that doubly charged helium particles (alpha particles) were projected onto very thin gold foil. Most of the alpha particles passed straight through, but some of the alpha particles bounced back because positive particles (protons) in the nucleus repelled them. Positive and positive always repel each other. If charge was the reason some of the alpha particles bounced back, then why didn't the electrons (negative) of the gold atoms which are much smaller then protons get attracted to the alpha particles? Positive and negative attract. I would have thought the doubly charged alpha particles would be attracted to the negative electrons.


What causes ballpoint pens to write intermittently?

After a while, a ball point pen doesn't write very well anymore. It will write for a little distance, then leave a gap, then maybe write in little streaks, then maybe write properly again. It seems to be worse with older pens, but I have observed this with new pens right out of the box too. Experiments I have done:   * Take the cartridge out and look at the amount of ink. There is still plenty.   * Inspect the ball with a jewler's loupe, no obvious damage, everything looks smooth and clean.   * Stored new pens unused tip-down to eliminate gravity slowly pulling the ink from the ball and leaving a air pocket. Some of the pens exhibit the symptom even when used the first time with the cap never removed before and stored this way for a year.   * Stuck a wire in the open end of the ink reservoir to see if maybe the end dried to a hard plug so that new ink couldn't move down as it was removed from the reservoir by writing. I have never found anything hard, and observed the same symptoms even after "stirring" the top of the reservoir with a wire a little.   * When a pen stops writing, shake it hard, like resetting a fever thermometer. That seems to help for a brief while, but so does just waiting a few seconds, so I'm not sure the shaking is relevant.   * Stored a pen ball-down in a glass of water overnight. The thought was if the ink just above the ball had dried, maybe this would re-constitute it. Some ink clearly dissolved in the water since it was colored, but once the pen was started up again there was no apparent change to the symptoms.   * Cold seems to exaggerate the symptoms, but warming to body temperature doesn't fix them. This is not just a single pen or a single model. I have bunch of different pens of different models that do this. I'm curious, what causes this? ## Added: I have done some more experimenting, and it seems Emile Jetzer was right. The cause seems to be that the ink is so viscous that new ink doesn't flow down to replace what is removed via the ball fast enough. Two experiments support this hypothesis:   1. A pen will write again after a while by just letting it sit ball-down, but the time is significantly decreased when you shake it, like you would resetting a fever thermometer.   2. Some stick pens are sealed except for a small air hole at the top. Putting lips around the top of the pen and applying pressure as if you were trying to blow into it resets the writing action quickly. Even better, I can write with such pens much longer than they would normally go by holding my mouth over the top and applying constant air pressure. So, I think the mystery is solved. Probably ink in the reservoir dries out slowly over time by losing water vapor from the top. That makes the entire ink more visous, which explains why old but unused pens also exhibit this symptom. The next experiment is to take such a pen and add a little water at the top of the ink reservoir, then let it stand for a week and see what difference that makes.


Why does material resistance increase with temperature but electrolyte internal resistance decrease with increase in temperature?

The question is pretty much in the title itself, why does the internal resistance decrease with the increase in the temperature of electrolyte? Is it related to the electrolyte bring a fluid or something?


Find acceleration along normal and tangent

A small block can move in a straight horizontal line a along AB. Flash lights from one side projects its shadow on a vertical wall which has horizontal cross section as a circle. Find tangential & normal acceleration of shadow of the block on the wall as a function of time if the velocity of the block is constant (v). ![enter image description here](http://i.stack.imgur.com/qaMRn.png)


Is there a simpler way to calculate the diminishing fringe pattern of a single edge?

A single edge diffraction pattern is different than a slit or multiple slit diffraction pattern. The pattern is not equally spaced and the fringe spacing's get smaller and smaller diminishing infinitely. The equations I have found for calculating these patterns are very complex and complicated. "Is there a simpler way to calculate the single edge diffraction pattern?" That is the question but what I'm really wondering is why the equations I use below are not used to work this out? I have found a way to calculate the diminishing single edge fringe pattern but explaining it can be quite involved especially when introducing a new or different way so I posted a link to my paper https://www.dropbox.com/s/lu5irtlxxe4hpis/Single%20Edge%20Certainty.docx?dl=0 If there is a better place to post and discuss this please let me know because I would love to talk about it. I can show that all slit diffraction patterns come from single edge diffraction patterns and calculating these unique diminishing fringe patterns is much simpler than the equations I have found anywhere else. Depending on the wavelength and distance from edge to screen one simple equation based on The Pythagorean Theorem will calculate the single edge fringe pattern infinitely and accurately. From there it can be shown how slit patterns can be constructed from overlapping single edge fringe patterns. Below is a few pages taken from my paper. If you could please take a look at the whole paper I would appreciate it and as I said would love to discuss it. Thanks Bill Alsept The following is based on 500nm light and a distance of one meter (1,000,000,000nm) edge to screen. The first time I calculated the distance (Y) (measured from the shadows edge) to the 1st bright fringe (B1). I tried adding ½ wavelength (250nm) to L to create the hypotenuse and then I calculated: Y=√((L+250nm)²-(L)²) or Y=√((1,000,000,000nm+250nm)²-(1,000,000,000nm)²) or Y=707,106.82nm But this was not the observed Y distance for the first bright fringe B1. I had nothing to compare my findings with because the equations I did find for single edge fringe patterns where extremely complicated. One day I came across this site about lunar occultation’s http://spiff.rit.edu/richmond/occult/bessel/bessel.html and discovered another way to look at single edge fringe patterns. This was the first time I had read about occultation. By chance I happened to recognize a sequence of numbers that were similar to the calculations I was doing for single edge fringe patterns and realized I was on the right track. Based On what I had learned about lunar occultation I could now calculate the distance of the fringes the moon would make on the earth. The creator of this web site had already calculated these distances so I finally had something to compare to. For my experiment all I had to do was convert (L) back to 1 meter instead of the 184,000 kilometers their experiment had used for the distance to the moon’s edge. Using Pythagorean Theorem and calculating backwards I converted the occultation’s measurements for Y and L and applied to them to my experiment and then calculated for the hypotenuse. What I discovered was that no matter what wavelength of light or distance you input (when calculating for the 1st bright fringe B1) the hypotenuse will always be L+3/4*1/2λ or L+3/8λ. Not quite 1/2λ This told me that these photons (maybe all photons) were emitted at the same phase of their amplitude cycle. This also told me that at the point of emission the photons were already part way into their cycles or more precisely 1/8 beyond the negative amplitude. That's why 3/8λ is added to (L) so that the first hypotenuse is the exact length needed to complete an oscillation just as it’s impacting the screen in a positive amplitude. The distance Y from the shadows edge to the first bright fringe is calculated as: Y=√((L+.75*.5*λ)²-(L)²) or Y=√((L+.75*.5*500)²-(L)²) or Y=√((1,000,000,000nm+187.5nm)²-(1,000,000,000nm)²) or Y=612372.46nm See Fig 12 below Figure 12


Is the liquid/solid line infinite?

Starting from the triple point, is the melting line between solid-phase and liquid-phase infinite? If not, why does it end? Because pressures are so high that classical inter-molecular interactions break down? Is this related in any way to the concept of degenerate matter? Can such pressure (at which the line ends) be easily computed?


What happens at the end of Coriolis Deflection

Consider we launch a cannonball due south from a point at 45 degrees latitude in the Northern Hemisphere (e.g the point defined with the co-ordinate system on this diagram). The cannonball travels for 5 minutes (through vacuum - ignore air resistance), experiences Coriolis deflection and drifts 'right/west' ending up at 40 degrees latitude. A few nanoseconds before it hits the ground, the rotational/tangential speed of the cannonball is slower than the tangential speed of the ground below. In order for the cannonball to stick/stay at the point of impact, its tangential speed must match the speed of the ground. What effect does this difference in speed have on the cannonball on landing/impact? Does it experience a sudden west-east acceleration due to friction? If yes, would this effect be even more pronounced in this case where the project is fired from North Pole?


Understanding the effective low-energy Lagrangian for hadrons

My course in Higgs Physics is discussing a two-nucleon low-energy effective theory of hadron interaction. With $\psi=(p,n)$, the pion is defined as $\vec{\pi}= i \bar{\psi}\vec{\tau} \gamma_5 \bar{\psi} $ and the sigma as $\sigma = \bar{\psi}\psi$ Then the Lagrangian is: $L=$ $\bar{\psi} i \partial_\mu \psi + \frac{1}{2}(\partial_\mu \vec{\pi} \partial^\mu \vec{\pi} + \partial_\mu \sigma \partial^\mu \sigma$) (kinetic) $+ g_\pi (i \bar{\psi} \gamma_5 \vec{\tau} \bar{\psi} ) \vec{\pi} + g_\pi \bar{\psi}\psi\sigma$ (nucleon-pion-interaction) $+ \frac{\lambda}{4}((\pi^2+\sigma^2)^2-f_\pi^2)^2$ (potential) First, I am surprised that the nucleon current in the nucleon-pion-interaction $ (i \bar{\psi} \gamma_5 \vec{\tau} \bar{\psi} )$ is almost the same as the definition of the pion $(i \bar{\psi}\vec{\tau} \gamma_5 \bar{\psi})$. In fact they seem to be equal since $[\tau_i,\gamma_5]=0$. ( $\tau_i$ acts on the SU(2)-space while $\gamma_5$ acts on the left-right space.) _Why are we reading the same term once as a pion and once as a nucleon current?_ Secondly, $f_\pi$ is referred to as the pion decay constant. It is supposed to be proportional to the coupling of a pion to the W-Boson. _Why is $\pi$ included into the potential in the way it is? How do I see that it is related to the W-Boson coupling?_


Reynolds number and inertial force

The Reynolds number is defined as the ratio of ´inertial´ forces to viscous forces. $$ Re = \frac{\text{Inertial Forces}}{\text{Viscous Forces}}$$ Now, viscous forces make sense to me. They are frictional shear forces that come about due to the relative motion of the different layers in a flowing fluid, resulting in different amount of friction, hence, different viscosity values. However, I am not really sure how to think about the 'inertial force'. This, to me, is somewhat of a dynamic effect since large Re numbers indicate turbulence in most cases, where there is a lot of motion, vortices and eddies. But what exactly is the inertial force and how can it be explained physically?


How can one build a multi-scale physics model of fluid flow phenomena?

I am working on a problem in Computational Fluid Dynamics, modeling multi- phase fluid flow through porous media. Though there are continuum equations to describe macroscopic flow (darcy's law, buckley leverett equations, etc), these models do not apply for heterogenous media (with transport properties). We could, however, try to use the microscopic model (lattice boltzmann, or pore network models) which would be more faithful to the dynamics of the macroscopic heterogeoneous media. But any computational simulation of this model would run too slowly to be worth it. The principals of conservation laws apply at both scales (conservation of mass, momentum, energy), but the equations that describe these laws differ at each scale. How then, can we upscale microscopic physics in a computationally efficient manner? Are there any techniques for describing microscopic phenomena at the macroscopic level without such a heavy computational cost? Is there any technique to build a continuum description at all scales of the problem?


Starting a physics Ph.D. at 60?

Decades ago I was in a physics Ph.D. program, made a hash of things, and quite rightly was dropped from the program. Since then I've had a very satisfying career as a computer programmer, and picked up an M.S. in Applied Math. I'm currently working in a bioinformatics research lab, and really enjoying myself. I think I might want to retire in five or six years, and lately I've been daydreaming about returning to my first love, and entering a physics Ph.D. program. This would be entirely for my personal satisfaction. I have no intention of seeking a further career in physics, I'd just like to spend a few years working in a physics research group. I've been out of physics too long to really know what area I'd like to work in, but I'm thinking quantum computing or quantum optics. While I wouldn't require any financial support, I recognize that taking on a Ph.D. student is still a major investment of resources for a researcher. My hope is that my years of experience in software development would be enough of an asset to make it worth their while. I'm just curious if there is anybody in the audience who's done something similar, or any faculty who can give an opinion about whether this is a realistic notion. Update: Thanks to all for your answers and comments. I think nibot's comment that the thing to do is to establish relationships with some faculty I might like to work with is the most "actionable" answer. I really appreciated all the other anecdotes and advice though.


Does the equivalence between inertial and gravitational mass imply anything about the Higgs mechanism?

For example: the role it might play in a theory of quantum gravity (ie causing space-time curvature)? I realize that inertial mass can result from binding energy alone. Has the equivalence principle been tested on elementary particles (like the electron) whose mass would be entirely due to the Higgs coupling? It seems like an obvious question, but I have never heard it discussed before.


Is there a definition of force?

Well, Newton's three laws talks about forces, but no definition is given. In truth, Newton's second law gives an idea of what **total** force is: the time change rate of momentum. But, if we have a force amongst others, this force alone isn't the time rate change of momentum, so that we cannot use Newton's second law to define force, but just to try defining total force. Knowing all of this, is there a definition of force or we take force as a primitive concept like that of time? Thanks very much in advance!


Isn't Newton's Second "Law" merely a definition of force?

Is it just me, or is Newton's Second "Law" in fact no law at all, but a mere definition of force? The concept of force does not exist prior to the statement of N2, and thus it would seem that the law is entirely analytic and does not in itself improve our knowledge about nature. Things were different if the concept of force had been introduced separatedly, and if N2 then postulated a _proportionality_ between $\mathbf F$ and $d\mathbf p/dt$. The fact that N2 is a statement of _equality_ now has the consequence that no better unit can be given for force than the traditional Newton = metre kilograms per second squared. As I see it, the only synthetic statements that could be considered part of N2 are: 1) all behaviours of and interactions between bodies may be described in terms of forces, which are applied _to_ bodies _from_ other bodies (note that this fact is not at all evident in itself); and 2) the values of the individual forces are independent of any other simultaneous forces acting from other objects. 1) now allows us to state _real_ laws about concrete interactions, like gravitation. 2) gives rise to the _superposition principle_ , while 1) and 2) together allow us to extend the concept of force from N2, which only defines the _net force_ acting on a body. The _force_ applied from a body on another body is then defined as what _would_ be the net force on the body if all other objects did not exist. In fact, it is not even evident that the concept of force is necessary for the physical science and its ability to describe the nature around us. The use of force is probably a consequence of its great practical applicability, which is in turn the result of a desire to describe nature in terms of linear differential equations. For entirely mathematical reasons, the study field of linear differential equations is rich, and thus its use in physics gives us invaluable tools that allow us to easily analyse physical systems. Work and energy are examples of concepts with properties that are again direct consequences of the mathematical framework (at least I think most of them are). However, were it not for those rich mathematical systems, it is not obvious that the mathematical framework for classical physics is even the most natural one. For example, a concept like _velocity_ (central to the force- oriented mathematical framework) seems less natural in a relativistic context, where _rapidity_ in some ways seems more in accordance with the physical system itself. So any thoughts about whether Newton's Second Law actually contains synthetic information about the physical world?


Operator norm of creation and annihilation operators

Are the creation and the annihilation operators $a(f)$ and $a^{\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked Wikipedia article.


Spinor inner products

The spinor inner product in particle physics is given by $\overline{\psi} \psi = \psi^{\dagger} \gamma_0 \psi $, where I take the convention that the zeroth gamma matrix is hermitian while the rest are anti-hermitian. This is invariant under spin group transformations, $\psi \rightarrow e^{\omega_{ab} S^{ab} }\psi$, with $\omega_{ab}$ real parameters and $S^{ab} = \frac{1}{4}[\gamma^a, \gamma^b]$. However, there is a second invariant inner product given by $\psi^{\dagger} \gamma_0 \gamma_5 \psi$, with $\gamma_5 = i\gamma_0 \gamma_1 \gamma_2 \gamma_3$. My question is, why not the other one? Is it down to the fact that there are observable differences between the two, and one is favoured by experiment?


What is the appropriate scaling for a Fermi-liquid fixed point?

I am puzzled with the RG description of Fermi liquids. Consider non- interacting spinless fermions in 2D that form a circular Fermi surface, with an energy cutoff $\Lambda\ll E_F$ ($E_F\equiv |\mathbf{k}_F|^2$ is the Fermi energy). Now, what is the appropriate transformation under which the observables (e.g. $\mathcal{G}(\mathbf{k},\omega)\,=\,\langle \psi_{\mathbf{k},\omega}\bar{\psi}_{\mathbf{k},\omega} \rangle$) remain invariant? Following Polchinski's and Shankar's RG approach, one is tempted to say, scaling $\mathbf{k}-\mathbf{k}_F$, i.e., scaling momenta measured from the Fermi surface. In the present case of a circular Fermi surface, this means scaling $r-r_F$, where $r$ is the radial coordinate in the momentum space ($(k_x,k_y)\to (r,\theta)$) and $r_F=|\mathbf{k}_F|$. I seem to recall that Shankar mentions in his paper that the fact that there is a scale, namely $k_F$, does not mean that Fermi-liquid fixed-points are not scale invariant. Strictly speaking, the scaling that maintains the Fermi surface is a non-linear scaling. Does this mean that for systems that possess a Fermi surface (not necessarily Fermi liquids) we should always cook up a non-linear scaling transformation? Is there a general recipe to deal with cases where the "zero-energy manifold" is not an isolated point?


Question about equation 2.27 from Pachos's Introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the subsection "derivation of the holonomy". On equation 2.27 he splits up the integral from the exponential of a time evolution operator into finite steps (that are infinitely small). Because they are within an exponential the infinite series becomes an infinite product. I get all of that. WHat I don't get is how the two Unitary matrices (the $U$ and the $U^{\dagger}$) come out of the exponential. This has been vexing me for a while. This is as he goes from the center series to the right-hand side product. I'm wondering if this is a mistake or not. I'm not asking for a complete solution since I want to work it out but a hint would be nice.


Naive unification of scalar QFT and GR is possible?

I am thinking on the Klein-Gordon equation with curved (non-diagonal) metrics. Is it possible? Doesn't have it some inherent contradiction? If yes, what? If no, what is this combined formula?


Lorentz force, multiple conductors?

3 conductors are placed in a magnetic field $B$, that have equal current $I$ flowing, is the Lorentz force totaled as the sum of all those conductors? Assuming that all of them have the same direction of current.


Prove scalar product is distributive

The scalar product is defined as r*s = the sum of all r*s. Using this definition, prove that r*(u+v) = r*u + r*v. Also, if r and s are vectors that depend on time, prove that the product rule for differentiation applies to r*s. Ok, so I'm new to proofs and I literally do not know where to even start. Any suggestions?


4-vector velocity to Newtonian?

The four-vector condition for a particle free of forces is: $\frac{du}{dτ} = 0$ and the equivalence of this to the statement of newton's first law follows from the expression for four-velocity: ![enter image description here](http://i.stack.imgur.com/DtUxu.png) Then I came across this part in my lecture notes: ![enter image description here](http://i.stack.imgur.com/01vnu.png) Now this got me curious. How exactly does one get to the Newtonian statement i.e. what algebra steps are involved? Thank you to anyone who can enlighten me~


Gauge symmetry for p-forms

It is well known that the Lorentz invariance of the S-matrix implies Gauge redundancy for 1-forms,'photons'. Does this argument go through to p-forms? That is does lorentz invariance of s-matrix of these fields imply that the following is a symmetry of the action? $A_{a_1 \dots a_p} \to A_{a_1 \dots a_p} + \partial_{[a_1}\lambda_{a_2 \dots a_p]} $ where $\lambda$ is a (p-1) form.


Is there a magnetic field that satisfies $\nabla \times B = 0$?

In electrostatics Maxwell's equations for the magnetic field are $\nabla \cdot B = 0$ and $\nabla \times B = \mu_0 J$ Now, take $B = xi-yj$, where $i$ and $j$ are the usual unit vectors, then one can show that $\nabla \times B = 0$ which consequently means that $J=0$. But in Maxwell's equations, isn't that $J$ supposed to be the source of the magnetic field B? Then how come $J$, the source of the magnetic field, is zero yet $B$ is not zero? What am I missing?


Experimentally measuring coherence length of laser

So I am trying to figure out the coherence length of my laser source. I am using a mach zehnder interferomter (image attached). I have theoretically calculated $L_c$ to be somewhere between 800-1200 $\mu m$ (equation on wiki page). So anyways. When I scan the movable mirror through a large range (50 mm), I observe interference fringes everywhere. I thought that outside of the coherence length, there should be no interference. Am I doing something incorrect?![Interferometer](http://i.stack.imgur.com/ULNEk.jpg) NOte: The top part of the MZI interferomter is an adjustable mirror. NOT TO SCALE.


Quantum death like heat death possible?

Quantum decoherence is an irreversible process which is the result of interaction of the system with its environment. It prevents interference due to lack of coherence. Environment acts just like a heat bath. Now my question is, is the different branches of the wave function of the universe becoming gradually more and more decoherent so that in a far future no trace of interference can occur? In other words I would like to know whether there will be any quantum death (like heat death of thermodynamics) of the universe if one waits for a sufficiently long time?


How to feel or sense magnetic fields from earth?

I know that earth has magnetic fields/forces, but why we don't feel them? So if I hold a magnet and earth's magnetic field is positive and my magnet is from the positive side, then why the magnet does still fall down? Shouldn't be there a pullsive force since they will be of the same polarity?


How stellar aberration is measured?

A simple calculation shows that stellar aberration due orbital motion of earth is roughly 20 arcseconds. My questions are:   1. Practically how this small value is measured?    2. Does this value is in the range of accuracy of a 11 inch reflective telescope with a camera?    3. And how this measurement originally was done by Bradley in the 18th century?


Regarding Non-renormalizatibility of GR

I've been doing some reading trying to get to a better understanding of some renormalization issues with the Einstein-Hilbert action. But, something odd came into mind that I'm hoping some users may have the background to comment on it. Specifically, most of the sources I have seen are based on expanding the Einstein Hilbert action about a flat background to achieve a Lagrangian of the form $(\partial h)^2+{1\over{k}}(\partial h)^2h+{1\over k^2}(\partial h)^2h^2+...$with terms that become more and more divergent the further out one goes. But, for some reason I find I'm a little pre-occupied with the notion that perhaps the particular expansion itself has issues with convergence that may or may not fundamentally have to do with the E-H action in its non expanded form. We commonly entertain the notion of a cut-off energy range to deal with the renormalization issues in the 'effective field theory' speak. How do we know that the expansion of the E-H itself isn't only valid as a low energy approximation though? Do we know that there isn't an expansion of an alternate form which has convergent terms for a range of energies where the first runs into problems?


How can airplanes fly upside down?

I've read many times, including here on this very site that the commonly known explanation of flight is wrong, and that airplanes can fly because the shape of their wings deflects air down. This makes sense, but as far as I can tell it doesn't explain upside down flight or symmetric wings. The images I've seen show an inclined wing, which forces the air to go downwards. But how can planes fly upside down then?


Static Electricity and Grounding Confusion

1. In an electroscope, you take metallic balls, and rub them with either a comb or a glass. Touching the balls with your finger is said to undo the effect of rubbing them with the comb or glass, since your finger is connected to ground, and ground is so enormous that it absorbs or releases the electrons to neutralize the metallic ball.   2. When you rub your shoes against a carpet and then touch a metallic doorknob, a shock results, since the build-up of electrons that was built up by rubbing against the carpet is released via the doorknob to ground. What I don't understand in #2 is how the buildup of electrons occurs in the first place. We've already seen in #1 that you are connected to ground, so in #2, why don't any excess electrons released from the carpet at the time of the rubbing immediately go back to ground? (I guess I'm asking a larger question: how can any rubbing of anything create a surplus or deficit of electrons if the thing being rubbed is connected to ground -- why doesn't that connection immediately discharge any imbalance of electrons created by the rubbing?)


Can the Lorentz force stabilize the Hydrogen atom?

I've recently been working on relative equilibria for some systems of particles. (ie. studying equilibrium solutions in a rotating frame. Saturn's rings for example.) This has evolved into some classical notions of atomic physics and questions of stability for Coulomb interactions there. Entertain for a few moments a classical(planetary) model for the atom. Take an isolated hydrogen atom for example. It seems to me that the $q\left(v\times B\right)$ term of the Lorentz force provides an intrinsic stabilization for the two particle system. Naively play the right hand rule game with the system and you'll see. If we assume that the electron is in a classical orbit and not radiating, then could we establish the stability of atoms with the Lorentz force? The non-radiating electron is a topic of its own and appears to be far from solved. For example, if you would like to make the statement that accelerating charges radiate, therefore...... Try to go one step further and show why this must be true in general and you imediately run into difficulties. Gyro-stabilization might be a relevant topic here. Order of Magnitude investigation: \begin{align} &\text{Proton Charge:}\qquad&q &\approx 1.602\times 10^{-19}\: C\\\\[2mm] &\text{Permeability of free space:}\qquad &\mu_0 &\approx 4\pi\times10^{-7}\frac{volt\cdot s}{amp\cdot m}\\\\[2mm] &\text{Permitivity of free space:}\qquad &\epsilon_0 &\approx 8.854\times10^{-12}\frac{farad}{m}\\\\[2mm] &\text{Electron Speed: }\qquad &v &= \alpha c\approx c/137 \approx 2.19\times10^6\frac{m}{s}\\\\[2mm] &\text{Electron Radius:(Bohr) }\qquad &r & = \frac{\hbar}{m_ec\alpha}\approx 5.29\times10^{-11}m\\\\[2mm] \end{align} Maximum Fields(Im thinking of the fields generated by the electron in a linear approximation for an infinitesimal section of the circular orbit: \begin{align} \text{Biot - Savart}\qquad{\bf B}_{\text{max}} &= \frac{\mu_0 q v}{4\pi r^2}&\approx& 12.537\:\text{T}\\\\[2mm] \text{Coloumb}\qquad{\bf E}_{\text{max}} &= \frac{q}{4\pi\epsilon_0 r^2}&\approx& 5.145\times 10^{11}\frac{N}{C} \end{align} With Force ratio: \begin{align} \frac{q(v\times{\bf B})_{max}}{q{\bf E}_{max}} =\frac{\frac{\mu_0 q^2 v^2}{4\pi r^2}}{\frac{q^2}{4\pi\epsilon_0 r^2}} = v^2\mu_0\epsilon_0 = \frac{v^2}{c^2}\approx\frac{(2.19\times10^6)^2}{(3\times10^8)^2}\approx 5.329\times10^{-5} \end{align} This shows the magnetic force is 5 orders of magnitude smaller than the electrostatic force for this crude model of the ground state hydrogen. Not sure what this conclusively says about the internal dynamics but at least I think I can foresee some problems with this force stabilizing the system. A colleague of mine who does work with NMR(nuclear magnetic resonance) informed me that the magnetic fields they work with in the lab are on the order of 3$T$ and that they do not observe the large fields I am envisoning in practice. The concept he quoted as the reasoning behind this was "orbital quenching". This appears to be a sort of averaging to zero of this field. However, it was also stated that the magnetic field that I am interested in can be measured in beam type experiments.


Why does the coriolis effect dissapear at the equator?

I'm studying from the book "Classical Mechanics" by Goldstein and from a coursebook my Professor provided me. In the coursebook, it says that "the Coriolis effect disappears at the equator (Where the rotation vector $\vec\omega$ of the Earth is horizontal)" Now here's my reasoning: Take the Coriolis term: > $-2m(\vec\omega\times\vec v_r)$ Nearby the north pole, this would cause a force pointing towards the equator if you were moving from west to east. The closer you get to the equator, the more this force starts pointing "upwards" if you were to describe the vector from the surface of the Earth.   At the equator, this vector is perpendicular to the tangent of the equator and pointing outwards.   Intuitively, this would mean that if you move from west to east on the equator you would be accelerating away from Earth. Now if my reasoning is correct, this doesn't mean the Coriolis effect disappears, only it turns into some sort of centrifugal force, but this seems weird because it's described by another term. Where did I go wrong?


Bullet: Piercing vs. Deflection

I am trying to create a simulation where a projectile (imagine a bullet) is fired at a surface of some kind. When it collides with the surface, I need to figure out what happens between the bullet and surface. > Example: > > Bullet fired at extremely sturdy object, bounces off the object. > > Bullet fired at very soft object, pierces into object. > > Bullet fired at thin sturdy object, dents the object but does not pierce it. 1) Does what I'm looking for have a name? I am having problems finding any type of equations that may relate to my examples. 2) Can somebody please explain to me what the forces are in this example and how they interact? 2b) What are the variables that I need to consider?


Coleman-Weinberg potential: resum at 2 loops?

Say we want to compute the Coleman-Weinberg potential at 2 loops. The general strategy as we know is to expand the field $\phi$ around some background classical field $\phi \rightarrow \phi_b + \phi$, and do a path integral over the quantum part of the field, $\phi$. We can retrieve the effective action by doing a path integral, something like eq.42 in this reference. There are 2 ways to do this at 1 loop, we can either evaluate a functional determinant or do the classic Coleman-Weinberg thing where we sum up all diagrams we get by inserting any number of background fields $\phi_b^2$ into the loop integral. This is eq. (56) of that same reference again. My question is, why do we not need to do this resummation over background field insertions at 2 loops? For example, in this (quite standard) reference, as well as in chapter 11 in Peskin and Schroeder, the authors seem to claim that the 2 loop contribution to the path integral are simply the "rising sun" and "figure 8" vacuum diagrams, and no summing over classical field insertions is even mentioned. What am I missing? EDIT: To give some more details, in perturbation theory, each diagram contributing to the path integral is spacial integral of some functional derivative acting on the free field path integral with a source: the loop diagram with n insertions of external field $\phi_b$ is the term: $$\left( \phi_b^2 \int dx \left( \frac {\delta}{\delta J(x)}\right)^2 \right)^n Z_0[J]$$ The 2 loop figure 8 is $$\int dx \left( \frac {\delta}{\delta J(x)}\right)^4 Z_0[J]$$ The 2 loop diagrams that it seems like the papers cited above are excluding are contributions like $$\left( \phi_b^2 \int dx \left( \frac {\delta}{\delta J(x)}\right)^2 \right)^n\int dx \left( \frac {\delta}{\delta J(x)}\right)^4 Z_0[J]$$ It seems to me that these terms will indeed arise in the exponential expansion of the interacting lagrangian, so it seems that a resummation over $n$, as in the 1 loop case, is still necessary. Where is my error?


Capacitance and energy

So I've been reading about capacitance (Specifically in capacitors) and I'm wondering if this reasoning is sound, and why/why not: C = Q/U U = Q/C We also know that: U = E/Q So: E/Q = Q/C Q^2 = EC E = Q^2/C However, if I try to use this it conflicts with some of the answers in my textbook, but I don't see where I made any mistake.


Does using batteries in a series raise the overall Wh's of the batteries used? Why/Why Not?

If one has 2 batteries of 6 Volts each with 500 mAh each, then separately the batteries should provide 6 Volts * 500 mAh = 3 Wh each or 6 Wh total. However would it be the case that if someone where to connect the two batteries in a series that they would be able to double the Wh, as now the equation would be 12 Volts * 1000 mAh = 12 Wh. Why/Why not?


Can a very thin sheet of any material float inside a liquid?

When a body is immersed in a liquid,buoyancy is the net force of all the forces acting on it. Now the forces are equivalent to those which will act on the same volume of liquid.Rightly so,but considering the chaotic motion of the molecules of the liquid,for every force acting on that particular volume of liquid,there will be another force in the opposite direction but of different magnitude.The magnitude differs because of the difference in height. Now let's consider a very very thin sheet immersed in the liquid.The opposite forces now will be of the same magnitude as difference in height is negligible and hence will cancel each other!so the only force acting will be its weight. So the thin sheet will never be balanced!So does that imply that no thin sheet will ever float inside a liquid??


Is it possible to mechanically isomerize an sp3 hybridized carbon center?

Imagine I have an sp3 hybridized carbon attached to four separate polyethylene chains. By pulling on the polyethylene chains in some manner, is it possible for me to mechanically isomerize the chiral center prior to breaking any carbon-carbon bonds? Perhaps a more realistic scenario would involve shining a laser on a compound similar to 2,2-dimethylpropane (i.e. a fully saturated carbon compound where four carbons are attached to an sp3 hybridized carbon), but with asymmetric functionalizations on the bonded carbons to allow for detection of isomerization. Just as in the previous scenario, can mechanical isomerization occur prior to breaking any carbon-carbon bonds? * * * Quoting from Georg's answer: "In "practice" there is a big problem, You would need two Laplacian demons (with two hands each) to do the experiment..." It's not clear to me that this is an impossible experiment. For example, you could attach one polyethylene chain to a surface and then try to make a covalent bond between the other relevant chain and, say, an AFM tip. If this is successful, one would then proceed in a manner similar to that reported in "How Strong is a Covalent Bond?" by Michel Grandbois et. al. (http://www.sciencemag.org/content/283/5408/1727.short), which estimates a ~4.0 nanonewton rupture force for a C-C bond (See Fig. 4) and directly measures the rupture of a single Si-C bond to be ~2.0 +/- 0.3 nanonewtons. In the case where one wants to look for mechanical isomerization of the sp3-hybridized carbon, one would look for an earlier sub-4.0 nN peak in the force vs. extension curve (or whatever critical force is experimentally measured for breaking a C-C bond), perhaps allow relaxation at some critical force to complete the isomerization process, and then compare the observed displacement with a geometrical model. Sounds ridiculously hard, but I see no fundamental reason it couldn't be done.


Simplest interferometer

I want to build simplest interferometer which should be able to measure movements down to fraction of wavelength. What is the simplest scheme for that, and what are the requirements for a laser? I have a bunch of laser diode-based ones, and I guess they might be not coherent enough... Are green DPSS ones any better?


What is this form of notation called?

$$^{14}_6C \rightarrow ^{14}_7N + E^{-} + \bar{\nu}_e$$ Just curious!


Optimal tunnel shape for travelling inside the earth

Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (independently from where both cities are actually) is a constant (about 42 minutes). It goes without saying : only gravitationnal potential and no friction. But let's upgrade on the idea. Fix the two cities : **what is the optimal shape of the tunnel** , so that the time it takes for going from one city to the other is shortest ? I assume a planar movement, where the plane contains both cities and the center of the earth. With Gauss' law, it is easy to compute that the gravitationnal potential on a mass m inside the earth at any distance $r$ from the center : $V(r) = m\alpha r^2$ where $\alpha$ is a constant. I parameterize the tunnel by a polar function $r(\theta)$ where the origin is taken at Earth's center. The problem reduces to minimizing the following functional $\mathcal{T}(\dot{r},r,\theta) = \int \limits_{\theta_1}^{\theta_2} \frac{ds}{|v|} = \int \limits_{\theta_1}^{\theta_2} d\theta \frac{\sqrt{\dot{r}^2 + r^2}}{|v|} = \sqrt{\frac{m}{2}}\int \limits_{\theta_1}^{\theta_2} d\theta \frac{\sqrt{\dot{r}^2 + r^2}}{\sqrt{E - m\alpha r^2}} \equiv \sqrt{\frac{m}{2}}\int \limits_{\theta_1}^{\theta_2} d\theta G(\dot{r},r,\theta)$ $\theta_1$ and $\theta_2$ are the angular coordinates of both cities. $\dot{r}$ is the derivative of $r$ with respect to $\theta$. With this parametrisation, the 'Hamiltonian' is conserved : $\frac{\partial G}{\partial \dot{r}}\dot{r} - G = c$ Which leads to $\frac{-r^2}{\sqrt{E-2\alpha r^2}\sqrt{\dot{r}^2+r^2}} = c$ And here I get stuck. Even if I try to play with special initial conditions, I cannot solve this equation. The easiest way is to perform separation of variables, but the integralcannot be carried out. Any ideas on how to solve this problem ? I am sure the solution is a nice well known symmetric function. I just don't know how to tackle this problem. Would you tink it'd be easier to solve this in cartesian coordinates ?


How do I write the Hamiltonian for a 3-level system?

I came across following types of three-level systems like V-system, Λ-system and 2-photon absorption ![enter image description here](http://i.stack.imgur.com/Xq1Lo.png) It seems that their Hamiltonians can be written intuitively by checking out the coupled levels and the gap between levels both in general and in the dressed picture (rotating frame approximation). Do V and Λ systems have the same Hamiltonian due to the symmetry in the level structure ? By following the analogy, I designed the Hamiltonian for 2-photon absorption process to be same as the other 2 cases but it is not so :- What are the coupled levels in the 2-photon absorption process and how can the Hamiltonian for this process be written ? How do I extend the design to N-level system -lets say for N-photon absorption in N-level system ?


Can you get a playground to swing from stationary

I feel this might be a FAQ but I would love a definitive answer. Imagine a frictionless stationary idealised child's playground swing. If you are sitting on the seat of the swing, is it possible in principle to set it into motion by simply moving your body and pulling on the chains of the swing? If so, is there a simple explanation for how you can change your center of mass without an external force acting on you?


extracting energy from cosmological expansion

This question is a more concrete reincarnation of an old question about energy conservation in GR. Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation? assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms?


Quick question on Perturbation theory - How do I evaluate this probability?

We know that hamiltonian for interaction between an electron and external field along $z$ is: $$\hat H = \frac{e\hbar B}{2m}\hat \sigma_z = \frac{\hbar \omega}{2} \hat \sigma_z $$ This has energy eigenvalues of $\pm \frac{\hbar \omega}{2} $ Time evolution operator is given by: $$ exp(i\frac{\omega}{2}\sigma_z) = \left( \begin{array}{ccc} e^{-\frac{i\omega t}{2}} & 0 \\\ 0 & e^{\frac{i\omega t}{2}} \end{array} \right) $$ At time $t=0$, the electron is in state $ \left( \begin{array}{ccc} a_0 \\\ b_0 \end{array} \right) $ and at $t>0$, the state of the electron is: $$ |\psi_{(t)}\rangle = \left( \begin{array}{ccc} a_0 e^{-\frac{i\omega t}{2}} \\\ b_0 e^{\frac{i\omega t}{2}} \end{array} \right) $$ Suppose now we have a small perturbation field in the x-y plane: $$B_1= B'\left( \begin{array}{ccc} cos (\omega t) \\\ sin (\omega t) \\\ 0 \end{array} \right) $$ Thus the perturbation is: $$V_1 = \frac{e\hbar}{m}\vec B_1 \cdot \vec S = \frac{e\hbar}{2m}B' \left( \begin{array}{ccc} 0 & e^{-i\omega t} \\\ e^{i\omega t} & 0 \end{array} \right) $$ I now want to find the probability the electron transits from the lower energy state to the higher energy state. By first order perturbation theory, $$\frac{\partial a_m^{(1)}(t)}{\partial t} = e^{\frac{i\Delta E}{\hbar}t}\frac{1}{i\hbar} \sum_n a_n^{(0)} \tilde V_{mn} \space dt $$ . Thus for this case, it is represed by this matrix: $$i\hbar \frac{\partial}{\partial t} \left( \begin{array}{ccc} a^{(1)} \\\ b^{(1)} \end{array} \right) = \left( \begin{array}{ccc} \langle-|V_1|-\rangle & \langle -|V_1|+\rangle \\\ \langle+|V_1|-\rangle & \langle+|V_1|+\rangle \end{array} \right) \left( \begin{array}{ccc} a_0 \\\ b_0 \end{array} \right) $$ Since the particle is originally in the lower state, and we want to find the chance electron jumps to higher (+) state, we want to find first order perurbation $|a_{+}^{(1)}|^2 = |b^{(1)}|^2 $ with $b_{(0)} = 0 $ $$ i\hbar \frac{\partial}{\partial t} \left( \begin{array}{ccc} a^{(1)} \\\ b^{(1)} \end{array} \right) = \frac{e\hbar B'}{2m} \left( \begin{array}{ccc} 0 & e^{-i\omega t} \\\ e^{i\omega t} & 0 \end{array} \right) \left( \begin{array}{ccc} a_0 \\\ 0 \end{array} \right) $$ Thus $b^{(1)} = \frac{e\hbar B'}{2mi\omega}a_0e^{i\omega t} $ and thus probability of jumping from lower to higher state is: $$ (b^{(1)})^2 = \left(\frac{e\hbar B'}{2m\omega}a_0\right)^2 cos^2 (\omega t) $$ Is it okay to simply take the real value? Because if I don't, it will be $|bb^*| = const.$ which is wrong, as it should oscillate with time.


A change in the mass of the moon

Would a change in the mass or radius of the moon have any effect on its speed? When asked this question in an assignment, I stated that it did not. My reasoning being that a change in the mass or radius of the moon would in no way change the acceleration or velocity of the moon. The moon would travel along the same path, the only difference being that the gravitational force being exerted by/on the Earth and Moon would be of a lesser magnitude. As the mass of the Moon decreases by ½, does the gravitational force. F=ma ½F=½m*a The original gravitational force being exerted is double that of the new gravitational force 2(G MiMii/r2)=G Mi(1/2Mii)/r2 I got this answer wrong and I am unsure why. Was I wrong in my approach? Are there other forces that I should have taken into account when coming to my conclusion? Edit:The mass of the moon simply vanishes.


How can Kaluza-Klein particles be told apart from winding modes at the LHC?

I`ve already asked this in the comments below this article http://profmattstrassler.com/articles-and-posts/some-speculative-theoretical- ideas-for-the-lhc/extra-dimensions/how-to-look-for-signs-of-extra-dimensions/ from Prof. Strassler but did not get a satisfactory answer... In the article Prof. Strassler explains that a possible sign of extra dimensions at the LHC could be the detection of Kaluza-Klein particles. The energy (or mass squared) spectrum of the corresponding particle tower would have a step size proportional to 1/r (where r is the radius of a compact extra dimension). Now I`m wondering how such a (hypothetical) KK particle candidate can be told apart from a winding mode particle (with the step size of the energy spectrum proportional to r) which is probably more easily produced depending on the size r of the extra dimension ? By the way what's the actual state of these searches for extra dimensions considering the data already available ? **Clarification** Considering an extra dimension that is much larger than the particle, the KK modes with the 1/r spectrum arise from the quantized angular momentum a particle can have due to it' movement in the extra dimension. If the extra dimension is smaller the particle can wind around the extra dimension leading to the winding mode particles. For these the energy spectrum is determined by the potential energy of stretching. The spectra of the KK and winding modes are T-dual to each other.


Is there a non-perturbative remormalization? If so, how does it work?

Is there a method to renormalize a theory without using perturbative expansions for the divergences? For example, is there a method to get masses and other renormalized quantities without using expansions and counterterms? O have heard about Lattice Gauge theory but what other solutions of examples of non perturbative renormalization (numerical or analytical ) are there?


Acceleration and Circular Motion

Lets assume that there is a force that makes our body moves in circular motion. We know that the acceleration of a body that moves in circular motion is Velocity ^ 2 / Radius . How is it possible that if we grant velocity to a body with mass M , Its acceleration is not dependent on M ?


How sharp is the direction of magnetic field of a cone magnet?

I heard recently that cone magnets can be used to focus and direct the field sharply at one particular point in a particular direction. I jus want to kno how accurate and sharp this field is? And if its so sharp that it does not influence the area surrounding that point but only straight above it, how small can the cones be fabricated and whaether we can fabricated cone electromagnets?


By how much will water resistance and upthrust slows down a fully submerged object?

Consider the following example: While running, I do work and I have a certain velocity and acceleration. When I'm in a pool (my feet are touching the bottom), I do the same work but, I have less acceleration. As more of my body is submerged I begin to be slower (doing the same work) . So I require an explanation, by how much water slows me.


Scales and Hooke's Law

Bob weighs $B$ kg and when he steps on a scale, the spring inside gets compressed by $c$ meters. Assuming the scale obeys Hooke's Law, if Alice weights $A$ kg, how much work is done compressing the spring when Alice stands on the scale? There is obviously a flat surface serving as the interface between the spring of the scale and the part of the scale Bob and Alice step on. For simplicity, I will assume that this interface has no mass, so that the only forces acting on it are the weight of the person and the spring force. When Bob is on the scale, the interface moves an amount $c$ at which point the spring force equals Bob's weight, i.e. $B = kc$, where $k$ is the spring constant. Solving for $k$ yields $k = B/c$. When Alice is on the scale, the interface moves until the spring force equals Alice's weight. If we call the displacement the interface undergoes $d$, then $A = kd = (B/c)d$, hence $d = (A/B)c$. The work done by Alice's weight to move the interface $d$ meters is $Ad$. The work done by the spring force to move the interface $d$ meters is $\int_0^d kx \cdot x \, dx=kd^3/3$. The net work on the interface is therefore $Ad - kd^3/3$. Apparently I am wrong because the answer is $Ad/2$. Where did I go wrong here?


Suggestions of a non-linear example for a small research project on numerical solution of ODEs?

I'm a first year undergrad and I'm doing a small research extension on numerically solving ODEs. I have done the main ODE course at my university, as well as physics. The second part of the project constitutes using a method (linear multistep in my case) to solve a system of our choice and writing a report about it. I spoke to my supervisor and she said that a nonlinear example would be instructive since I am interested in plasma physics in the future. Also I'd like to do something involving fluids. Problem is I have no idea where/how to find a specific example of a system modelled by a nonlinear ODE (I have found plenty of nonlinear PDE examples). So I was wondering if someone could tell me or guide me to an example, or how to find an example. I am aware that this style of question may be frowned upon, but I've been looking for a while without success (and posted on other forums) so I thought I'd give it a shot.


SUSY and the Large hadron collider

Did the LHC rule out SUSY as a solution to the hierarchy problem ? What's the common belief among theorists regarding the possibility of discovering SUSY in the 2015 run of the LHC ? Did the absence of a SUSY signal so far depress the enthusiasm that SUSY is an important element of our universe ?


Is observable universe an explanation against Olbers' paradox?

First of all, let me tell you that I'm not a physicist but rather a computer scientist with a mere interest in physics at nowhere near a professional level so feel free to close this question if it doesn't make any sense. I remember a physicist friend mentioning me about an argument about the finiteness of the universe. I have looked it up and it turned out to be Olbers' Paradox. We computer scientist like to use astronomical numbers to help us imagine the complexity of an algorithm. One of the most common one is the number of atoms in the Observable Universe (which we take as $10^{80}$) so I have a crude understanding about the observable universe concept. I had known these two for some time hence I woke up with the dilemma today. So my question is, how come it can be argued that universe is finite just because it is dark if we know that we can only observe a finite portion of it? Can't it be the case that the universe is infinite even if the sky is dark because not all the light from all the stars reach the earth? I have searched this a little bit but I think I need an explanation in simpler terms (like popular physics). A historical perspective would also be welcomed.


Path integral vs. measure on infinite dimensional space

Coming from a mathematical background, I'm trying to get a handle on the path integral formulation of quantum mechanics. According to Feynman, if you want to figure out the probability amplitude for a particle moving from one point to another, you 1) figure out the contribution from every possible path it could take, then 2) "sum up" all the contributions. Usually when you want to "sum up" an infinite number of things, you do so by putting a measure on the space of things, from which a notion of integration arises. However the function space of paths is not just infinite, it's _extremely infinite_. If the path-space has a notion of dimension, it would be infinite-dimensional (eg, viewed as a submanifold of $C([0,t] , {\mathbb R}^n))$. For any reasonable notion of distance, every ball will fail to be compact. It's hard to see how one could reasonably define a measure over a space like this - Lebesgue-like measures are certainly out. The books I've seen basically forgo defining what anything is, and instead present a method to do calculations involving "zig-zag" paths and renormalization. Apparently this gives the right answer experimentally, but it seem extremely contrived (what if you approximate the paths a different way, how do you know you will get the same answer?). Is there a more rigorous way to define Feynman path integrals in terms of function spaces and measures?


Drag/air-resistance impulse

I'm simulating snowfall in a discrete-time physics engine, using this formula for the drag force on a snowflake: $F_d=CAv^2$ (where $A$ is the flake's cross-sectional area and $C$ is a catch-all constant). I need to calculate the drag impulse over a given $dt$, and that's where I'm stuck. I've read this similar question, but it's been so long since I've dealt with differential equations that I can't follow the accepted answer well enough to adapt it for my purposes.


Not satisfied with "trick" in zeta function regularization

I am not satisfied with the explanations of the $\sum_n \log \lambda_n = - \frac{d}{ds} \sum_n \lambda_n^{-s}|_{s=0}$ "trick" used in zeta function regularization, discussed here and here, or the references therein. If someone could give a step-by-step explanation or a reference if the explananation is more complicated than I'm guessing, that would be greatly appreciated.


Can a photon exiting from a gravity well ever reach a frequency of zero / wavelength of $\infty$?

In reading another question about gravity's effects on a photon, I wondered if it were possible for a photon to ever be redshifted to zero wavelength. I know that black holes have a gravity field strong enough to keep light from escaping but what about the affect on the photon that is escaping. Say we have a single photon escaping on a path directly away from a non-spinning singularity. Also conjecture that we can release a single photon at various distances from the singularity (still on a trajectory directly away) so as to select the strength of the gravity field it must traverse. I know the distance has already been calculated (the event horizon I believe) but what of the photon itself? Is there a point where the amount of redshifting (e.g. gravitational) causes the wavelegth of the photon to become infinitely large? This answer mentions the change in potential and kinetic energy. Is it possible for all of a photon's kinetic energy be transformed to potential energy? What are the implications, both if it were possible and/or impossible? More importantly if impossible, what makes it impossible? This would beg the question, what is the smallest amount of energy possible (or quanta) for a photon? If that minima-photon attempted to escape from within a gravity field, it can't slow down so what does it do? As the wavelength increases approaching infinity, so must it's frequency decrease to approaching zero.


Stress vs Strain for mild steel

![two muppets](http://i.stack.imgur.com/mELg2.jpg) After Yield point on stress strain diagram the under curve occurs what does it mean what will happen for the mild steel at that particular time and again why the curve goes to up and reaches ultimate point i am not getting that WHY that under curve comes


Argument for quantum theoretic conformality of $\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions -Part 2

This is in continuation to what I was asking here earlier - Argument for quantum theoretic conformality of $\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions Or one can look at this paper - http://arxiv.org/pdf/0704.3740 I am mainly having in mind the $\cal{N}=2$ lagrangian that would be obtained by combining the equations 2.4, 2.5 and 2.7 of that paper.   * In that light I would like to understand the meaning of the comment made in that paper just below equation 2.9    * I have heard that one way of arguing the existence of the $\cal{N}=3$ fixed point in the flow of the matter couplings of the $\cal{N}=2$ theory is to see that for large values of the coupling the theory is effectively behaving like the Wess-Zumino model and for small values it is like pure-super-Chern-Simons theory.  What is known about the signs and the derivatives of the beat functions of these two limiting theories? Like if it known that the sign of the beta function is different for these two cases then it might suffice to argue by continuity that the beta function does go to zero somewhere in between and then that is a strong candidate to be the $\cal{N}=3$ theory.


What happens to a conducting ring when exposed to an electric field?

It might be a silly question, but one of my friends just got asked this question at an oral exam, and he could not answer it, and didn't receive the answer either (Or at least he forgot). And I've been thinking a while, and I'm not sure what would really happen. If I take, lets say a wedding ring, made from a conducting material, placed it on a table, and turned on an external electric field, what would happen ? The inside of the ring is field free, right ? So does anything even happen, and if so, is it only on the surface of the ring ?


General motion of a cone on an inclined surface

Suppose that a solid cone is placed horizontally on an inclined surface and is initially at rest. How will the cone move when it starts motion due to its weight? I know that its motion depends on the incline angle and also on the friction coefficient of the surface (as I observed by doing some experiments), but I can't establish the relation between them. Can anyone help me?


Are there irreducible tensors of half integral degree in quantum mechanics?

According to Ballentine, an irreducible tensor of degree k can be defined as a set of $2k + 1$ operators $\\{T_q^{\;\;(k)}:(-k \le q \le k)\\}$ satisfying the following commutation relations: $$ [J_\pm, T_q^{\;\;(k)}] = \hbar \left( (k \mp q)(k \pm q + 1) \right)^\frac{1}{2} T_{q \pm 1}^{\;\;(k)} $$ $$ [J_z, T_q^{\;\;(k)}] = \hbar q T_{q}^{\;\;(k)} $$ The usual examples employ scalars, vector operators and the irreducible subspaces of cartesian tensor operators. The irreducible tensors that arise from these constructs all have integral degree. Is it possible to have irreducible tensors of half integral degree, for example $k=\frac{1}{2}$? If so is there a simple example?


Value of the Hubble constant over time

There is something I don't understand about the Hubble "constant" H, as it seems to clump two concepts together that I can't quite unify in my head. On the one side, we have $$V = D H$$ which means that for a given distance D, there is a certain amount of new space created over time - and H is simply the factor that makes this relationship work. So, for example, say we have two points 1 Mpc apart, this would mean they recede at about 70 km/s away from each other (given our current approximation of H). Now the thing I can't wrap my head around is that $$T =\frac{1}{ H}$$ is also the age of the universe. Contrary to claims made on, say, Wikipedia this means that H cannot possibly have been a constant throughout the past 13 billion years, because mathematically 1/H means that H _must be_ continually shrinking as the universe ages. So if H _did_ start out as some huge value and is now shrinking over time, doesn't this mean the expansion of the universe is slowing down? Because if H is shrinking, I'll get a lower value of V today than I'll get tomorrow. Shouldn't the notation then be more like $$V = DH(t)$$ So which one is it? If 1/H is simply the solution for D=0, how can we use it as the expansion-velocity-per-unit-of-distance at the same time? What's worse, how can literature say H has probably been more or less constant forever and simultaneously assert that 1/H is the current age of the universe? What am I missing?


resistance of a flywheel to change of angular momentum

I've got a flywheel with angular momentum $\vec L$. Now I'd like to change the orientation of $\vec L$ by an angle $φ$ (in degrees). How do I calculate the angular momentum $\vec M$ for that change?


Differentiate wave speed, don't understand

The speed $v$ of some wave is $ω/k$ and I want to differentiate this with respect to $k$. Apparently this equals: $dv/dk = d(ω/k)/dk-ω/k^2$ But I don't understand why. Isn't this just saying "the derivative minus the derivative"? Should it not simply be: $-ω/k^2$?


Can a photon exhibit multiple frequencies?

Can a photon be a superposition of multiple frequency states? Kind of similar to how an electron can be a superposition of multiple spin states.


Semiconductor doping and conductivity

Why is the conductivity of Si dopped with P and As much bigger than the conductivity of Si dopped with In and P?


Do we exert more force when we walk?

When we walk, is the magnitude of the earth's force on us greater than if we were standing still?


Could a Dyson sphere destroy a star?

Freeman Dyson proposed that the power needs of an advanced civilisation would eventually require the entire energy output of a star to be collected, so that the star would end up surrounded by a dense network of satellites extracting power from the radiation. In science fiction this idea has often mutated into a solid shell that completely surrounds the star. My question is, if it were possible to build such a shall, what effect would it have on the dynamics of the star inside? I suspect the effect would be catastrophic, but I'd like more details. Here's my reasoning: Wikipedia implies that the temperature gradient between the core and outer layers of a star plays an important role in its stability. The shell around the star would reflect or re-emit a lot of the star's radiation. (I'm assuming the shell isn't composed of perfect solar collectors that would simply absorb all the radiation.) This reflected radiation would reduce the loss of heat from the outer layers of the star. This would reduce or even eliminate the temperature gradient, which I guess would cause the star to expand. My question is whether this is correct, and if so whether it would be enough to disrupt the process of fusion in the star's core. Or would there be some other, less obvious effect on the star's dynamics? A closely related question is, would stars be stable in a static or contracting universe? In this case all of space would be filled with the radiation emitted from other stars, and I'd be interested to know what effect this would have on stellar dynamics.


Ball on a slope

I have a lab report, but I can not write it in a correct way. The lab experiment was about a ball rolling on a slope, I have a height of the slope, the distance, and the time in which the ball spent to reach the end of the slope. I have data from 4 trials. My question is: How I can know the gravity is constant in all these trials, and which equation do I need to use?


A thermodynamic transformation that can be represented by a continuous quasistatic path in its state space may still be irreversible. Why?

A thermodynamic transformation that has a path (in its state space) that lies on the surface of its equation of state (e.g., $PV=NkT$) is always reversible (right?). However, if the path is a continuous quasistatic path in state space but not on the surface of equation of state, it is considered irreversible. Why is this so? In this case the gas still has a uniform thermodynamic variables (because all the intermediate states are points in the state space). Why can it not be reversed exactly along the same path?