title: Application of Ampère's law in low $Re_m$ MHD

text: Ampère's law states that $$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} \tag{1}$$ Simultaneously, from Ohm's law, we know that $$\mathbf{J}=\sigma\left(\mathbf{E}+\mathbf{u}\times\mathbf{B}\right)\tag{2}$$ When equating both currents $\mathbf{J}$, and applying Faraday's law, and knowing that the magnetic field $\mathbf{B}$ is solenoidal, one arrives at the transport equation of the magnetic field $$ \frac{\partial\mathbf{B}}{\partial t}=\nabla\times(\mathbf{u}\times\mathbf{B)}+\frac{1}{\sigma\mu_0}\nabla^2\mathbf{B}\tag{3}$$ In the low magnetic Reynolds numbers limit $Re_m=\mu_0\sigma u L \ll 1$, the externally imposed magnetic field does not change and equation $(3)$ is not explicitly solved, because the diffusional term becomes negligible. In this so-called one-way coupled magnetohydrodynamics (MHD) (valid for most liquid metals), the electric field $\mathbf{E}$ has a potential $\Psi$, which is calculated from the divergence free ($\nabla\cdot\mathbf{J}=0$) condition, using equation $(2)$, with a Lorentz force $\mathbf{J}\times\mathbf{B}$. My question is how should one look at Ampère's law in the context of one-way coupled, or low $Re_m$ MHD? Specifically considering the following two examples **Example 1** Suppose a situation with an insulating boundary (with $\mathbf{n}=\mathbf{\hat{y}}$). This means that $J_y=0$. Furthermore, the imposed magnetic field is constant in time, but varies in space. From $(1)$ $$J_y = \frac{dB_x}{dz}-\frac{dB_z}{dx},$$ which, for an arbitrary, non-uniform, steady magnetic field, would be non- zero. **Example 2** Suppose the externally imposed magnetic field $\mathbf{B}(x,y,z,t)=\mathbf{B}_0$. According to $(1)$, $\mathbf{J}=\mathbf{0}$, and thus there would be no Lorentz force? What is exactly happening at the interface, and am I applying Ampère's law correctly. What am I overlooking?


title: Outdoor Active Noise Canceling Speaker

text: I would like to understand how to create an active noise canceling system to reduce the noise coming from a street road bump. The advantage is that the area the noise is coming from is very small.


title: Electrical potential difference?

text: What is the electrical potential difference and why we have to talk about a difference and not about the electrical potential itself? What is the electrical potential difference in practical terms (can you explain it using concrete example?) Please help me with a very clear explanation. (sorry for the numerous questions)


title: Light formed by the sun?

text: This is an extract from the astrophysics chapter in my book: > Hydrogen atoms fuse to form helium. At the same time, lots of of gamma > photons and neutrinos are produced. The photons take thousands of years to > "fight" their way to the surface of the Sun, but then escape into space as > visible or near visible photons at the speed of light. I am really confused about how the sun produces light. I understand that through nuclear fusion, energy is formed, but isn't this energy in the form of thermal and gamma photons? I do not understand how visible light is emitted which is the reason to why we see the sun. Could someone please elaborate the paragraph from my book.


title: Parallax Cloud Displacement - Angular Distance from Geostationary Satellite

text: When observing clouds from a geostationary satellite, they appear displaced because of the parallax effect. The satellite looks at the clouds "at an angle" and it projects them on a point of the earth that is not exactly "under" the cloud. This image shows the Normalized Cloud Offset as a function of the Angular Distance from Nadir. ![Offset](http://i.stack.imgur.com/WY2ym.gif) $\text{Normalized Cloud Offset} \times \text{Cloud Height} = \text{Offset}$ (because of displacement) Given a geostationary satellite at a given longitude at 35,800 km over the equator, how do I calculate the Angular Distance to a generic point on the earth (given its latitude/longitude)?


title: What proportion of a star's hydrogen is consumed in its life?

text: I've heard in a lecture that a star like the sun would burn 3% of its Hydrogen before expiring. I would have thought it would be much more.


title: Mixing mild and cold water, which one to pour first?

text: Suppose for example that a person like me likes his water in-between. A bit colder than the room temperature but not very cold. If you have a water dispenser that pours rtp water and cold water, which one to pour in the cup first to ensure the temperature I'm looking for? I was just thinking of it as I was pouring water today. Could someone provide a general abstract answer, and also a scientific detailed one?


title: How literally should you take "The Higgs boson gives other particles mass"?

text: A standard phrase in popular discussions of the Higgs boson is that "it gives particles mass". To what extent is this a reasonable, pop-science, level of description of the Higgs boson and it's relationship to particles' masses? Is this phrasing completely misleading? If not, what would be the next level down in detail to try to explain to someone?


title: What is the minimum magnification needed for an instrument to see all the Messier objects?

text: I know someone that wants to buy a 60cm (1.5 inch) refractor and I thought about recommending what to look at the Messier objects. And I was wondering, can he see all these objects with that refractor ? what about with a 10x magnification binoculars ?


title: Falling into Saturn or Jupiter, would we pass through it until we hit the nucleus?

text: If we fall into Saturn or Jupiter, would we pass through it until we hit the nucleus? Or would we hit the surface and stop there?


title: Young's DS experiment and most light sources

text: The reason why two independent bulbs cannot be used to create the young's two slit interference pattern is that the phase difference between those two sources varies very rapidly and therefore, the interference pattern cannot be sustained and shifts rapidly producing general illuminance. But by using a single source and two nearby slits illuminated by the common source eliminates this problem, because the phase of the light being emitted may change as rapidly as possible, but the phase difference between the two slits is "locked" and remains constant (only geometric path difference).   * Considering the generation of light by the common incandescent bulb, it generates light as the electric energy is converted to heat and at a high enough temperature, it emits a characteristic light due to excitation and de-excitations of the constituent atoms emitting photons. All the photons which constitute the emitted light have varying phases (all of the emitted photons are not in phase). Why then does the phase relation get fixed between the slits, since it is not necessary that a single photon must simultaneously pass from both the slits (I appreciate the quantum mechanical property of a single photon to _pass through_ both the slits simultaneously)? Two different photons with an arbitrary phase might pass through not allowing the interference pattern to form.


title: Difference between B and H in magnetic fields?

text: I'm confused between them. Can someone explain the difference between them? Is $\vec H$ field ONLY relevant during magnetization or demagnetization? $\vec H$ is just that value needed to magnetize/demagnetize or what is it useful for? Is $\vec H$ only relevant to solenoids or magnets as well? Does it make sense to state a magnet has a field of $0.5T$ and a field strength $\vec H$ of $9\times10^5 A/m$?


title: Subjectivity of decoherence

text: I read that quantum decoherence is subjective, in the sense that two observers may not have the same "environment" and after each one has traced over those degrees of freedom they will end up with a different density matrix for the couple system-apparatus. I kind of understand the idea, but has an example of this calculation already been done? With the same system looked at by two different observers, and both density matrices calculated/compared.


title: Neutrino beam energy

text: Neutrino is one of the most mysterious particles in todays physics. Even when values of some parameters like for example mass associated with it are not known (or there is great range of possible values), its existence is not questioned. There is a field of research named neutrino astronomy. Because neutrino interaction with ordinary matter is very weak, sensitivity of detectors is minuscule. The only natural sources of neutrinos we could detect are sun and SN1987A supernova that exploded in year 1987. I am curious what is the total energy of neutrinos flowing to earth from the Sun? For convenience I want this value as power density [W / m2]. For comparison electromagnetic energy flowing to Earth from Sun in upper limits of atmosphere in equator areas is about 1360 W / m2.


title: Maximum speed of a water skier

text: Water skiing is a sport where an individual is pulled behind a boat or a cable ski installation on a body of water, skimming the surface. Consider an idealized case where the boat is moving at a constant velocity $\overrightarrow v_0=\text{const}$ (relative to the water), independently of the skier. The boat and skier are connected by a massless, unstretchable rope. Surface of the water is assumed to be smooth. Then, what is the maximum possible (instantaneous)speed of the skier $ v_\text{max}$ (relative to the water)? My first guess, based on intuition, is that $v_\text{max}=2v_0$. But I'm not at all sure. **Edit:** Skier's velocity may vary. Consider vector projections of velocity vectors $\overrightarrow v_1 $ and $\overrightarrow v_2 $ of the boat and skier in the direction of the rope. Because the rope is unstretchable, these projections must be equal. That means, must hold the equality: $$v_1\cos{\alpha}=v_2\cos{\beta}$$ $\alpha$ is an angle between $\overrightarrow v_1 $ and the direction of the rope   $\beta$ is an angle between $\overrightarrow v_2 $ and the direction of the rope So for example, if $\alpha<\beta$ then $v_2>v_1$. I.e. the skier is moving faster than the boat relative to the water surface.


title: Expected result of vertical Michelson-Morley Experiment

text: What would the result of a vertical variation of the Michelson-Morley experiment be? I.e., if one were to compare light traveling along a horizontal arm with light traveling along a vertical arm (perpendicular to the surface of the Earth), what would the result be? My understanding is that light traveling along each arm will both travel at c and consequently the result of such a variation of the experiment will be exactly as all of the previous (fully horizontal) versions of the experiment. Namely, that there is no directional dependence in the speed of light. Is this understanding correct?


title: value of $\omega$ in nonlinear equation

text: IN this article the author written a nonlinear equation as $$-\frac{\partial^2 \phi}{\partial t^2}+ \nabla \phi= \phi+ \sum_{k=2}^\infty g_k \phi^k$$ They have scaled the time as, $$\tau=\omega(\varepsilon) t$$ they have said in equation (8) is $\omega=1$ because of threshold and in the equation (14) they have said that $\omega_1 =0$ because ofbounded solution and absence of bounded solution no resonance occur. Can you explain the two conditions please that why is this?


title: Can you help me solve a difficult kinematics problem?

text: I've been working on this problem for a while now, but I still don't know how to solve it. It involves 2D kinematics, which aren't in themselves hard, but this one threw me for a loop. A ball is thrown straight up from the edge of the roof of a building. A second ball is dropped from the roof 1.00 seconds later (ignore air resistance). (a) If the height of the building is 20.00 meters, what must be the initial speed of the first ball if both are to hit the ground at the same time? (b) Consider the same situation, but now let the initial speed of the first ball be given and treat the height of the of the building as an unknown. What would the height of the height of the building for both balls to reach the ground at the same time? Now I know for (a) the initial position of each ball, the acceleration and velocity of the second ball (gravity), and final position of each ball, but that's it! I have no idea how to find out the velocity of the first ball. I'm pretty sure that you have to calculate the time it takes for the first ball to reach 20 meters in a second, to be dropped at the same time as the first ball, but I have no idea. And in (b), the initial speed isn't given! How do I solve that?!


title: Big Bang Physics/Cosmology

text: Anyone care trying to explain how there is supposedly no center to the universe? Quantum Holography implies a center. Any flowery language of proto atoms or cosmic eggs does as well too. Even water droplet-like fission computer simulations of inflation can mathematically assign a center point. Plainly speaking, QFT hold spherical symmetry as well. And even our everyday common intuition lets us know things inflate symmetrically. What gives?


title: Will it ever be possible to observe the cosmic neutrino background?

text: Is there any foreseeable technology which would facilitate the direct observation of the cosmic neutrino background? Would the ability to warp spacetime have any application here?


title: Physics textbooks that distinguish between laws and definitions?

text: Often when I am learning physics I start to think about whether the laws I'm learning are mere definitions or experimentally determined, and usually the textbook does not make this clear. As Thomas Kuhn writes in _The Structure of Scientific Revolutions_ , > These generalizations look like laws of nature, but their function for group > members is not often that alone. Sometimes it is: for example the Joule- > Lenz Law, $H = RI^2$. When that law was discovered, community members > already knew what $H$, $R$, and $I$ stood for, and these generalizations > simply told them something about the behavior of heat, current, and > resistance that they had not known before. But more often, as discussion > earlier in the book indicates, symbolic generalizations simultaneously > serve a second function, one that is ordinarily sharply separated in > analyses by philosophers of science. Like $f = ma$ or $I = V/R$, they > function in part as laws but also in part as definitions of some of the > symbols they deploy. Furthermore, the balance between their inseparable > legislative and definitional force shifts over time. In another context > these points would repay detailed analysis, for the nature of the > commitment to a law is very different from that of commitment to a > definition. Laws are often corrigible piecemeal, but definitions, being > tautologies, are not. For example, part of what the acceptance of Ohm's Law > demanded was a redefinition of both 'current' and 'resistance'; if those > terms had continued to mean what they had meant before, Ohm's Law could not > have been right; that is why it was so strenuously opposed as, say, the > Joule-Lenz Law was not. Another example is conservation of momentum; is momentum conserved because it is defined as a quantity that is conserved or because it has been observed to be conserved? Are there any physics textbooks (maybe for classical mechanics) that take a more axiomatic approach, clearly distinguishing definitions from laws determined by experiment?


title: Why are sine/cosine always used to describe oscillations?

text: What I am really asking is are there other functions that, like $\sin()$ and $\cos()$ are **bounded** from above and below, and **periodic**? If there are, why are they never used to describe oscillations in Physics? EDIT: Actually I have just thought of a _cycloid_ , which indeed is both bounded and periodic. Any particular reason as to why it doesn't pop up in science as much as sines / cosines?


title: Conservation of momentum collision problem

text: The problems is as follows: > A 2000kg Ford was travelling south when it collided with your 1000kg sports > car travelling west. The two badly-damaged cars became entangled in the > collision and leave a skid mark that is 20 meters long in a direction 14◦ to > the west of the original direction of travel of the Excursion. The wealthy > Excursion driver hires a high-powered lawyer who accuses you of speeding > throughthe intersection. Use your knowledge, plus the police officer’s > report of the recoil direction, the skid length, and knowledge that the > coefficient of sliding friction between the tires and road is $\mu=0.6$, to > deduce the original velocities of both cars. Were either of the cars > exceeding the 30mph speed limit? My attempt: $$\sum \vec p_{initial}=2000v_1(-\hat x)+1000v_2(-\hat y)$$ $$\mu=\frac{F_f}{F_N},F_{N,(after collision)}=3000g,F_f=\mu g(3000)$$ So then using F=ma, $$\mu g(3000)=(3000)a,a=\mu g$$ $$d=\frac 12 at^2+v_0t$$ In this case, I know that d=20meters, and $a=\mu g$. However, I do not have a $v_0$. My thought was to solve this for t, and then ssee what the cars final velocity was. After the collision, I know that the momentum is: $$\sum \vec p_{after}=3000v_{after}(-\cos(14^\circ)\hat x-\sin(14^\circ) \hat y)$$ So I get lost on how to combine all of these equations to solve for the initial velocities of the two cars.


title: Why are turns banked?

text: I have seen everywhere that where on road, on railway track when there is a turn then one side of that road or railway track is below the surface or some what down then actual surface. Any one tell me why this type of mechanism is used on turn ?


title: Torque on a Box

text: I think I'm missing something with torques. I seem to have gotten myself confused. I have a box that's centered at ( 0 , 0 , 0 ) with length ( $x$ dimension ) = 1 , width ( $y$ dimension ) = 0.25, and height ( $z$ dimension ) = 0.5. The edges are parallel to the axes. The $x$ axis is left(-) and right(+), the $y$ axis is up(+) and down(-), and the $z$ axis is into(+) and out of(-) the page. A force [ 0 , 50 , 0 ] is applied at the point ( 0 , 0 , -0.25 ). To find the torque, we would apply $\tau = r \times F$ and so $r$ is [ 0 , 0 , -0.25 ] and $F$ is [ 0 , 50 , 0 ]. And the crossproduct is [ 12.5 , 0 , 0 ], so the torque is in the $x$ direction and the box should rotate clockwise? If I were holding that box in my hand, then ( 0 , 0 , -0.25 ) would be on the side facing me. If I were capable of applying an upward force at ( 0 , 0 , -0.25 ), wouldn't the box start spinning away ( into the page ) from me - and not spin clockwise? Thanks for taking the time to read. I would really appreciate any help with this.


title: Tensor Operators

text: **Motivation.** I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. I then skimmed the Wikipedia page on tensor operators, and felt similarly dissatisfied. Here's why In these discussions, one essentially defines an indexed set of operators $T_{i_1\cdots i_k}$ to be a "cartesian" tensor operator of rank $k$ provided $$ U(R)\, T_{i_1\cdots i_k}\, U^\dagger(R) = R_{i_1}^{\phantom{i_1}j_1}\cdots R_{i_1}^{\phantom{i_1}j_1}T_{j_1\cdots j_k} $$ for each rotation $R\in\mathrm{SO}(3)$ where $U$ is some unitary representation of $\mathrm{SO}(3)$ acting on a Hilbert space (usually that of some physical system whose behavior under rotations we with to study). Similarly one defines a "spherical" tensor operator of rank $n$ as an indexed set of operators $T^{(n)}_{q}$ with $-n<q,q'<n$ for which $$ U(R)\,T_q^{(n)}\,U^\dagger(R) = \sum_{q'=-n}^n D_{q'q}^{(n)}(R)T_{q'}^{(n)} $$ where $D^{(n)}$ is the irreducible representation of $\mathrm{SO}(3)$ of dimension $n$. Based on these standard definitions, I would think that one could define something less "coordinate-dependent" and extended to representations of any group, not just $\mathrm{SO}(3)$, as follows. > **Candidate Definition**. Let a group $G$ be given. Let $U$ be a unitary > representation of $G$ on a Hilbert space $\mathcal H$, and let $\rho$ be a > representation of $G$ on a finite-dimensional, real or complex vector space > $V$. A $k$-multilinear, linear operator-valued function $T:V^k\to > \mathrm{Lin}(\mathcal H)$ is called a **tensor operator** relative to the > pair of representations $U$ and $\rho$ provided \begin{align} U(g) T(v_1, > \dots, v_k) U(g)^\dagger = T(\rho(g)v_1, \dots, \rho(g)v_k) \end{align} for > all $g\in G$ and for all $v_1, \dots, v_k\in V$. Notice that if a basis $u_1, \dots, u_N$ for $V$ is given, and if we define the components $T_{i_1,\dots i_k}$ of $T$ in this basis by \begin{align} T_{i_1 \dots i_k} = T(u_{i_1}, \dots, u_{i_k}) \end{align} and if $\rho(g)_i^{\phantom ij}$ denotes the matrix representation of $\rho(g)$ in this basis, then by using multilinearity the defining property of a tensor operator can be written as follows \begin{align} U(g) T_{i_1\cdots i_k} U^\dagger(g) = \rho(g)_{i_1}^{\phantom {i_1}j_1}\cdots \rho(g)_{i_k}^{\phantom {i_k}j_k} T_{j_1\cdots j_k} \end{align} So this definition immediately reproduces the cartesian tensor definition above if we take, $V =\mathbb R^3$, $G=\mathrm{SO}(3)$, and $\rho(R) = R$, and similarly for the spherical tensor definition if we take $V=\mathbb C^{2n+1}$, $G=\mathrm{SO}(3)$, $\rho = D^{(n)}$ and $k=1$. **Question.** Is the sort of object I just defined the "proper" formalization/generalization of the notion of tensor operators used in physics; it seems to contain the notion of tensor operator used in the physics literature? Is there any literature on the sort of object I define here? I would think that the answer would be yes since this sort of thing seems to me like a natural generalization a mathematically-minded physicist might like to study.


title: Is squared motor gearbox ratio proportional to inertia ratio?

text: I read an interesting article http://m.machinedesign.com/news/motor-sizing-made-easy It is very interesting, but I can not follow the 2nd last paragraph. I don't understand why it is true. _**Gearboxes offer a significant benefit in that they affect the inertia ratio by a factor of the gearbox ratio squared._** Please help. Thanks


title: What would earth look like if traveling at 77% the speed of light?

text: I've been having a bit of trouble wrapping my mind around this. If I were in a ship going 77% the speed of light (enough to experience reasonable time dilation) would I see earth going in fast forward? Would people on earth see my ship moving very slowly? I have this question because I was reading Physics of the Impossible (Micho Kaku) and he stated that if we were to travel this fast, we would get to a nearby star in about 10 years, but on earth it would have been many more years. This leaves me to suspect that if many years passed on earth even though I would experience time as normal, that I would see people in fast forward. Or would they be moving slowly because relative to me _they_ are moving the speed of light?


title: Is Λ-CDM and no inflaton field compatible with the observed value of the scalar spectral index?

text: From the cosmic microwave background, one can extract the scalar spectral index $n_s$. It is measured to be smaller than 1 by several standard deviations. Wikipedia says that it is a parameter of the Λ-CDM model. This is puzzling me. I thought that Λ-CDM includes a cosmological _constant_ (i.e. no inflaton field necessary), standard model matter, radiation and WIMP dark matter. How do these parameters determine $n_s$? Note: I don't want to get into the argument "Oh, but to explain the cosmological constant, we need an inflaton field", I think that's not relevant to this question. Edit: If the answer is "You're wrong, Λ-CDM is always with an inflaton and a constant Λ has been ruled out", that's a perfectly valid answer to me, if you can point at some resources.


title: Finding superpotentials and central charges in $AdS_3$

text: In text "Covariant theory of asymptotic symmetries, conservation laws and central charges" is given an example of finding central charges and superpotential (among other things). I am interested in $AdS_3$ case, since there is a lot of literature on this space-time, and I'm going to need to preform the same analysis for 4 dimensional Near-Horizon Extremal Kerr (NHEK) metric, so I would like to know how to do that in a simpler case. In a given example, on page 48, after specifying boundary conditions, they go on to find the linear part of $\delta[(1/16\pi)\sqrt{-g}(R-2\Lambda)/\delta g_{\mu\nu}]$, which is, as far as I understand it, a kind of variation of Lagrangian. The formula is given: $\mathcal{H}^{\mu\nu}[h;\bar{g}]:=\frac{\sqrt{-g}}{32\pi}\left[-h\bar{R}^{\mu\nu}+\frac{1}{2}h\bar{R}\bar{g}^{\mu\nu}+2h^{\mu\alpha}\bar{R}_\alpha^\nu+2h^{\nu\beta}\bar{R}_\beta^\mu-h^{\mu\nu}\bar{R}-h^{\alpha\beta}\bar{R}_{\alpha\beta}\bar{g}^{\mu\nu}+\bar{D}^\mu\bar{D}^\nu h+\bar{D}^\lambda\bar{D}_\lambda h^{\mu\nu}-2\bar{D}_\lambda\bar{D}^{(\mu}h^{\nu)\lambda}-\bar{g}^{\mu\nu}(\bar{D}^\lambda\bar{D}_\lambda h-\bar{D}_\lambda\bar{D}_\rho h^{\rho\lambda})+2\Lambda h^{\mu\nu}-\Lambda \bar{g}^{\mu\nu}h\right]$ I also have the background metric ($\bar{g}_{\mu\nu}$) which is that of $AdS_3$, $\bar{D}$ is covariant derivative, $h$ is trace, given by $h=\bar{g}^{\mu\nu}h_{\mu\nu}$. Ricci tensor and scalar are known. Now, what confuses me is: how did they get the results (for example $\mathcal{H}^{tt}\to\mathcal{O}(r^-{3})$)? I don't understand this, since they only give the boundary conditions as leading orders in $r$ ($h_{\mu\nu}\to \mathcal{O}(r^{m})$, $m\in\mathbb{Z}$). I can raise and lower indices, I know about summation, and I can find out what terms need to be present in this expression. But how do I preform calculation with $\mathcal{O}$ notation? I've been baffled by this every time I read similar articles. They all do these calculations, but I cannot find a single example where everything is explained in detail :\ So any help on clarifying this is welcomed. Any mathematics books that explain this or something...


title: Achieving almost the speed of light

text: I am much interested in physics. I have been caught up with one serious question related to speed of light. Hearing about Large Hadron Collider (LHC), it is said that protons in there are accelerated to almost the speed of light, i.e., $3\times10^8$ metre per second. One can be sure that the length in which the above experiment is conducted cannot exceed even $50~\text{km}$. (LHC at CERN is a $27$-kilometer ring buried deep below the countryside on the outskirts of Geneva, Switzerland.) So how can they achieve almost the speed of light in such a small place? Also, out of curiosity, I would like to know what instrument they use for measuring that speed?


title: Acceleration of bouncing ball with air resistance

text: Here is a strange test question I really don't get: > A ball falls freely from a certain height, and due to air resistance it will > reach a certain terminal velocity (assuming there is a normal gravitational > acceleration going on all the time). Then it hits the ground, and bounces > back without loss of energy. > > Question : what is the acceleration on the ball immediately after it > bounced? The answer is $2g$ (2 times gravitational acceleration), but the explanation is not included. _Note: This is a question from the Physics Olympiad!_


title: Is reparameterization invariance some kind of gauge symmetry?

text: On page 116 of this book it is said, that reparameterization invariance of the string action is analogous to the gauge invariance in electrodynamices. Whereas Maxwell's equations are symmetric under a gauge transformation which allows to describe the same $\vec{E}$ and $\vec{B}$ fields by different potentials $A_{\mu}$, reparameterization invariance makes it possible to use different grids on the world sheet to describe the same physical motion of a string. So is reparameterization invariance some kind of gauge invariance too? What would take the role of the gauge potential $A_{\mu}$ and what would the corresponding "gauge invariant" action look like?


title: Physical and dynamical components the four potential

text: I have a question regarding the four-potential and its gauge symmetry. We have a gauge freedom: $A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}\chi$      Such a transformation does not alter the EM field. Now my questions are: How do I find the gauge invariant components of $A_{\mu}$? I guess it is everything which cannot be written as a total derivative, but how do I make this explicit? Also which components are dynamical and which non-dynamical? What is the physical part of $A_{\mu}$? None of the components individually I guess, these can be altered by the gauge. But what part is physical? I realize the answers to these questions probably overlap to some degree but I am clueless how to go about finding the answers and interpreting them. Thank you!


title: Which twin produces more widgets?

text: Two twins are identical in every way, including their widget production. One twin goes on a long space trip at near the speed of light to produce widgets during the trip. When space-twin returns he has aged less than earth-twin. But, who produced more widgets?


Several bodies in the Solar System are possessed of a magnetosphere - Sol, Jupiter, Saturn, Earth to name a few. With these we have the proof of spacecraft/readings from Earth. For exoplanets being discovered lately magnetosphere is apparently reported to exist by virtue of a bow-shock observed from Earth/Earth Orbit.   * How does bow-shock constitute evidence of a magnetosphere? Does it spring from empirical evidence in the Solar System? As a corollary -   * What is the weakest magnetosphere within the Solar System that may be detected from Earth/Earth Orbit?


title: Using sinusoids to represent sound waves

text: Having some free time, I've begun a mini-project attempting to understand how music works in more detail, which has so far only involved music theory and a bit of set theory. I'm now trying to get my head around the physics of sound waves, and in particular, I'm trying to understand how any sound can be analysed as a sum of sin waves. I'm studying for a mathematics degree, so I'm familiar with Fourier transforms, trigonometric functions etc., but I've never really needed to apply them to sound waves in any detail. In this question I'm not really asking how the mathematics of Fourier transforms work, I'm more concerned with the specifics of how a sine (or cosine) function can model a sound wave. Apologies in advance for the long question, it was originally almost twice as long but I've condensed it as much as I could. I know that a sound wave is an longitudinal oscillation of particles in a medium, and that a wave might be generated by something moving back and forth repeatedly such as a guitar string, or a piece of cardboard inside a speaker. I've had a bit of trouble understanding how these oscillations are passed on, so as context to my questions I'm going to try and explain my current understanding of it (and hopefully this will make it easier to highlight where I've gone wrong). * * * **My conceptual understanding so far:** (Having read this question) I'm imagining the first oscillation of a speaker membrane (i.e. a piece of cardboard) in air, but it could be a vocal chord, a guitar string etc. The membrane imparts a certain initial velocity to the particles it hits, but this velocity is quickly absorbed by the surrounding particles of air due to inefficient collisions occurring at angles different to the direction of wave propagation. The nearby particles are pushed very close together, and due to the nature of gases/liquids, spread out, but this causes a gathering of particles somewhere further down the line, and the chain continues until the movement of particles reaches my ear. When the speaker membrane moves backwards, an area of space is freed up for the particles to move into; the particles begin to move in the reverse direction. Since this movement is not at all governed by the initial velocity imparted to the particles, the speed of sound is constant (within a given medium). If what I have said above is correct, then (as far as I can tell) to define the wavelength $\lambda$ we have to stop time for a moment, then measure the distance between two points where the particles are maximally dense. The frequency $f$ of the speaker membrane is defined to be how many times it oscillates back and forth per unit time, so that if the time taken for one oscillation is $T$, we have $f=\frac{1}{T}$. Clearly the speed of the wave will be $v=\frac{\lambda}{T}=f\lambda$. However we know $v$ is a constant, so increasing the frequency of the speaker membrane means decreasing the wavelength of the wave. Every time the membrane oscillates it sends another chain of dense particles on its way, so that the frequency of the membrane is equal to that of the wave. * * * **Question 1:** _Does there exist a velocity such that the membrane moves too quickly for the air particles?_ In order for the above model to work, the air must fill the gap left by the speaker membrane as it oscillates almost instantaneously, else the membrane will have no particles to move on its return journey. The speed of sound in air is $343$ $m/s$. If the membrane moves at say twice this speed, then it will have returned before the air particles have returned to their original positions. Obviously this puts no limit on the frequency, as you can make the velocity of the membrane arbitrarily small by reducing the distance it has to travel, but am I right in saying that this will occur? **Question 2:** _In which cases can we model sound with a sine function, and why?_ This is my main problem; I don't understand how a sine function corresponds to a sound wave in reality. Suppose $f(x)=\sin(x)$ where $x$ is the axis of wave propagation. Does $\sin(x)$ (the amplitude) represent the displacement of the speaker membrane from its equilibrium position, or does it represent air pressure (for a given moment)? I'm not quite sure how we even know that air pressure varies in accordance with a sinusoid, I can't seem to find an explanation anywhere. Having googled this quite a lot, I find most courses either just state that the sinusoid will represent air pressure or some other quantity, or worse still they neglect to comment and don't label the axes etc. Also, the function $f(x)=\sin(x)$ is not time dependent, and only represents variation in one axis. If this is the case, how can it fully model the propagation of a wave in space. When you look at wave forms on programs like audacity, they only graph time and amplitude, so a simple pure tone would be a function like $A(t)=\sin(t)$. This function is completely position independent, so I'm assuming it is created by detecting pressure variations at a point and graphing it. It doesn't fully describe the wave though! So I'm wondering what kind of function will represent a pure tone in three dimensions? Thanks for any help!


title: What experiment would prove string theory?

text: The conceptual model of multidimensional vibrating strings provides an incredibly rich and diverse "language" that almost certainly is encompassing all of known physics as a singular point somewhere within its almost unfathomably large parameter space; e.g. see What experiment would disprove string theory?. However, since in our human-sized world the fundamental organizing concept of "vibrating strings" comes to us from observing complex, multilevel interactions of condensed matter, and fields within space and time (e.g. see What are the strings in string theory made of?), another possible interpretation for this compatibility exists. Perhaps the inclusiveness of string theory emerges not from the simplicity of vibrations, but from the mathematics of vibrations being some sort of beautifully succinct emergent expression of of the many diverse physics threads that must come together to create a universe rich enough for both observers and string-like vibrations to exist within it. If so, string vibrations would be more like the outer crystalline facets of some incredibly rich and complex compound, or perhaps the mouth of Krishna: An outward expression of seemingly simplicity within which the entire universe lies hidden. So, after all of that, my question is this: If the all-inclusiveness power of string theory is _emergent_ rather than primary -- that is, if any working string vibration model _already_ contains all of known physics, even if in redundant or highly distorted form -- then shouldn't there exist experimentally tractable methods by which the quality of diverse string models and string model categories can be compared and selected? Such selections would be based on a sort of Occam's razor or signal extraction approach that seeks to minimize their complexity and maximize their fit to known physics, I would think. (I may elaborate more on that later; it's late.)


title: Liquid Nitrogen Expansion to Desired Temperature

text: I want to take liquid nitrogen in a pressurized container and spray it through a hose onto objects in a room. The container has a pressure regulator so I can adjust pressure as desired. The requirement is that a particular temperature is reached on exit from the hose. I understand that the final temperature will of course be room temperature but that is after reaching equilibrium. I desire a temperature above liquid but still very cold no where near room temp upon exiting the hose (T_desired or T_3). There would be a phase change from liquid to gas, and then a gaseous expansion to the desired temperature that can be achieved by regulating the pressure valve. What are the equations for this process, particularly if I want to know the volume of nitrogen gas emitted from the hose at the desired temperature? Note: This is for a practical application so the answer can contain simplifying assumptions, but please list them if used. Ex. Constant volume as liquid. Edit: If temp cannot be regulated by the pressure valve alone, then I would like to know what temp it is at, as well as the volume. This container is a good example: princetoncryo.com/cylinder/cryo-cyl/cryo- cyl-35l-lp.html


title: Lateral forces on earths surface

text: If a giant stone sphere was sitting on the earths surface, would it experience a later force due to the earths rotation? For example my intuition tells me that if there was a sphere a mile high outside my window I would see it moving very slowly, rolling along the ground as the earth rotated in the opposite direction. Any pointers to formulas or discussions on this topic would be highly appreciated. :)


title: Neutron star references?

text: I'm looking for contemporary reviews on neutron stars. Seems like this area is pretty active, so even reviews from five or ten years ago are somewhat lacking, though certainly not worthless. Does anyone have recommendations? Newer is better. Books are okay too, but I'd prefer one of those 70-ish page reviews you find in journals.


title: Electromagnetic induction of an uniform magnetic field

text: EDIT: Thanks to the first answer, I may see that there is differences between a giant solenoid and a completely uniform magnetic field. Therefore it would be great if one can explain to me both the case. Let's say there is an uniform magnetic field inside a very long and large solenoid. The magnetic field is slowly decreased. How can we find the electric field? From the Maxwell-Faraday Equation we have $\nabla \times E=-\frac{\partial B}{\partial t}$ and it's can be solved by a curly electric field with the value of electric field is constant (proportional to $|\frac{\partial B}{\partial t}|$) and directed to circles around a centre. But the centre is arbitrarily (one may say it's the central of the solenoid but I find it uncompelling since any centre will satisfy Maxwell equation). So how can one find the electric field in this situation?


title: Mutual Inductance of co-planar (but not concentric) loops

text: Two small wire loops, each with radius b, are placed a distance $d >> b$ apart. The loops are oriented as shown below; i.e., both loops lie in the x-y plane. Each loop carries the same current I. How would your answer change if the loops carried the same magnitude current, but were oppositely directed? ![enter image description here](http://i.stack.imgur.com/O5fTr.jpg) Should I directly use Neumann formula? How should I deal with the double integral? Why does it indicate that $d>>b$ here?


title: Alignment of a system in equilibrium and its potential energy

text: Consider a flexible free conductor which is placed near a strong long bar magnet. How will the the conductor arrange itself if current is passed through it? I asked my professor: he gave me a hint that always a system has tendency to be in a state such that its potential energy is minimum. Why is this so? PS. I feel the conductor will wrap itself around the wire. But I don't have any justification for it. Please tell me if I am correct. Also if yes, what is the reason?


title: Delta to Star/Y Conversions and vice versa in Electric Ciruits

text: We all know the basic rules for conversion of $"Delta"$ circuits to $"Star"$ circuits and vice versa. We also know that this is needed for simplification of circuits in complex cases. Can anyone please explain HOW the concept of such conversions came about? To be clearer, can anyone show the derivation of the Conversions? for both Capacitors and Resistors?


title: Velocity at certain point?

text: I seem to have misunderstood something in velocity. There is a path from point A to point B, that gap between is 100 meters. An object moves from point A to B with a velocity is 10 m/s. $V=\frac{\Delta x}{\Delta t}$ $ x = 100$ $meters$ $ t = 10$ $seconds$ Now, within that gap is **area x** , is the velocity the same knowing that $x$ now is smaller than $100$$m$? The velocity would be less correct? ![enter image description here](http://i.stack.imgur.com/IFrkE.png) Also, given the acceleration I still can find the velocity of the object correct? Does it matter what formula I should use? **_Edit_** : The object has a constant acceleration that is: $1$ $m/s^2$


title: Symmetry factor of Feynman diagram

text: What is the symmetry factor for the following Feynman diagram if we assume that the external points are held fixed?![enter image description here](http://i.stack.imgur.com/AyDGM.jpg) Please ignore the arrows in the diagram. I am referring to the second diagram on the third row of Figure 9.7, page 62, of Srednicki's book, but, in contrast to Srednicki, I want the external points to be held fixed.


title: What happens at the exact moment an electron absorbs a photon

text: For example let's have the hydrogen atom in its ground state and the electron gets excited by a photon. If we consider relativity the wave function cannot change instantly, right? The process has to be continous. What happens during this process....


title: Demo of Classical Example of a Spin 1/2 System

text: I was wondering if there are demos of spin 1/2 systems that anyone knows of. Are there any classical systems that are symmetric under a rotation of $4\pi$ and not $2\pi$?


title: Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

text: The question is about: * * * **(1) whether giving a Lagrangian is sufficient enough to (uniquely) well- define a Hamiltonianian quantum theory with a Hilbert space?** The answer should be Yes, or No. If yes, then I suppose one can write any of its state function $|\Psi \rangle$ in the Hilbert space, with a Hamiltonian $H$, satisfies: $ i \partial_t |\Psi \rangle=H |\Psi \rangle $ Then, whether the situation changes if one replacing **Lagrangian $L$** to **an action $S$** or a **path integral** or partition function **$Z$** , i.e. **(2) whether this $L$, $S$ or $Z$ to which level (uniquely) well-defines a Hamiltonianian quantum theory with a Hilbert space? Or, they cannot well- define it yet?** **(3) The even further question is whether a Lagrangian is enough to well- define Hamiltonianian quantum theory with a Hilbert space on the discrete lattice**? From this arXiv paper 1305.1045, it is obvious that even if we have a Lagrangian for the standard model, it is **NOT enough** to well-define a Hamiltonianian quantum theory with a Hilbert space on the lattice **NON- PERTURBATIVELY**. * * * [comments] It will be important to explain why this is so. I may be wrong, but I had impression reading some post comment about questions by an expert at Physics Stackexchange(Dr. Luboš Motl?) stating that knowing Lagrangian/action is everything about knowing the physical system. Sorry if I made a mistake here. But my interactions with other experts in the field of condensed matter, often stated that Lagrangian is not yet enough. The quantization, Hamiltonian and Hilbert space etc is needed. It will be nice to touch the issues of **gauge theory** , whether giving a **gauge theory Lagrangian** can **define a Hamiltonianian quantum theory with a Hilbert space?** As I know the 2+1D $Z_N$ gauge theory written as 2+1D U(1)xU(1) Chern-Simons theory is discussed in this post. In that sense givging a U(1)xU(1) Chern-Simons theory can mean $Z_N$ gauge theory (with discrete $Z_N$ symmetry) or something else(two copies of U(1) symmetry). Thanks.


title: Is it possible to extract nitrogen from the air via a centrifuge

text: I was thinking about how they have cans of compressed nitrogen for filling paint cans, and also that they sometimes will blow nitrogen in to food containers before sealing them, and that got me interested in extracting nitrogen for a nitrogen filled refrigerator. I read up on it and it appears the main way that nitrogen is extracted from the air now is via liquidification of the air and then relying on the different boiling points. I was curious why they didn't use something like a centrifuge to separate the nitrogen out. I'm guessing from what little I know of centrifuges use in nuclear refinement is that it's not energy efficient since the weights are not different enough to cause easy separation. But I was curious if that was the real reason, or if there were other forces that prevented a centrifuge from being used in this kind of gas separation.


title: ``What is life?'' by a physicist definition

text: The question is about defining **``What is life?'' in the field of Physics**. Whether there is any (insightful) way of defining ``What is life?'' from physicists. There are pioneer works, including Erwin Schrödinger's nice book _What is life?_, cf. Wikipedia. But it will be helpful to hear from someone here if there are alternate new viewpoints. Starting from as basic concept as possible. Probably involving concepts of entropy, thermodynamics, randomness, order- ness, emergence, aging/time-evolution, etc. But not necessary. Since this question has been criticized being opinion based as [on hold] (I doubt there is a way to save this issue to be non-opinion-based for a less- developed frontier land like this). Let me focus on **this sharp question.** The sharp question, I would like to know, is that, suppose only (quantum) mechanical/physical robotic/computer measurement is available, **how can one detect whether an object is being defined as life or lifeless by some physical measurement or computer algorithm?** I imagine, if answered by a quantum physicist, perhaps it will involve with the measurement on a (ground) state wavefunction of the living/lifeless object? Detect certain electronic/magnetic/response signal, etc?


title: Entangled or unentangled?

text: I got a little puzzled when thinking about two entangled fermions. Say that we have a Hilbert space in which we have two fermionic orbitals $a$ and $b$. Then the Hilbert space $H$'s dimension is just $4$, since it is spanned by \begin{align} \\{ |0\rangle, c_a^\dagger |0\rangle, c_b^\dagger |0\rangle, c_a^\dagger c_b^\dagger |0\rangle\\}, \end{align} where $c_i^\dagger$ are the fermionic operators that create a fermion in orbital $i$. Say we have a state $c_a^\dagger c_b^\dagger |0\rangle$. Then if I partition my Hilbert space into two by looking at the tensor product of the Hilbert spaces of each orbital, i.e. $H = H_a \otimes H_b$, then my state can be written as $c_a^\dagger |0\rangle_a \otimes c_b^\dagger |0\rangle_b$, from which it is obvious that this state is unentangled ($|0\rangle = |0\rangle_a \otimes |0\rangle_b$). Now I was thinking about writing the state in first quantized i.e. a wavefunction. Let $\phi_a(r), \phi_b(r)$ be the wavefunctions of the orbitals $a$ and $b$. Then \begin{align} \psi(x_1,x_2) = \langle x_1 x_2 | c_a^\dagger c_b^\dagger |0 \rangle = \phi_a(x_1) \phi_b(x_2) - \phi_a(x_1)\phi_b(x_2). \end{align} This is where I got confused. What object is $\psi(x_1,x_2)$, i.e. what Hilbert space does it belong to? What exactly are we doing when we do $\langle x_1 x_2 | c_a^\dagger c_b^\dagger |0 \rangle$? We seem to be changing/expanding our Hilbert space by taking the position representation? Written in this way, and assuming the same partition $H_a \otimes H_b$, the unentangled nature of the original state is no longer manifest. I'm not sure what the partition $H_a \otimes H_b$ even means in this context. Would that be saying $\psi(x_1, x_2) = \psi_a(x_1, x_2) \times \psi_b(x_1,x_2)$ where $\psi_i(x_1,x_2)$ is a linear combination of $\phi_i(x_1), \phi_i(x_2)$? This does not seem right to me. Regardless, now I have a state written in two different but supposedly equivalent ways, with the same partition of the Hilbert space, yet it is unentangled in one way and entangled in the other. Help?


title: Can speed be defined in the complex plane?

text: This question cropped up while I was playing with the equation for time dilation. If I set the speed to be $i$ (imaginary unit) the answer from the equation still makes sense, but does that matter if the input data doesn't make sense? Or rather does the input data make sense? Let $v=i$ and $c=2$ then $T' = T \sqrt{1 -\frac{i^2} {2^2}}$, or $T' = T \sqrt{1 -\frac{-1} {4}}$, or $T' = T \sqrt{\frac{5}{4}}$. Am I completely out of bounds here or is it something that can be explained?


title: If outside a cylindrical solenoid exist an electrical field what does that mean to the Aharonov-Bohm Effect?

text: To the question "What is the electric field outside a cylindrical solenoid when inside is turned on a magnetic field" the answer is that outside exists a electric field. Does that mean that the fringes shift in the double slit experiment with electrons could be explained with electromagnetic fields and it is not necessary (but of course possible) to explain it with quantum mechanics?


title: Where does $R/\gamma^3$ come from in relativistic E&M?

text: I am starting to read many papers on electron beam physics; more specifically its motion in a magnet i.e. curved motion. The term $R/\gamma^3$ comes up a lot, but I don't know where they are coming up with it. Any help or guidance would be very appreciated! Here is a link to a paper that uses the term (see paragraph 2 on page 2).


title: What is the most general definition of a bosonic Gaussian state?

text: I am reading this paper where the definition of the bosonic state is mentioned on page 2 here :- http://arxiv.org/pdf/0806.1625.pdf . From a general definition of any density operator in terms of Wigner functions, it reduces to a Bosonic Gaussian for Gaussian Wigner functions. Is this a more general definition of the Bosonic state or is it equivalent to $\rho=\exp(\sum_{ij}M_{ij}a_{i}^{\dagger}a_{j}) $ where $a_{i}^{\dagger}, a_{j}$ are bosonic creation and annihilation operators. It will be sufficient to get a reference that can describe the Wigner representation more explicitly and relate it to my representation- all the references I found start with the Wigner function.


title: How would wormhole-based FTL violate causality?

text: We already have an answer why physically traveling faster than light would violate causality (the clock on board our hypothetical FTL spaceship would tick backwards to some outside observers). However, would a wormhole-based FTL violate causality? In this case, never does an object actually move faster than light. If this violated causality, then how could I send back information from, let's say, 2014 Earth to 2004 Earth?


title: Does FTL signaling allow signaling in the past at the cost of arbitrary precision?

text: In a recent paper (http://link.springer.com/chapter/10.1007/978-3-642-30870-3_53#page-1), G. Szekely and P.Nemeti wrote the following. Let us consider two reference frames _A_ and _B_ initially, at event I, at rest w.r.t. each other. _B_ is then set in uniformly accelerated motion w.r.t. _A_. There would be an event _O_ on _A_ -worldline such that it would satisfy the three following properties :   1. Any event _E_ on _B_ -worldline would be simultaneous with _O_ in the inertial referentiel comoving with _B_.   2. Let us fix an event _M_ on *A-wordline that is posterior to _O_ according to _A_. If supraluminal signaling were possible, _B_ could send a signal to _A_ that would reach it between _O_ and _M_ from any point on its worldline, no matter how far in spacetime.   3. As the distance in spacetime between _A_ and _B_ becomes larger, the sender will have to compute the speed of the faster-than-light signal with greater and greater accuracy to reach _A_ between _O_ and _M_. Are these three properties (especially #3) actually verified by superluminal signals in Special Relativity?


title: Does the sun's bearing at a given time vary, and if so, how?

text: More an astronomy question, but Physics is the closest stack exchange category I could find and Google doesn't have the answer. Granted, the sun's elevation changes for a given time of day throughout the year, easily explained by the earth's axial tilt. Does it's bearing vary, by how much, and by what mechanism(s)? Unless my admittedly crude observations are flawed, I believe that the sun's bearing for a given time of day has increased since earlier in the year (my crude observations being its angle to my street at the consistent times that I leave to and return from work). Have raised the question with my cousin who is a navigator in the Royal Navy, after a couple of pints and a pitcher of Pimms, and we're now very much eager to hear an intelligent answer!


title: Can dark matter and energy be formulated as local perturbations of the metric

text: Note, my formal physics education ended over ten years ago so I may be missing some obvious piece of understanding. The relationship between space-time and matter/energy distribution is described by Einstein's field equations: $$G_{\mu\nu}=8\pi T_{\mu\nu}$$ G is a tensor that describes the geometry of space time, and T describes the distribution of matter, how it's moving, etc. When I read about Dark Matter I presume these are inserted as terms in the stress energy tensor on the right and are then interpreted as mysterious missing matter. Similarly, Dark Energy terms may be added as extra terms on the left, such as the cosmological constant, or further terms in T. My question is, then, why these cannot be captured as local perturbations to the global curvature. Ie, scale-dependent terms in the Ricci tensor? This would allow for localized gravitational lensing effects and apparent extra mass around massive objects such as galaxies and would just be a property of space-time.


title: Why does one of the extra dimensions of F-Theory have to be a temporal dimension?

text: F-Theory, as I understand it, is a realisation of Type IIB String Theory as a 12-dimensional theory in such a way that the $SL(2,\mathbb Z)$ symmetry becomes natural because Type IIB String Theory is F-Theory compactified on a torus. _Question 1:_ However, this explains only why F-Theory needs to be 12-dimensional. Why does one of these extra infinitesimal dimensions need to be temporal, though. _Question 2:_ I've also heard that F-Theory isn't exactly a 2-time theory. However, that doesn't make sense to me, as, since the metric signature needs to be $(-1,-1,1,1,1,1,1,1,1,1,1,1)$, i.e. mostly plus, with 2 temporals and 10 spatials. Or am I missing something obvious, here?


title: 3 Questions about modern Physics

text: First i'd like to apologize for both my writting skills (i'm not english) and for my physics knowledge (being them very basic and/or naive).   * With general relativity from Einstein, gravity is no longer labeled as a froce. However, i find it hard to understand what gravity is if it's not a force. I mean, if we ignore the energy of a system and we calculate the gravity field using the newtonian formula, the result would be a force since it's true that "$F=ma$" when we calculate the force with $F=mg$.  But now we know energy also affects gravity so i wonder if it wouldn't be posible to still use newtonian gravity model adapted with Einstein's famous "$E=mc^2$" and so to calculate the gravity field we would use: "$g = G(m+E/c^2)/r^2$" And to calculate the force resulting from the interation with the field and another entity (be it mass or energy) we would use: "$F = (m+E/c^2)g$".   * My other grip with modern physics is string theory. Every other theory untill now could be proven to be true or false sooner or latter, however, string theory is just a mathematical model that requires our universe to have 10 (or 11) dimensions from witch we can only appreciate 3 of them. Will this theory ever be proven to be true or false? Or will we remain playing by the pseudo-science rules?   * Higgs Boson. Why do we need the hadron colisoner to detect something that's everywhere.... I mean, if mass comes from the interation of the quarks with the higgs boson, then, my body is in contact with the higgs boson.


title: Compressilibility and the form of Newton's second law in fluid mechanics

text: In deriving Euler's equations for fluid mechanics, in particular $$f=\rho \partial_t v +\rho v\cdot \nabla v$$ for some body force $f$ (e.g Landau & Lifschitz 2.3) one assumes the continuity equation $\partial_t\rho=-\nabla\cdot\left(\rho v\right)$ and Newton's second law in the form $F=ma$ so $$f=\rho D_t v$$ where $D_t=\partial_t+v\cdot \nabla$ is the total time derivative (e.g Landau & Lifschitz 2.1). If one instead uses $F=\dot p$ so that $$f = D_t\left(\rho v\right)$$ then in order to get to get our initial equation back one must assume $D_t \rho=0$ but of course $D_t \rho=\partial_t\rho+v\cdot \nabla\rho=\nabla\cdot\rho$ and so we have assumed the flow is incompressible. Landau implies that he has not assumed incompressibility at this point so how is it that one can choose the first form for Newton's second law without loss of generality? _(If your answer would appeal to stress-energy tensors or Navier-Stokes etc., if possible please show how the assumption isn't implicitly made.)_ To be more clear, as I have shown above, if you say that $f = D_t\left(\rho v\right)$ and also that $f=\rho \partial_t v +\rho v\cdot \nabla v$ then it follows that the flow is incompressible. I don't think that this does imply incompressibility so why is the assumption false?


title: Potential energy in $E_f^2=(mc^2)^2+(pc)^2$?

text: Let's consider $$E_f^2=(mc^2)^2+(pc)^2$$ where the $mc^2$ is the rest energy due to the rest mass -- in Finnish "lepomassa". $$ \sqrt{(mc^2)^2+(pc)^2} - mc^2~=~(\gamma-1)mc^2$$ is the kinetic energy due to the movement because of momentum $p=\gamma mv$. Now where is potential energy if $E_f=\gamma mc^2$ is the total energy?


title: How did Goddard's rocket reach 41 feet in under 2.5 seconds?

text: I'm working on a simple rocket flight simulation, and am trying to reproduce aspects of Goddard's first successful rocket flight. My sources put the empty weight at 5.75-6lbs, loaded weight 10.25-10.4 lbs, and thrust 9 lbs (40N), using a fuel-rich gasoline/LOX propellant with Isp of about 150. The rocket is supposed to have reached an altitude of 41' (12.5m) and turned to cover 184' (56m) horizontally in a 2.5 sec flight after burning 20 seconds without leaving the launch stand. Fully fueled, the rocket didn't reach 1:1 thrust-to- weight, so couldn't lift off; my calculations agree that 20-25 seconds of fuel consumption is about where the weight would have dropped to 9 lbs.; so far so good. However, in order to reach 12.5m altitude in no more than 2.5 seconds, the rocket would have to accelerate at about 4m/s on average (d = 1/2 at²), implying TWR of 1.4; this thrust could be achieved only very briefly right before fuel-out, and that would leave no time at all for the fall (actually a powered dive!) from that peak altitude. If we take the widely reported 41' altitude as counting the rocket's 10' height, that still leaves us with a 9.5m apogee to account for. My simulation doesn't yet include rotation off the vertical, so my final apogee is much higher (Goddard's rocket had another 40 or 50 seconds of propellant available after liftoff; it flipped over and crashed quite quickly), but 2.5 seconds after liftoff I see only 0.04m of vertical travel, nowhere near the reported figure. Am I overlooking something obvious? My model includes drag, but at the low speeds in the first seconds of travel, it's negligible.


title: Store up cheap electricity to use later?

text: My utility company charges me more for electricity in the summer. Is there any way I could "save up" non-summer electricity (eg, a giant battery?) and then use it during the summer, saving myself money? Other power companies charge more for electricity during "peak hours" daily, so even saving electricity for 16 hours or so could be useful. I realize that it's probably not possible to build such a device/system in practice right now, but is this at least theoretically possible? How much more would peak/summer electricity have to cost to make this useful?


title: How does nuclear war look like from space?

text: As today's the 29th anniversary of possible Armageddon, had Mutual Assured Destruction taken place, with the USA and USSR exchanging most of their nuclear arsenal, what would someone   1. In low earth orbit   2. On the moon   3. On Mars observe with the naked eye, given the right conditions e.g. they are on the surface of the moon/Mars facing Earth, both   * During the exchange, and   * After the exchange (compared to before it)? Would only detonations be visible, or the launch and trajectory of the rockets too? Presumably a war would span the entire northern hemisphere, so some parts will be in daylight and some at night. According to this thread and _The Martian Chronicles_ by Ray Bradbury, Earth may appear to twinkle from Mars. If there were aliens equipped with only technology we have today, how near must they be to detect that something unusual was happening (ignoring the speed of light)?


title: 4D instantons and the moduli space of N=2 on R^3 x S^1

text: I am reading the paper arXiv:0807.4723 by Gaiotto, Moore, and Neitzke on wall- crossing. I would like to understand whether if the Darboux coordinates in the mutually non-local case contain the information on the corrections to the N=2 Seiberg-Witten prepotential due to 4D instantons or not. If they do, does anyone have an understanding of why this is the case?


title: Ocean waves, and surfing

text: If waves only move up and down, why is it that in the ocean you can move forward along a wave? I suppose it's a bit different then a normal wave because you have the action of the crest falling down, but I don't know how that happens either. Where is the forward momentum coming from?


title: Why do we use spinors for describing fermions?

text: I.e., what properties of the spinors gives us a reason for using them for describing of wavefunctions of fermions?


title: Projectile motion maximum distance to origin

text: Is there an elegant or easy way to derive the maximum distance from the origin to a point of the parabola created by projectile motion (assume initial position is the origin)? Other than differentiating with respect to $x$ and equate to zero the distance given by $\sqrt{x^{2}+y^{2}}$?


title: Potential energy graphs of chemical systems

text: ![enter image description here](http://i.stack.imgur.com/ggTmN.png) I really wish someone could dissect this for me in the language of physics. The system is that of two atoms that approximate each other along 'r' (or distance themselves apart). My concerns are these:   * What does negative potential energy mean here?   * Where is the "zero" reference defined?   * Why is there reference to repulsion and attraction in the PE graph?   * If $F=-dU/dr$, shouldn't the graph of the attractive force be flipped upside down? (Specifically they say the Energy of attraction is $-1.436/r$ and the energy of repulsion is $7.32x10^-6/r^8$.)   * My textbook makes reference to $U=\int F\,dr$. Why isn't it defined as the negative work in this system?


title: Pair annihilation - how to generaly solve these types of problems?

text: Simple question regarding pair production: Can a collision of an electron with a positron result in two photons? I have this problem to solve and I doubt this is even possible. Please provide some theoretical background (basics for start and if I need more explanation I will say so). * * * EDIT: After getting confirmation from @Michael Brown i decided to extend this question. I am new to solving anihilation problems and i want to know:   * what equations can i apply here and   * how do i generaly solve an example where an positron with known kinetic energy colides with a stationary electron. Is there a general aproach? * * * EDIT: I added this problem as requested from @leongz > Positron with a kinetic energy $20MeV$ colides head on with a stationary > electron. What are the energies of emerged photons if first one is headed in > an original direction and second one in an oposite direction? At what angle > from original direction would photon fly if they had same energies? What is > that energy? This is my first problem and i have had a very bad explaination - our professor was really really bad. So i need you to be descriptive along the way as you solve this problem. _I didn't want this to become a homework question at first but used the oppurtunity (i added the tag)._


title: Mixed symmetrization and antisymmetrization / Combinatorics

text: I have the following sum of 10 terms: $$ \delta^{ab}f^{cde} + \delta^{ac}f^{bde} + \delta^{ad}f^{bce} + \delta^{ae}f^{bcd} + \delta^{bc}f^{ade} + \delta^{bd}f^{ace} + \delta^{be}f^{acd} + \delta^{cd}f^{abe} + \delta^{ce}f^{abd} + \delta^{de}f^{abc} $$ In other words I consider all permutations of 5 indices, but only use those for which the first two indices and the last three are ordered (at the same time). On top op that, $\delta$ is symmetric and $f$ is fully antisymmetric. What I am looking for is some short-hand notation which would evaluate exactly to this sum. Consider the following sum as an easy example: $$ \delta^{ab}\delta^{cd} + \delta^{ac}\delta^{bd} + \delta^{ad}\delta^{bc} = 3\,\delta^{(ab}\delta^{cd)} $$ Normally $\delta^{(ab}\delta^{cd)}$ would evaluate to 24 terms, but because of the symmetry property of $\delta$, these simplify to three. I am looking for a similar notation for the first sum. Because $\delta$ is symmetric and $f$ antisymmetric, one has $\delta^{(ab}f^{cde)}=0$ and $\delta^{[ab}f^{cde]}=0$, so these don't fit. And $\delta^{(ab)}f^{[cde]}$ is incorrect as it doesn't mix the two sets of indices. I came up with some kind of "mixed symmetrization": $$\delta^{(ab\,|}f^{cde]} $$ where I defined: $$ \begin{align} T^{(a_1 \cdots a_m \, |\, a_{m+1} \cdots a_n]} &= \text{sum of all } n! \text{ permutations, where each permutation gets a sign depending} \\\ &\text{ on the number of permutations needed to put } \mathcal{P}\left(a_{m+1} \cdots a_n \right) \text{ in canonical} \\\ &\text{ order.} \end{align} $$ This indeed evaluates (up to a factor 10) to the first sum, but it feels a bit awkward to introduce a notation that is not generally usable (and for which properties have to re-derived). As these kind of "ordered" sums are for sure not uncommon, I expect them to be treated in some corner of combination theories.. Does anybody know whether such 'mixed symmetrisation" already exists in literature? Or even better, does anybody know of a simple way to rewrite the first sum, maybe in some combinatorics notation? Many thanks in advance!


title: Electric field at the centre of a charged disc of uniform charge density

text: The electric field at any point at a finite distance z from the centre of a charged disc of uniform charge density along the axis of the disc is given by the equation:   ![equation](http://phy214uhart.wikispaces.com/file/view/22_21.gif/58087210/22_21.gif) According to this equation, at z=0, the electric field is : ![electric field](http://phy214uhart.wikispaces.com/file/view/22_22.gif/58087212/22_22.gif) However, logically, I feel that at the centre of the disc, the field must be zero as all the field vectors get cancelled at that point. Why does this happen?


title: Identity of Operator Product Expansion (OPE)

text: I have one more s****d question in Polchinski's string theory book, Eqs. (2.3.14a) $$ j^{\mu}(z) :e^{ik \cdot X(0,0)}:~ \sim~ \frac{k^{\mu}}{2 z} :e^{ik \cdot X(0,0)}:,$$ where $j^{\mu}_a =\frac{i}{\alpha'} \partial_a X^{\mu}$, $::$ is normal ordered, defined as $$:X^{\mu}(z,\bar{z}): = X^{\mu} (z,\bar{z})$$ $$:X^{\mu}(z_1,\bar{z}_1) X^{\nu}(z_2,\bar{z}_2): = X^{\mu}(z_1,\bar{z}_1) X^{\nu}(z_2,\bar{z}_2) + \frac{\alpha'}{2} \eta^{\mu \nu} \ln |z_{12}|^2 $$. $\sim$ means equal up to nonsingular terms. I thought I have derived it in analogy of integration by part, but it turns out that not as I thought. Actually how to derive Eq. (2.3.14a)? Eq.(2.3.14b) will be expected in analogous...


title: How can I determine (measure?) if spraying my A/C condenser with water will make it operate more efficiently?

text: My brother had an idea for a DIY project and I am curious if the science works out well enough to try it. The idea is to collect rainwater then mist it onto the air conditioners condenser coils (the hot side). The hypothesis is that the water mist will provide additional cooling because water is a better conductor of heat and that evaporating water has a cooling effect. My question for Physics.SE is how can I work this out on paper first to determine if its remotely useful? I think what I need is to understand how to compute heat transfer from one medium to another. That would let me compare the difference between the coil to air and to water. I suppose the simpler solution is to actually build this contraption and measure how long the A/C runs during a given time period and noting both the indoor and outdoor temperatures.


title: Are there bounds or measurements on the derivative of acceleration (jerk)

text: The title says it all. Is there a physical maximum value to the 3rd derivative of position? Common Lore says that there is not and that jerk does not play any role in physics. My guess is that there is not as well. But my question is looking to stir the pot a bit. Have there been experiments on this? Has anyone tried to detect perturbations to Newtonian kinematics when high values of the 3rd derivative are involved?


title: Function describing an adiabatic process

text: I understand that an isothermic process, plotted on a PV diagram, can be described by a rectangular hyperbola. However, now I am wondering, what about an adiabatic process, what function best describes it?


title: Integrating factor $1/T$ in 2nd Law of Thermodynamics

text: How would you prove that $1/T$ is the most suitable integrating factor to transform $\delta Q$ to an exact differential in the second law of thermodynamics: $$dS = \frac{\delta Q}{T}$$ Where $dS$ is the change in entropy, $\delta Q$ is the change in heat energy, and $T$ is the equilibrium temperature.


title: preparing qutrit mixed state

text: I have a problem: Ok so I want to create a superposition qutrit state, like this: $$ |\Psi\rangle=\frac{1}{\sqrt{3}}(|0\rangle+|1\rangle+|-1\rangle) $$ is it possible to go there, starting with an initial state of $|0\rangle$ and appropriate $\pi/2$ (or any other pulses like $\pi$ $3\pi/2$ or whatever) pulses? Is this impossible?


title: Is single photon perfectly monochromatic?

text: Now, we do have equipment to generate single photon at a time, and LASERs are nearly monochromatic. While typing the question, am realizing that successive photons in case of single photon generation may have slightly different wavelengths, but are its single photons essentially monochromatic? Because we may be able to use the phenomena to make highly coherent lasers.


title: Asymmetry of induced e.m.f due to a bar magnet falling through a coil

text: It is well known that as a bar magnet falls freely through a coil, an electromotive force is induced in the coil by Lenz's Law. However, it appears that the peak e.m.f induced (ignoring the direction) is higher after the passing through a coil, than before it passes through the coil. Why so? My explanation was that after passing through the coil, gravity causes it to accelerate after the coil in the same direction in which the magnetic flux linkage through the coil was changing. By Faraday's Law, e.m.f induced will be greater. However, under this hypothesis, gravity would also have the same effect on the bar magnet before it passes through the coil. Is there any theory that can explain this asymmetry well?


title: Hot water freezing faster than cold water

text: This question has puzzled me for a long time. There is already a question like this on Physics.SE. John's answer to the question seems quite satisfying. But when I googled the cause I found this and this explanation. They maybe wrong but I think How Stuff Works is a reliable source. And here's the original paper. I am quite confused now reading the different explanations. Can anyone please shed some light on the issue?


title: Capillary action meniscus height in a tube fitted inside another tube?

text: I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved). Height of liquid column: ![enter image description here](http://i.stack.imgur.com/P5s6S.png) where: $\gamma$ = liquid-air surface tension $\theta$ = contact angle $\rho$ = density of liquid $g$ = gravity acceleration $r$ = radius I tried my best to draw the examples I'm interested in order to help my explanation. I didn't consider the capillarity inside the smaller tube in both example #2 and #3 because I'd like to assume that $a/2$ in example #1 is close to $c$ in example #2 and #3 (drawings not to scale). ![enter image description here](http://i.stack.imgur.com/Au1cp.png) Since from what I understand the column height is given, among other things (most of which can't be changed, like liquid-air surface tension, contact angle, density of liquid and gravity acceleration), by the tube radius, I'd like to know if $c$ in example #2 can be considered as $a/2$ in example #1 to calculate column height using above formula. Also I'd like to know how having beads of slightly smaller diameter than $c$ between the two tubes (example #3) would affect the column height. If said beads were less dense than water, could they still improve column height or would they just form a floating mat on top of 1 unit thickness? What'd be the column height of example #2 and #3 assuming $c$ as 1mm? I'm quite sure that given the same reached height $h$ in example #2 and #3, $c$ of #2 has to be smaller than $c$ in #3.


title: How does the kinetic energy of quarks contribute to the mass of macroscopic objects?

text: As we know, most of the mass of ordinary matter comes from the kinetic energy of quarks. This means kinetic energy of quarks contributes to the mass of any object. However take a look at this question. Does a moving object curve space-time as its velocity increases? A moving star does not bend space-time any more than if it were not moving. (One example given was that a moving star won’t collapse into a black hole just because its moving). This seems to be a contradiction. In the first case motion of quarks do contribute to bending (by contributing mass). In the second case, the motion of the star does not. Both make sense. But how do we resolve the contradiction? Let's take a more macroscopic example. If we had a many asteroids moving in random directions in a region of space, would the motion increase the bending of space-time (say the observer far away from the asteroid field). Imagine that you shrink down the asteroids and increase their number so you have a gas of small particles. This “motion in random directions” translates to internal energy and that contributes to the bending of space time, just as the kinetic energy of quarks contributes to the bending of space-time. And in fact, enough of this can cause a black hole. The interesting thing is you can’t transform to a frame where none of the asteroids or small particles are in motion. In the case of a single star in motion, you can transform to frame where the star is not moving. This question seems related: Does relativistic mass exhibit gravitiational effects? Article by Wilzcek on "mass without mass". I think this article is a good answer to the question. http://www.aip.org/pt/nov99/wilczek.html Other notes: In the case of a star, the stress-tensor determines how space-time curves. The stress-tensor is a coordinate independent object. So it doesn’t matter what particular coordinates you use to do your calculations. In other words, whether you are in some frame co-moving with the star or in spaceship moving by the star, calculated metric tensor should be the same. In the case of a gas, there is no frame in which the particles are all standing still with respect to that frame. You stand still with respect to some particles, but other particles are then moving and so on. The component of the stress-energy tensor change, but the result is the same. So motion does add to the curvature of space time, but only if you can’t transform it away.


title: What is a degenerate Fermi gas?

text: In ultracold atoms, it is generally talked of a degenerate Fermi gas. What does _degenerate_ mean here?


title: How far from the Sun is a photovoltaic effective?

text: Just what the title states. Photovoltaic cells convert incident light to electricity; this probably means they create less electricity when it is dark - as in deep space. How far can a solar cell be taken from the Sun, and still be effective?


title: The Lorentz Transformations in the Micro-World

text: Two particles[or micro-observers] A and B are in relative uniform motion wrt each other in the x-x’ direction. The “observer” A decides to deduce[or interpret] the Lorentz Transformations wrt to B. Accurate knowledge of the position of B makes its[that of B] momentum highly uncertain.[$\Delta x \Delta p\ge h/2\pi$] How does the observer A go about his job? Rather how do we interpret the Lorentz Transformation in the micro-world?


title: Is it possible to lift yourself off from the ground?

text: Say for instance a person who was strong enough to lift double his body weight. If he placed his hands under his bottom and tried to lift$^1$ himself$^2$ off the ground, could he? \-- $^1$In a slowly, controlled motion; sudden jumps etc. are excluded. $^2$ All parts of himself without touching or being supported by the ground in any way.


title: Calculating conductivity from Green's functions

text: I am trying to calculate the conductivity in the linear response regime of a disordered electron gas. (or eventually of a mean field Heavy fermion system with known one particle green's functions). I trying to use method in "Quantum field theory of non-equilibium states" by J.Rammer. Specifically I'm looking at section 6.2. Honestly, I don't understand very much of what he is doing. Have somebody done those calculations and could give guidelines on how to think and perhaps some more steps in the derivations? My question in short would be: If I know the known free Green's functions, how do I calculate linear response when applying an electrical field?


title: Do interaction-free measurements require a physical collapse or splitting in order to be truly interaction free?

text: Interaction-free quantum experiments like Renninger's experiment or the Elitzur-Vaidman bomb tester are often taken to be examples of interaction-free measurements of a system. Unfortunately, such assumptions presuppose the ability to post-select in the future just to make sense. Interpretationally speaking, it is hard to see how post-selection can possibly be made without some form of physical collapse of the wave function or a preferred physical splitting of the wave function into branches. Without either a physical collapse or splitting, is it possible to gain information about a system of which we are totally ignorant about the preparation of its properties we are interested in without an actual interaction with it? Basically, does the idea of interaction-free measurements only make sense within some interpretations of quantum mechanics and not others? Is there a philosophical reading of the two-state formalism which does not presuppose a collapse or a splitting? In the two-state formalism, does the necessity of normalizing the overall probability factor to 1 entail the ontological reality of the other outcomes because we have to sum up over their probabilities to get the rescaling factor? The other outcomes where the bomb did in fact go off?


title: Why do phonons cause excellent heat conduction in diamonds?

text: Phonons are the quantum of lattice vibrations in crystals and are not to be confused with photons, the gauge bosons of the electromagnetic force. Apparently, they contribute to heat conduction, but I don’t understand why, as this is not explained in course 2062 of Aalto University. So why is diamond a good heat conductor and how do phonons contribute/are related to this?


title: Calculating Reynolds number for a viscous droplet

text: I'm trying to develop a very basic scaling law/unit analysis for viscous droplet formation, and I'd like to get some rough numerical values of the Reynolds number to play with. To be specific, I'm looking at the behavior of the smaller of the two droplets shown in the picture below (experimental setup shown sideways): ![Experiment setup \(sideways\)](http://i.stack.imgur.com/BmGVV.png) The darker fluid is a glycerol/water mixture and the lighter fluid is mineral oil - so it's a viscous fluid dropping into another (dissimilar) viscous fluid. I'm trying to understand how the viscosity of the dark fluid effects the size of the small droplet - the large droplet pretty much remains the same size, but the small one gets smaller for lower viscosities. Since I'm trying to work only with dimensionless parameters, I'd like to work with something like Reynolds number instead of viscosity. I know that $\text{Re}=\frac{\rho \mathbf{v} L}{\mu}$ is the "standard" formula, but I want to make sure this applies here, and if so, that I'm using the correct values for the parameters. I can calculate $\bf{v}$ from my high speed video, and I can calculate $\rho$ and $\mu$ for both fluids using a formula, but I'm wondering:   * Do I need to use a _ratio_ of the two densities and viscosities, or do I just use one? Would I use the viscosity of the stationary fluid or the moving fluid?    * Does the width of the channel make sense for the characteristic length $L$, or should I go with something more related to the droplet? Feel free to suggest a book or online resource if this doesn't have a simple answer. I have to admit I'm very hazy on the "physicist's" viewpoint on fluids (I'm a math grad student). Thanks in advance!


title: How do two rigid bodies with different 3rd moment of inertia rotate differently?

text: If rigid bodies $R_1$ and $R_2$ has exactly same total mass $M$, central of mass, and rotational inertia $I$, but different third moment of inertia $M_3$, how would they move/rotate differently? What's the observable phenomenon of the difference in their third moment of inertia?


title: What are some publications which continue Schrödinger's "What is life?" discussion?

text: I'm looking for modern publications on the physical nature of life in which the primary reference is to the discussion started by Schrödinger in 1944 in his book "What is life". For example, publications that suggest amendments (which have been generally accepted) to his ideas, similar books (not text books) that take into consideration new discoveries and offer a modern follow up, compiled works, reviews, and the like.


title: Do unmeasured particles function the same as when measured?

text: If there are particles that we are uncertain of are in a state of superposition, do they still function as they would when they are measured? For instance say there is a mechanical clock, and the gears are quantum in superposition, would the clock still function? Or would it be impossible since there is not a measurement to bring the particles into a specific state?


title: The boundary for quantum mechanical behavior and classical mechanical behavior

text: To what size and how does "quantum weirdness" such as entanglement and superposition stop applying to larger objects (mere unions of these quantum particles). How do these macro objects that behave as classical mechanics not function like the very particles that they are made of?


title: What is the percentage of stars that are massive enough to end their lives in a supernova?

text: I have been searching for the percentage of stars that are massive enough to end their lives as a supernova but couldn't get any result. As far as I know, a star has to be at least 8 times more massive than the sun to go supernova, so what is the percentage of stars that are more than 8 solar masses of all the stars in the observable universe ?


title: Find Inductance in a Coil

text: Question: An inductive coil has an internal resistance of 20 $\Omega$. When an AC Voltage source with a frequency of 100 Hz is connected to the coil, the current lags the voltage by 30 degrees. What is the value of the inductance L. My Work: I am unsure where to begin here. If we know that $\omega_d=\frac{1}{\sqrt{LC}}$ and we also know that $f=2\pi \omega_d$, then we can just find the capacitance and solve for L in the first equation and we will have our answer. However, I am not sure how to find capacitance. Any help is appreciated.


title: Is synchrotron radiation considered incandescent?

text: Synchrotron radiation is produced via the acceleration of charged particles, much like incandescence. However, all information I've seen state that incandescent light is produced exclusively from the thermal motion of the charges. Is there therefore a better term to describe the light produced via synchrotrons? Thanks!


title: Deriving spring oscilation period

text: We have a spring attached to a wall and at the other part an object on a frictionless surface. I tried to calculate the spring oscilation period. I used the conservation of energy and kinematics equations for the average acceleration. $$\frac{1}{2}kA^2 = \frac{1}{2}mv^2 \Rightarrow v^2=k\frac{A^2}{m}$$ $$v^2=2aA+v_0^2=2aA \Rightarrow \frac{kA^2}{m}=2aA \Rightarrow a= \frac{kA}{2m}$$ $$v=at \Rightarrow \sqrt{\frac{kA^2}{m}}=\frac{kA}{2m} t \Rightarrow t= 2 \sqrt{\frac{m}{k}}.$$ This only the motion from the amplitude to the $x=0$ position. The forward 3 parts are the same because the kinetic energy is constantly transformed into elastic potential. $$T=4t=8 \sqrt{\frac{m}{k}}$$ I used the kinematics equations for some average acceleration of out varrying acceleration. The formula said $T=2 \pi \sqrt{\frac{m}{k}}$. $\pi$ doesn't equal 4. Can anybody explain how to derive the formula ?


title: Is there a quantum state for a large system

text: My understanding of quantum mechanics is that the state of a system is represented by a vector in multidimensional complex vector space. Is there, in principal, a state vector that represents a large, classical object such as, say, a cheeseburger, at an instant in time? If so, what is the physical meaning of that "state"?


title: What is the specific meaning of "Fourier frequency" (as opposed to simply "frequency")?

text: I've noticed that many journal articles (in optics) use the phrase "Fourier frequency" to describe, well, the frequency of something.   * Google scholar search for "Fourier frequency". Example: > _To compare with previous sensitivity curves of a single LISA Michelson > interferometer, we construct the SNRs as a function of **Fourier frequency** > for sinusoidal waves from sources uniformly distributed on the celestial > sphere._ This phrase seems redundant to me. Does the adjective "Fourier" add anything? Is it trying to make a distinction of audio frequencies from optical frequencies? Or temporal frequencies from spatial frequencies?


title: Do atoms have lifespans?

text: From _A Short History of Nearly Everything_ by Bill Bryson: > Atoms, however, go on practically forever. Nobody actually knows how long an > atom can survive, but according to Martin Rees it is probably about 10^35 > years Can anybody explain this result to me? I am told that the original atoms in the universe were likely Hydrogen, condensing out of primordial big bang stuff. Lighter atoms are said to come around as part of the fusion cycles that keep stars running, and heavier ones out fusions that occurs in energetic events like supernovae. We also have radioactive decay to consider. I'm fine with all of these things, but it sounds like Bryson is talking about something altogether different. I see how you could make a statistical argument involving the likelihood of a given atom to be involved in future such events, but I don't think that is what's going on here as the result is twenty-five orders of magnitude older than the universe (as far as we know). Any ideas?


title: Path integral and geometric quantization

text: I was wondering how one obtains geometric quantization from a path integral. It's often assumed that something like this is possible, for example, when working with Chern-Simons theory, but rarely explained in detail. One problem I run into is that, when trying to repeat the usual derivation of the path integral, we want to insert a complete set of states, but typically here the Hilbert space is finite dimensional, so I don't see how to interpret this as the integral over some manifold, as in the usual case. The simplest case I can think of is $S^2$ with $j$ times its usual symplectic form, which gives the spin $j$ representation of $SU(2)$. Is there a way to recover this from an integral over paths on $S^2$?


title: Swinging onto a Hill

text: I was just watching the movie "Frozen" (not the Disney one), and they were trapped in a ski lift 50 feet off the ground. They had to jump. My question is, when they jump, should they swing on the lift towards up the hill or down the hill? If they swing up the hill, it will be a shorter drop. If they swing down the hill, they will not have such a hard collision. Physics is not my thing, and I am just looking for a general answer, so any variable you can just assume to be "generic". I guess the slope of the hill is 45 degrees and the swinging momentum is...whatever.               \      \      o       \  <-   ->        \         \          \


title: A particular notation about fermions

text: I am not sure that this notation is specific to supersymmetry theories but I ran into this while studying that. I see people talking of component fields of a chiral superfield as $\phi$ and $\bar{\psi}$ in the adjoint representation of some gauge group. (One sometimes also seems to use the notation of $\bar{\psi}_+$ to denote which component of the spinor is being picked up) One also denotes the covariant derivative operator as $D$ (which is sometimes written as $D_{++}$ to denote which component of it is being chosen..but I don't understand this notation) Now one talks of "single trace operators" of the kind $Tr[\phi \bar{\psi}]$, $Tr[\phi \bar{\psi}^2]$, $Tr[\phi^3D \bar{\psi}^4]$ and stuff like that..basically taking arbitrary combinations of powers of the scalar and the fermion and then taking a "gauge trace". There are lot of things that I don't understand.   * I am puzzled by the notation of taking powers of a fermionic field. How does one define powers of a spinor? (What is square of a fermion?) Also here in components $\psi$ would have a decomposition like $\psi = \psi_A t_A$ where $\psi_A$ are fermions and $t_A$ runs over the generators of the Lie algebra of a gauge group in the adjoint representation. I guess one is taking a tensor product between the the spinor $\psi_A$ and the matrix $t_A$. But then I don't understand what is a square or any other power of that tensor?    * I am told that $\bar{\psi}^2 \neq 0$ but $Tr[\bar{\psi}^2] = 0$ I don't understand what is this supposed to mean.    * One way I can think of the powers could be that $\bar{\psi}$ are after quantization operators on the Hilbert space of the theory and these are powers of that Hilbert space operator. But in this way of thinking I am confused how to intprete the trace over the gauge indices.    * In this language one wants to think of the supersymmetry operators $Q$ to act on the fields in the following way,  $Q\phi=0$ $Q\bar{\psi} = 0$ $Q(DO) = [[\phi,\bar{\psi}],O\\} + D(QO)$ where $O$ is some operator and in the first term on the RHS of the last of the above equations the symbol $[,\\}$ means that one will take the commutator or the anticommutator depending on whether $O$ is bosonic or fermionic. The above looks like a very different way of thinking about supersymmetry transformations than the language I am familiar with from the books like that of Weinberg where the action of $Q$ on $\phi$ and $\psi$ is through commutator and anti-commutator respectively or one thinks of infinitesimal supersymmetry transformations as $\delta \phi$ etc.   * Definitely the last of the above list of commutators is totally unfamiliar to me!    * What the above notion confuses is how I am supposed to think of the action of $Q$ on composite operators like say $Tr[\phi\bar{\psi}^3]$. Apparently one is supposed to think of this as a fermionic operator since the fermion is being raised to an odd power. (I can't see the full argument here!) Then the action of $Q$ is supposed to be (dropping the overall trace), $Q(\phi \bar{\psi}^3) = (Q\phi)\bar{\psi}^3 + \phi (Q \bar{\psi})\bar{\psi}^2 - \phi \bar{\psi}(Q \bar{\psi})\bar{\psi} + \phi \bar{\psi}^2(Q \bar{\psi})$ The signs alternate depending on how many $\bar{\psi}$s has the $Q$ skipped over.   * I would like to know of explanations for the above way of writing powers of fermionic fields and doing supersymmetry transformations on them. I would be glad if I am directed to some reference which explains the above way of thinking which I haven't come across anywhere else in any book.    * {Eventually one interested in calculating the "cohomology of $Q$" on the space of all such single trace operators. (for some reason not clear to me people want to drop from this list of operators which are total covariant derivatives) This depends on the theory one is looking it by what the auxiliary field integrates to and appearing on the RHS. I would like to know references along that too and why this is "cohomology" and why is this calculated.}


title: Indicate if objects after collision will stick

text: Is it possible to indicate if objects after collision will stick together knowing their properties(materials,hardness,etc)?


title: How many subatomic particles can absorb/emit photons?

text: Is the electron the only subatomic particle that can absorb and emit a photon?


title: How is viscosity described on the molecular level?

text: What is the ' ** _molecular_** ' origin of the **_viscosity_**? The molecular origin of elasticity is almost clear for me: at the very bottom the 'elasticity' comes from the attraction and repulsion between atoms when they are reasonably far from each other or very close (like Lennard-Jones potential). Is there a similar description for the viscosity as well? Where does it come from?


title: Is it possible to create a vacuum ballon using concentric partially pressured spheres?

text: The old saying goes, nature abhors a vacuum, but that isn't exactly true. The universe is _full_ of vacuum; what nature abhors large energy differentials. So for example, a vacuum chamber has to be strong enough to withstand the ~101kPa of pressure on the surface, and typically the materials required to build such a structure are heavier than the volume that contain. I've often heard references to Neal Stephenson's Diamond Age, and people saying that perhaps one day we'll be able to manufacture small strong nano structures that are lighter than air (indeed, it's quite plausible: http://www.euronews.com/2012/11/01/lighter-than-air-material-discovered/). However, I'm curious why concentric partially pressured spheres wouldn't work as an approach. If you consider a structure which is in cross-section:               Outside | P1 | P2 | P3 | ... PZ ... | P3 | P2 | P1 | Outside      Where the value of P(n) is something like, say, P(n-1) * 4 / 5 (or some other fractional amount), and PZ ~= 0, it seems to me there's no reason why it wouldn't be possible to build the structure out of relatively light, relatively weak materials with many concentric layers that wouldn't collapse. Perhaps there would be some minimum gap distance required and there are also practical considerations like, 'what stops the spheres from falling due to gravity and touching; when two surfaces touch the pressure difference is effectively the delta over the two open spaces; which will eventually be external vs. internal vaccum and break the structure', 'how would you depressurize the spheres', etc. ...but lets say for an ideal situation in which the structure floats in a gas without a strong gravity source to mess things up, and the pressure has been magically set in advance in each pressure zone. Does this work? If not, why does the structure collapse? Could it be possible to actually fabricate such a structure? (for example, a single pipeline running from the edge to the center could maintain the position of the spheres and depressurize them one at a time as required; the amount of material to do this (despite being dense) would be related to the radius of the sphere, while the buoyancy would be related to the _volume_ of the sphere. Plausibly a net positive lift could be generated, I imagine...)


title: Where does $p^i/p^+$ come from in the EOM of an open string?

text: I have a stupid question about Eq. (1.3.22) in Polchinski's string theory volume 1. In the equation of motion for an open string, Eq. (1.3.22), $$X^i (\tau, \sigma) = x^i + \frac{ p^i}{p^+} \tau + i \bigl(2 \alpha'\bigr)^{1/2} \sum_{\substack{n= -\infty,\\\n\neq0}}^{\infty} \frac{1}{n} \alpha_n^i \exp\biggl( -\frac{ \pi i n c \tau}{ l}\biggr) \cos \frac{ \pi n \sigma}{l} $$ How do I get the factor $ \frac{ p^i}{p^+} $?