Function for curved line in log-log-plot

I've done experiments on the thermo-viscoelasticity of liver tissue and the following results               temp shift     1   20     4     2   30     8     3   40    20     4   45    49     5   50   300     6   55  8000     7   60 60000      ![shift vs temp plot](http://file2.npage.de/013648/02/bilder/curved_loglog.jpg). show a curved line in a loglog-plot, which means I can't use a power-law function to approximate my data. Do you have any idea what kind of function might work for my data? I've tried superlinear functions of the type y(x) = a^x^b but that doesn't seem to lead to a good fit. I'm adding a plot of the same data plus 2 additional temperatures (70 and 80). temp shift 1 20 4 2 30 8 3 40 20 4 45 49 5 50 300 6 55 80000 7 60 600000 8 70 9000000 9 80 9000000 ![shift vs temp plot](http://file2.npage.de/013648/02/bilder/curved_loglin.jpg).


E&M question in example in my book

So I still not sure how to apply like Right-hand rule (RHR) in this setup in problem like the one in the following so I tried to do RHR in order to get the direction but it didn't work out. This is an example from the Halliday and Resnick book. > Figure 32-24 shows a wire segment, placed in a uniform magnetic field B that > points out of the plane of the figure. If the segment carries a current $i$, > what resultant magnetic force acts on it ? Here is image of the problem at hand; $F_1$ and $F_3$ are the force on the straight wire segments ![enter image description here](http://i.stack.imgur.com/WFe6M.jpg) Solution: $$F_1 = F_3 = -iLB $$ so its equal to $$iL = \langle iL,0,0\rangle \quad B = \langle0,0,B\rangle$$ $$iL\times B = \langle0,-iLB,0\rangle$$ So it points in the **negative y** direction I tried to do same thing now for circular part to get the directions: $$iL = i\langle r \cos \theta,r \sin \theta ,0\rangle \quad B =\langle0,0,B\rangle$$ $$iL \times B = \langle r \sin\theta B,r \cos\theta,0\rangle$$ So since its a circle, $\cos \theta$ will gets canceled by symmetry so it will be so it will have only a horizontal component equal to $r \sin \theta B$ of course this works out in the problem when you do the integral but its not right reasoning as the book have and book used RHR.


Why do I hear beats through headphones only at low frequencies?

I was recently playing with this Wolfram Demonstrations applet, which demonstrates beats. At first I thought the app didn't work because I couldn't hear any beats. Then I realized that the applet doesn't analytically add two sine waves of nearby frequencies and play the result; it plays different sine waves to the left and right audio channels. Since I had headphones on, I only heard one frequency per ear, and only consciously detected a single tone. When I took the headphones off and held the ear buds close together, I could hear the beats distinctly. Messing around a little more, I found that when I lower the frequency to around 600 Hz or lower, I begin to be able to hear the beats even with headphones on. What's happening? Are my ears for some reason picking up relative phase information only at lower frequencies? Are only lower frequencies transmitted effectively through my skull? I would guess it's a coincidence that I begin to hear beats when the wavelength of the sound (in air) is on the order of the distance between my ears, but is that right?


How do people work out the trajectories of planets and stars just by looking at them?

I've been thinking about how astronomers can look at bright dots in the sky and deduce a whole bunch of things from their movements. I'm particularly interested in how people like Kepler and Galileo, 400 years ago, were able to deduce that planets move in ellipses and work out laws like equal times means equal areas, among others. It seems really remarkable to me that people have figured out a lot of very detailed things about the universe just looking from our planet. There is a similar question that asks how people can determine what the different celestial objects are made of. I'm more interested in the mechanics side of it; in class we've worked out Kepler's laws theoretically using Newtonian mechanics, but of course that's not how Kepler did it. I'd be interested in a more-or-less detailed explanation of how can one deduce so much about celestial objects just by looking at the sky. I realize this is probably not trivial, so references to books or websites are fine.


Is acceleration due to gravity constant?

I was taught in school that acceleration due to gravity is constant. But recently, when I checked Physics textbook, I noted that $$F = \dfrac{G \cdot m_1 \cdot m_2}{r^2} $$ So, as the body falls down, $r$ must be changing, so should acceleration due to gravity.


What would the universe be like if Electroweak symmetry were unbroken?

> **Possible Duplicate:**   >  What happens to matter in a standard model with zero Higgs VEV? What if the Higgs did not have a "Mexican hat" potential and the therefore it's vacuum expectation value were zero. Then the elementary particle masses would be zero, so the electron, W+, W-, Z and quarks would all be massless. Would a proton with non-zero mass be possible since most of the proton mass comes from the binding energy of gluons which are already massless? The weak force would be long range and fall off like $1/r^2$. I assume the Z would take the role of the photon. How many kinds of weak charge are there? Would like charges be repelled and unlike charges be attracted via the Z? Would there be something equivalent to magnetism? What would the massless W do? Would the long range weak "W" force change the electrical charge values of electrons/neutrinos and quarks? How weird is that! I know, we wouldn't exist to see all of this, but I am curious about how this would all work.


To find the center of charge in current waveform

I have a circuit for which output current waveform has both positive and negative points. Now I want to get a center of the charge for this waveform (means where my net charge is concentrated ?). Is there any formula or theorem to calculate this. My current waveform has time on $x$ axis and current values at $y$ axis.


Illegal gauge condition in electrodynamics

Just a quick sanity check here: I'm preparing a tutorial for a class on classical electrodynamics and I wanted to show an example of a gauge condition which leads to a contradiction, so I simply thought of $$(A^\mu)' = A^\mu - \partial^\mu \chi \stackrel{!}{=} 0\quad \forall \mu \in \\{0,1,2,3\\}\qquad (1)$$ For the 0th component this gives me the Weyl gauge ($\phi = 0$), which is allowed since I will always find a $\chi$ satisfying $$\phi = \frac1c \partial_t\chi.$$ I do, however, have a contradiction in the spatial components, since $$\vec{A} = \vec{\nabla}\chi$$ is only true for $\vec{A}$ with $\vec{\nabla}\times\vec{A}=0$ (on a simply-connected region), so if a given vector potential has non-vanishing rotation, I won't be able to find a $\chi$ so that the transformed field vanishes in all components. **Does anyone have some other examples?** * * * (1) - As mentioned by QMechanic: Here are actually four gauge-fixing conditions disguised as one; this is unusual since common gauges (e.g. $\vec{\nabla}\cdot\vec{A}=0$) generally eliminate only one degree of freedom.


High-spin neutron star

Suppose a large neutron star were to be spun-up by a particular pattern of mass-accretion. The increased centrifugal force could presumably mitigate the increased gravity thus delaying gravitational collapse. How far could this process be taken?


Why do some air-conditioned stores blast you with jets of air as you enter?

I went to a grocery store on a hot day that was very well air-conditioned, and I noticed as I went through the open entrance that there seemed to be a very powerful downward air current right at the doorway. After crossing the invisible threshold, the temperature immediately dropped a good 15 degrees or so. How does this process work?


Holes in a P-type semiconductor under external force E

Basically in almost every semiconductor texts, there will be all these concepts concerning electrons, holes, dopants, fermi-levels. However, I have been always confused about the picture of hole transport in semiconductor device, say, a simple PN junction. With a specific acceptor level and dopant concentration, we have some "holes" in the valence bands, which are in fact the absence of some electrons having gone to the acceptor levels, then all theses books seem to assume in the valence bands of the p-type region, there will be only holes that conduct current? My questions:   * aren't there still many electrons in the valence band? though they have some negative effective masses, but still do contribute to the transport?   * under a certain external force(say,E), the electron and holes in the valence band are moving in the same directions since electrons have negative effective mass and holes have positive one, so the corresponding currents cancel with each other in a P-type semiconductor?


What is going on in front of and behind a fan?

Why is it that when you drop paper behind a fan, it drops, and is not blown/sucked into the fan, whereas if you drop paper in front of a fan, it is blown away?


Can all systems be put in equilibrium?

I'm in a first year statics course. We have spent the whole semester solving for forces and moments so that the system is in equilibrium. When we are given a system, we immediately begin solving for all variables for the system to be in equilibrium. But how do we know that the system _can_ be put in equilibrium? I asked my prof about this and he said that all systems can be solved to be put in equilibrium, but some systems might possibly have multiple solutions. Why can all systems be put in equilibrium? If I misunderstood the prof and all systems _can't_ be put in equilibrium, is there a quick way to check if a system can be put in equilibrium? (Other than the obvious way of solving the system). Thanks.


What is the physical meaning of the affine parameter for null geodesic?

For time-like geodesic, the affine parameter is the proper time $\tau$ or its linear transform, and the geodesic equation is $$\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma_{\rho\sigma}^{\mu}\frac{dx^{\rho}}{d\tau}\frac{dx^{\sigma}}{d\tau}=0. $$ But proper time $\Delta\tau=0$ for null paths, so what the physical meaning of is the affine parameter for null geodesic?


Why does the law of increasing entropy, a law arising from statistics of many particles, underpin modern physics?

As far as I interpret it, the law of ever increasing entropy states that "a system will always move towards the most disordered state, never in the other direction". Now, I understand why it would be virtually impossible for a system to decrease it's entropy, just as it is virtually impossible for me to solve a Rubik's cube by making random twists. However the (ever so small) probability remains. Why does this law underpin so much of modern physics? Why is a theory that breaks this law useless, and why was Maxwell's demon such a problem? Does this law not just describe what is _most likely_ to happen in complex systems, not what _has_ to happen in all systems?


How can a human eye focus on a screen directly in front of it?

I am asking this question here because I think the answer has something to do with the way light is bent as it's captured through the eye. I saw a show a while ago about tiny screens on contact lenses to pull up data on objects you see in the real world, I also just saw this article about Google testing the same idea with screens in the lenses of sunglasses. The part I do not understand is how your eye would focus on a screen that is so close. My eyes (and I believe most others) cannot focus on anything closer than a couple inches away. Yet the lenses on glasses are much closer to the eye than a couple inches. So I would guess they would have to use some special technology to separate the light rays in a way that your brain could make an image from it. If that's correct, how would it work? if not, how would they get your eye to focus on a screen so close?


What are "packets"?

Paraphrased from Wikipedia: > Infrared sensing in snakes depends on a kind of natural thermography, by > which tiny **packets** of cellular water are raised in temperature by the > infrared radiation. What is "packets" refering to in this is example?


Type of stationary point in Hamilton's principle

In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all follow the path with the least resistance. However, a stationary point can be a maximal or minimal extremum or even a point of inflexion (rising or falling). Can anybody give examples from theory or practice where the action integral takes on a maximal extremum or a point of inflexion?


Are different frequencies of light lensed differently during gravitational lensing a bit like refraction?

So I was wondering about the event horizon on a black hole. And wondering if the point of no return for radio waves vs gamma rays would be different. I guess the logic being, since gamma rays have more energy than radio waves, their point of no return might be different. But I'm not sure if gravity has the same affect on all frequencies of electromagnetic radiation. I'm sure the answer for this would lie in gravitational lensing, ie are different colours of light lensed by the same amount? or do higher frequencies lens less?


Historical analysis of light interference - difference frequencies

It is well-known that light of two different frequencies illuminating a detector will produce an output with a component at the difference frequency. While such considerations are eminently useful (various heterodyne measurement techniques) it has not been practical until lasers came along. Regardless of practicality, the effect has been known for a long time. The question is: how long? And, specifically, can you provide a reference to an analysis which establishes it? A web-accessible reference, for preference. I'd expect it was determined shortly after Maxwell's equations were established, but I could be wrong. Ideally, this would be an acknowledged seminal paper, comparable to Skutt's explanation of the color of the sky.


Can a hydrogen atom sit still?

Can we take a single lone hydrogen atom (proton and electron?) and set it in one place and observe it staying there? How about just a proton? A neutron? How about just an electron?


Will an electron gun fire behind it?

If an electron travels as a wave, can it therefore not be aimed with any precision? If you fire an electron gun, can't you aim it at a particular slit? If the electron travels as a wave, will an electron gun hit anything at a 180 degree angle from the barrel? 360 degrees?


Effect of crystal growth on its scintillation

How does crystal growth affect its scintillation properties? What are possible ways in which growth of CaWO crystals grown via Chochralski method could be modified to avoid non-linearity in its scintillation output with respect to incident Gamma rays? Apparently this is relevant for rare event searches (eg. CRESST) using these scintillating crystals as detectors.


Generator phase noise influence on its averaged signal

I need to estimate averaged signal of a generator with known phase noise and the amplitude noise can be neglected. The averaged signal of the generator is expressed as: $$ \langle A\exp{i(\Omega t + \varphi)}\rangle + \mathrm{c.c.} = Ae^{i\Omega t} \langle \exp{i\varphi} \rangle + \mathrm{c.c.}, $$ with $\Omega$ the generator carrier frequency and $\varphi$ the phase noise, $\langle\varphi\rangle = 0$. I suppose that phase noise is stationary. Its spectral density $ S_{\varphi\varphi} = \frac 1{2\pi} \int_{-\infty}^\infty\mathrm d\omega \, e^{-i\omega t} \langle \varphi(0)\varphi(t) \rangle$ is given. To obtain $\langle \exp{i\varphi} \rangle$ I suppose $\varphi$ to have Gaussian statistics. It is easy to show for that case that $$ \langle \exp{i\varphi} \rangle = e^{-\frac{\langle\varphi^2\rangle}2}. $$ Then, one finds $$\langle\varphi^2\rangle = \int_{-\infty}^\infty\mathrm d\omega \, S_{\varphi\varphi}(\omega). $$ I've taken $S_{\varphi\varphi}$ of the Agilent N5173B at 20GHz carrier, see the picture below and the generator's datasheet. (Note the two-sided phase noise spectral density is called $\mathcal L$ in industry, see the IEEE standart.) ![Agilent N5173B phase noise ](http://i.stack.imgur.com/qd2p6.png) I approximate the integral of the spectral density as the square under the connected points shown in the table: $$ \begin{array}{|r|r|} \hline f, \mathrm{Hz}& 1& 10& 100& 10^3& 10^4& 10^5& 10^6& 1.1\cdot10^6& 1.4\cdot10^6\\\ \hline S_{\varphi\varphi}, \mathrm{dBc}/\mathrm{Hz}& -40& -50& -80& -90& -95& -95& -125& -130& -135 \\\ \hline \end{array} $$ First plotting the spectral density and approximating its integral by hand, than using Python scipy trapz integration, I obtain $$\sqrt{\langle\varphi^2\rangle} \sim 10\,\mathrm{rad}$$ which is hard to believe for such a costly generator. And, according to the abovesaid, it will actually diminish the generator output, as $e^{-\frac{\langle\varphi^2\rangle}2} \sim 0.01$. So, 0) What the hell? 1) Probably I've misunderstood the IEEE definitions? I guess there should be some narrower frequency cutoffs? 2) Is it valid to assume the phase noise of a generator to be gaussian? 3) Probably better approaches exist for estimating the averaged signal of a generator with phase noise?


History of the names "Feynman-gauge" & "Landau-gauge". How arised & how settled?

Warning: Students, stay away from antiquities. The aim to learn is to survive. Hi. Today the nomenclatures _Feynman gauge_ and _Landau gauge_ seem established, but could you explain the history? It's two-fold: 1. Who used first such gauges and how? 2. How did the terminology gain ground? It also inevitably asks some history of the Lorenz gauges in QFT. In particular, when was it realized that the freedom (and need) of the gauge parameter ξ lies in quantum theory? First I apologize that I have only limited access to literature on asking such questions. My preliminary studies revealed below. 1 **Terminology in literature** (Schweber 1961) is not gauge-parameter conscious, and "Lorentz[sic] gauge" (in Feynman gauge) is opposed to coulomb. But it also refers to today's Yennie gauge in sec 15g, saying (Fried and Yennie 1958) found it is possible to take gauge where the photon propagator ∝ $g + 2pp/p^2$. In (Nakanishi 1966), the word "Landau gauge" is seen, and it cites several near-past articles regarding Landau gauge quantization. (It is an important paper in canonical quantization in Landau gauge, together with (Lautrup 1967). Nakanishi was a strong proponent of the Landau gauge.) In p. 74 of (Nakanashi 1972), it reads "Feynman gauge or Fermi gauge" and "Landau-Khalatonikov[sic] gauge, or simply Landau gauge". (Landau and Khalatnikov, 1955) is listed in the bibliography section, but I couldn't find which part of Nakanishi actually cites it. (Nakanishi 1972) is a review article, one of whose main topics is canonical quantization of EM field in arbitrary Lorenz gauge, i.e., for any gauge parameter. In p 134 of (Itzykoson & Zuber, 1980), the words "Feynman gauge" and "Landau gauge" are used. Were the names settled at that time? Hmm, in p 389 of (Siegel 1999), "Fermi-Feynman gauge" is introduced. (Srednicki 2007) uses the word "$R_\xi$" for QED, remarking "[it] has been historically used only in the context of spontaneous symmetry broken [...] but we will use it here as well." 2 **Gauge parameter symbol** ξ is now usual. Is it due to (Fujikawa, Lee and Sanda 1972)? For other symbols, I mention $\alpha$. (Nakanishi 1972) uses it, and even after (Fujikawa, Lee and Sanda 1972) it is sometimes used, for example and in (Siegel 1999). 3 **Theory Timeline** 1930 - Fermi: P. 240 of (Schweber 1961) says Fermi proposed to add $-\frac{1}{2} (\partial A)^2$ to the Langrangian. (Fermi was the first to introduce a subsidiary condition, but it was not perfect. See also Gupta and Bleuler below.) Although I haven't checked Fermi's papers, it may be better to call "Fermi-Feynman(-'t Hooft) gauge." 1948 - Feynman: Feynman simply justifies the use of Feynman gauge in the section 8 of (Feynman 1949). Before Feynman, it wasn't Lorentz covariant, and transverse photons were separated. Feynman says it's not necessary, and it's ok to do $\gamma^\mu$...$\gamma_\mu$. 1950 - Gupta & Bleuler: They say Gupta and Bleuler succeeds in covariant canonical quantization in Feynman gauge, by discovering the correct subsidiary condition. 1956 - Landau & Khalatnikov: See (Nakanishi 1972) above. 1958 - Yennie gauge: It is said (Fried and Yennie 1958) uses the "Yennie gauge" of today, $\xi = 3$, in bound state problems. Early or mid 60's - Rise of interest in Landau gauge? See (Nakanishi 1966) above. 1966 - 67 Nakanishi & Lautrup: canonical quantization of EM field for any ξ. 1967 - Faddeev & Popov 1971 - 't Hooft: In 1971, 't Hooft used "Feynman-'t Hooft gauge" or simply "'t Hooft gauge" for broken gauge symmetry. (Fujikawa, Lee and Sanda 1972) generalized to any ξ. Its abstract uses the word "Feynman-'t Hooft gauge". (According to Weinberg. Haven't read both two.) 1972 - Still canonical quantization for arbitrary ξ is of interest, including massive vector field. See (Nakanishi 1972). 4 Loren't'z gauge (misspelling) You may know that in the 20th century, the common spelling was "Lorentz gauge", with extra "t". I couldn't find any exceptions at my hand. The turning point might be the errata of Peskin & Schroeder. Srednicki and Siegel spell it correctly. 5 **Bibliography**   * Fermi, E., Atti. Acad. Lincei. **9** (1929) 881, Atti. Acad. Lincei. **12** (1939) 431, Rev. Mod. Phys. **4** (1932) 87.   * Feynman, R., Phys. Rev. **76** (1949) 769.   * Fried H.M., Yennie, D.R., Phys. Rev. **112** (1958) 1391.   * Fujikawa, Lee and Sanda, PRD **6** (1972) 2923   * Itzykson & Zuber "Quantum field theory", 1980.   * Landau, L. D., Khalatnikov, I. M., J. Exper. Theor. Phys. USSR 29 (1955), 89 [English translation: Sov. Phys. J. E. T. P. 2 (1956), 69].   * Lautrup, B., Mat. Fys. Medel. Dan. Vid. Selsk. 35 (1967), No. 11.   * Nakanishi, N., Prog. Theor. Phys. **35** (1966) 1111 (Downloadable gratis)   * Nakanishi, N., Prog. Theor. Phys. Supple. **51** (1972) 1 (Downloadable gratis)   * Schweber, "An introduction to relativistic quantum field theory", 1961.   * Siegel, W., "Fields", arXiv:hep-th/9912205   * Srednicki, (2007) _Quantum field theory_ 6 **Revisions of this question** 26 Jun: Added the post 't Hooft era and the symbol ξ.


Forces on a simple flat slope

![enter image description here](https://docs.google.com/drawings/pub?id=1cm2nq5OKck4zKphMYgMIJuCREZDO9JlNriRA0_4pUTc&w=400) This diagram is of a block (b) moving down a ramp unaided. Taking gravity (g) as $9.8ms^{-2}$, and the mass of the block (b) as $0.2kg$, how can I find the force acting along the slope (f). BTW: the ramp is inclined 1.8 degrees, not radians. I thought I could use basic trigonometry, and $F=ma$: $$ Sin(1.8) = \frac{0.2*9.8}{f} $$ $$ f = \frac{0.2*9.8}{sin(1.8)} $$ However this results in: $f = 62.40$ Seeing as force is a vector I thought I could use trig to split into it's components but obviously I'm doing something wrong. (the answer should be ~0.06N) Sorry for the noob question, but it's got me clueless.


Is this simulation following real physics?

I am trying to simulate a game in Box2D(Physics engine). The game that I am trying to simulate is very simple and can be found here: http://www.makaimedia.com/#/speartoss What I want to know is that, is the spear in that game following real physics principles? In other words, if I went out and threw a spear, will it behave like the spear in this game? Specifically, I am taking about the rotation of the spear. In the game, the spearhead is point up first half of the trajectory and as soon as it hits the vertex, the spear head starts point down. I am simulating a spear in Box2d physics engine. The density of the spear is 15x more than the density of the handle. I set an angle (between 0, 90 degrees) and give it a linear velocity like this:                   b2Vec2 force;         force.Set(cos(angle) * 22.0f , sin(angle) * 22.0f);         _spearBody->SetLinearVelocity(force);      What happens is, the spearhead never points downwards. It stays at the exact same angle throughout the trajectory. Is this more realistic or the physics in speartoss game? Thanks.


Why aren't there compression waves in electromagnetic fields?

I just started learning about optics, and in the book I'm reading they explain how the electrical field caused by a single charged particle could be described by a series of field lines, and compare them to ropes, to provide an intuition of the concept. Then they say that and that if we wiggle the particle up and down, that would produce transversal waves in the horizontal field lines, but no waves in the vertical lines. I know that the physical analogy is not to be taken literally, but I don't understand why wouldn't that cause compression waves in the vertical lines. I mean, even though the direction of the field in the points directly above and below the particle doesn't change, the intensity does. And I assume it wouldn't instantly. So what am I missing?


When is the Hamiltonian of a system not equal to its total energy?

I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical interpretation in such a case?


Is there an analogous Gauss' law which is applicable for a gravitational field?

Consider the Earth to be a flat infinite plane having linear mass density equal to the mass density of the actual earth.   1. Can there be an analogous Gauss' law that can give the gravitational field for any point on a particular Gaussian surface.    2. If yes, what would be the substitute for the permittivity constant that appears in the Gauss' law for an electric field?


Which areas of physics are related to the act of playing drums?

I'm musician (drummer) and I'm trying to figure out what can I study (related to Physics) for better understanding of the drumsticks and wrist movements, the force applied and the better way to apply it and introduce some theory in my training. I'm really awful at physics, so I really don't know what areas are important (pressure, mechanics, acceleration?).


Yang-Mills CP violation

Why does a term proportional to $\left(F,\,\tilde{F}\right)\propto Tr\left[ F_{\mu\nu}\tilde{F}^{\mu\nu}\right]$ in the Lagrangian of the pure Yang-Mills theory violate CP?


Gibbs equation and third law of thermodynamics

I have seen a discussion of what would happen for an ideal gas expands irreversibly and adiabatically until absolute zero degree K. The entropy change is like that: Δs=Cv ln(T2/T1)+R ln(V2/V1) It is impossible for T2 to be zero K in the equation and so it becomes one justification of the third law. I wonder whether it is valid and whether an irreversible process can be represented by the above equation. For irreversible process the heat exchange is not equal to Tds so can the Gibbs equation be applied? The video is in https://www.youtube.com/watch?v=r4fGG_7NQr8 , 46:12


Problem understanding basic sail mechanics

I am trying to understand the physics behind a single sailed boat. I have read the following questions Sailing against the wind. Is this a fair model? What exactly is the 'lift' of a sailboat as explained by Bernoulli principle I've also been following this explanation which seemed to be helping but I am still confused. Full disclosure: my physics is pretty weak and I've never sailed, but my maths should be sound. I am (now) aware of the two components that provide the force that moves the boat: The _"Newtonian aspect"_ , where the wind collides with the sail and this produces a force and the _"Bernoulli aspect"_ resulting from pressure differences resulting from the air passing around the sail. (If people think this would be better as two questions, happy to break it down, but I suspect one or both will be silly) **Update:** It has been said by @CarlWittoff that the Bernoulli aspect is pretty trivial, so I would like to focus on how the Newtonian part of the force can achieve a force that goes into the wind. **Newtonian Aspect** I understand this in the most simplest sense, a sail perpendicular to the wind will produce a force on the boat proportional to the relative wind speed. As the sail is angled in to the wind, you get more "lift" and less "drag" (in plane terms). In the sailing explanation, it implies that the Newtonian aspect can be used to explain going into the wind. This is also hinted at in the first question. However, I don't understand how this can be. Surely the force applied to the sail, and hence the ship, is always in the direction of the wind. Hence it can never provide a force into the wind. The article seems to suggest you can get away without the Bernoulli effect, but I don't see how. ![Diagram of the air flow](http://i.stack.imgur.com/16B4F.png) This is my attempt at how this translates into sail. The air is pushed around the sail, which exerts a force (in purple) on the boat. However, I can see no way for this force to be into the wind. **Bernoulli Aspect** This is the area I am even more lost with. My understanding is that somehow, you can angle the sail in such a way that the air that goes past on one side knows to go faster which reduces the pressure, giving a force. However, I cannot seem to connect that intuition to any kind of applicable formula. Specifically, what determines the direction of the force and what is it proportional to? I'm trying to simulate a boat for a game and any help on which formulae I actually need to be plugging in would be hugely appreciated :)


Why does a flat universe imply an infinite universe?

This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all.


What happens to a rotating rod that breaks in two?

I know that the approximation for the moment of inertia of an infinitely thin rod of mass $m$ and length $L$ spinning around an axis perpendicular to its own axis at its center is $\frac{mL^2}{3}$: ![enter image description here](http://i.stack.imgur.com/LVgxB.png) What happens when this rod is instantaneously broken in the middle with no loss or gain of energy? Rather than some external event triggering the breakage, imagine some sort of latch that held it together before, and now it is simply released. Easier to pretend with than an infinitely thin rod with mass, at any rate. I'd like to know how I should determine the motion (absent gravity) of the two halves. Suppose I view the scene from directly above (viewing in the same the direction as the axis of rotation), the rod is rotating clockwise from that perspective (viewing from below, in the picture above), and the event occurs when the rod is vertically aligned from that view. Based on this I assume a coordinate system: The rod was rotating within the x-y plane with its center at the origin. The axis of rotation is the positive z axis. My camera is pointing in the positive z axis from some position $(0,0,-z)$. The top half of the rod travels to the right. Its center of mass is at its middle point, which is at position $(0,\frac{L}{4},0)$ at $t=0$. The bottom half is going to travel to the left, in a symmetrical fashion. What are their angular velocities? Do they spin faster or slower, or at the same speed as the original rod? The length has halved, which means the moment of inertia has been quartered. But conservation of angular momentum states each rod carries half of the rotational energy (is this correct application of conservation of angular momentum? Is there such a thing as conservation of rotational kinetic energy?) Is part of the original angular kinetic energy now split between resultant angular kinetic energy and linear kinetic energy? It seems so, because the original rod had zero linear kinetic energy (and momentum) but after breaking, the parts now both have linear kinetic energy since they are flying apart. At this point I'm almost positive I can stumble my way to the solution for the split happening exactly at the middle. How might things change if the split is for example 1/10th of the way down the rod? Surely the little piece would go off much faster but would it spin at the same rate also?


Spin-Position Coupling/Interaction

Firstly, is there a general procedure/algorithm for attacking Hamiltonians having spin-position coupling, e.g., $$\frac{\mathbf{\sigma_1}\cdot\mathbf{\sigma_2}}{|\mathbf{r_1}-\mathbf{r_2}|}$$ I do not know where to begin. Secondly, I would like you to recommend a textbook which discuss such a kind of systems. I looked out and found none.


Where to find detailed measured emission spectra of all chemical elements?

I'd like to have something like this, but for single atoms and with more extended range of wavelengths. All I could find e.g. for hydrogen was lots of talks about Rydberg formula etc. and plots of spectrum in visible region. But I need _experimental_ data to look at, not something plotted from formulas like Rydberg's one. Similarly for other elements, I mostly found only plots of spectral lines in visible range. So, where can one find detailed measured spectra for all (or most) chemical elements, preferably with UV and IR parts? It'd be most useful to have it in numeric form (spreadsheet etc.) instead of pictures.


Does the strength of a smell depend on the volume or surface area of an agglomerate of material?

I just switched from Timberland to Adidas shoes, and noticed a horrible trait of the Adidas tread: instead of being composed of externally-protruding bumps like the Timberland, it is composed of channels striated _internally_. This has the horrible trait of trapping and making difficult to remove small pebbles, mud, and other foreign objects that one might inadvertently step on. Assuming that one were to step on an unusually aromatic organic viscoplastic, **would the 'strength' of the smell depend on the surface area of the exposed agglomerate, or of its mass**? My instinct leads me to think that the surface area would be the major factor, but empirically I see that larger massed agglomerates, even if wedged deep within an Adidas tread with little exposed surface area, seem to be more easily detected by human olfactory organs. However I contend that my sensing apparatus and testng methodology are not scientifically sound, nor do I have access to facilities which would allow me to make more objective observations. Of course, the principle of "a month in the lab could save an hour in the library" applies, so I searched online but found surprisingly little (read: none) literature on the subject.


What Would Negative Mass Do To Spacetime?

It's known that positive mass bends space-time to create a curvature. But if something had negative mass what would it do? Make it flat or like a crest?


Static electrical attraction

Coulomb's law is used to calculate the electrical attraction between 2 charged particles, what formula do I use to calculate an electrical attraction magnitude between 2 plates? Let's assume the first plate holds a charge of +3x10^-4 coulombs and the other one holds the same negative amount of charge (-3x10^-4), both plates area is 5 square centimetres and they are fully exposed and facing to each other. Also let me assume that the distance between them is 2 meters.


Vibration transmission using spring and damper model

I want to simulate the vibration transmission between two colliding objects. I use a Finite Element Model to simulate the vibration on each object, but I haven't found any good mathematical model to transfer the vibration between two FEM models that collided and keep in contact. ![enter image description here](http://i.stack.imgur.com/LcbS8.jpg) In the figure, a sphere is dropped into a plate. the change in color represent the displacement on the FEM models. Currently, I am using a spring/damper model to transmit the vibration from one object to the other. But, I haven't found any good model that give any theoretical foundation to my simple spring/damper model.


Rod slipping against block due to gravity?

A uniform rod of mass $m$ and length $l$ is pivoted at point O. The rod is initially in vertical position and touching a block of mass M which is at rest on a horizontal surface. The rod is given a slight jerk and it starts rotating about point O. This causes the block to move forward. The rod loses contact with the block at $\theta = 30^\circ$. All surfaces are smooth.   1. The value of $M/m$?   2. The velocity of block when the rod loses contact with the block?   3. The acceleration of center of mass of rod, when it loses contact with the block?    4. The hinge reaction at O on the rod when it loses contact with the block? ![enter image description here](http://i.stack.imgur.com/uJWRv.jpg) Here is my attempt at the problem The component of angular speed of the tip of the rod in horizontal direction is same as that of the block. Let V be the speed of the block when rod loses contact with block. So,Horizontal component of angular speed of the tip of the rod = $\omega l\cos(60)=V$ So, $\omega l/2=V$ Thus, $\omega = 2V/l$ Applying Conservation of Energy, Loss in Potential Energy of Rod = Gain in Rotational Kinetic Energy of Rod + Gain in KE of Block $$\frac{Mgl}{2}(1-\sin60^\circ) = \frac{1}{2}\frac{ml^2}{3}(ω ^2) +\frac{1}{2}MV^2 $$ $$\frac{Mgl}{2}(1-\sin60^\circ) = \frac{1}{2}\frac{ml^2}{3}\frac{4V ^2}{l^2} +\frac{1}{2}MV^2 $$ $$\frac{Mgl}{4} = \frac{2mV^2}{3} +\frac{1}{2}MV^2 $$ The forces on the rod are Hinge force in x direction = $ H_{x} $ Hinge force in y direction = $ H_{y} $ Force due to gravity = mg Normal contact force from the block = $N$ The force on the block are Normal contact force from the rod =$ N$ Now, I am not able to proceed further.


Why do humans have bilateral symmetry?

About the eyes I know that it requires for gauging distance as in Modern 3D cameras have two sensors. And two ears for sound source localization using differences in levels and timing (But not yet two microphones in mobile phones/other devices). What about other organs? Even not all organs are in pair. If it is about about redundancy then why not more than two? Thanks for your answer.


Is there a proof from the first principle that the Lagrangian L = T - V?

Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it? On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the answer, and if possible, the primary source (who published the answer first in the literature). * * * Notes added on Sept 22:   \- Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of.   \- We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider.   \- But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion.   \- Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.


What is the current status of Pluto?

Pluto has been designated a planet in our solar system for years (ever since it was discovered in the last century), but a couple of years ago it was demoted. What caused this decision? And is there a chance that it could be reversed?


Decomposition of SU(N) adjoint representation under SU(2)

I am having trouble in demonstrating that under SU(2) transformations the adjoint representation of SU(N) transforms as one spin 1, 2(N-2) spin $\frac12$ and singlets. I am trying to demonstrate it from $N \otimes \bar{N} = 1 + A$ where $A$ is the adjoint representation; and the fact that an N vector of SU(N) decomposes as one $j=\frac12$ plus two $j=0$. So basically: $(1\oplus^{N-2}0)\otimes(1\oplus^{N-2}0)= ?$. But I end up with one spin 1, $2N-3$ spin $\frac12$, and $(N-2)^2+1$ spin 0. Embarrassing. Is my equation with the question mark wrong?


Gravitation in a space that is topologically toroidal

In my scant spare time I'm building an Asteroids game. You know - a little ship equipped with a pea shooter and a bunch of asteroids floating around everywhere waiting to be to blown up. But, I wanted to add a little twist. Wouldn't it be cool if Newtonian gravity was in effect and you could do thinks like enter into an orbit around an asteroid and fire at it, or shoot gravity assisted bankshots around a massive asteroid so that you can shoot one behind it. But the problem is that in asteroids, space is topologically toroidal. If you fly off of the top of the screen, you reappear at the same x coordinate at the bottom of the screen (and similarly for the right of the screen). So how does one calculate the distance between two bodies in this space? Really, I realize that this question doesn't make sense because body A would pull upon body B from a variety of directions each with their corresponding distances. But anyway, the main questions: How would Newtonian gravity work in toroidal space? AND Is there any applicability to the answer to this question outside of my game?


I think there is a mistake in solutions to this problem. plus and minus sign

![](http://img526.imageshack.us/img526/2593/49699289.jpg)! I pretty much had what the solutions had, but we disagreed with one thing, a minus sensitive sign. $\sum W = \Delta K$ $\int_0^{-s} - ks^2 ds = \frac{ks^3}{3} = \frac{1}{2}m(v^2-4^2)$ Rearranging, I get $v^2 = 4^2 - \frac{2k(-0.2)^3}{3m} = 19.2$ So v = 4.38m/s But the key did $v^2 = 4^2 - \frac{2k(0.2)^3}{3m} = 12.8$ where they let $\int_0^{s} - ks^2 ds = \frac{ks^3}{3}$ **EDIT** Apparently I am stupid and I made a question that didn't even need answering... I had this on paper and I typed up the wrong thing, that's why I was confused. $\sum W = -\int_{-s}^{0} ks^2 ds = \frac{1}{2}m(v^2-4^2)$ Sorry everyone


how to determine if a vortex is laminar or turbulent

In a cylindrical chamber with a high diameter-to-height ratio; a fluid is tangentially injected. there is an axial exit to the cylinder. how do I determine if the vortex so formed is laminar or turbulent? what parameter will indicate this? (this is a simulation; so I can measure whichever velocity/turbulent KE that is needed.)


If 100% of the energy from the sun is reflected back into space

100% of the energy from the sun is reflected back into space, it's just shifted from a low-entropy state to a high-entropy state, and from a high frequency (ultraviolet) to a low frequency (infrared). But does the total number of photons change as a result? Are there more low frequency photons, then, in order to make the same amount of energy?


How to get the angle needed for a projectile to pass through a given point for trajectory plotting

I am trying to find the angle needed for a projectile to pass-through a given point. Here is what I do know:   * Starting Point $(x_0,y_0)$   * Velocity   * Pass-through point $(x_1, y_1)$ I also need to incorporate gravity into the equation. Anyone have any ideas? I haven't had much luck so far, so any ideas/suggestions would be great.


Do all the 4 forces of nature act at the same speed?

It is believed that gravity, the weakest of the four forces propagates at the speed of light, cf. e.g. this Phys.SE post. One would expect (perhaps erroneously) that the other, stronger, forces acted at an even quicker speed. Of course, it is not like that. At least, it's not for electromagnetism. What do we know about the speed of the other two?


Wall visible after light source removed

Let's say you have a lamp pointed at a wall, in a dark room with no other light source, if you turned off that lamp #  Would the wall still be visible for any amount of time?


How long does a supernova last?

Just what the title indicates. Is a supernova over instantaneously? Or, does the (for want of a better word) explosion continue for a while? What is/are the order of timescales involved? EDIT: Following the posted replies/comments I figure what I mean to ask is the duration for which the supernova continues to release copious amounts of energy.


Energy stored in fields

How to intuitively think of energy stored in a magnetic/electric field? Kindly answer in a bit simple terms without referring to mathematics.


Can the laws of quantum mechanics be derived from a more fundamental theory?

String theory takes quantum mechanics and tries to make it compatible with gravity. If it turns out to be a theory of everything then would it explain why our world is described by the laws of quantum mechanics? I want to understand if the laws of quantum mechanics are the only logical possibility for a low-energy theory potentially derivable from a better understanding of string theory. The other possibility is that they are just strange laws that are discovered through experiments that can't be reached by pure theoretical reasoning. Which of the two possibilities is more reasonable from a physics point of view and why?


Vacuum and repulsive gravity

How can one show from General Relativity that gravity is attractive force, and under which conditions it becomes repulsive, also why positive energy vacuum drives repulsive gravity?


Measuring flow rate of gases

Flow rates of liquids could be easily measured with a stopwatch and a known volume container. But gases, if entrapped in a container will create a back pressure, and may lead to decreased flow. I wonder how they measure the quantity of gases generated by a reaction or flow rate of fluids in turbines, etc., without creating back pressure. What/how back pressure corrections are providedif this cannot be prevented.? Are there flow meters for gases at all pressures similar to fluids.?


Chiral perturbation theory: what is the Quark Condensate? why expand in $U$ rather than Goldstone fields?

I'm studying Chiral Perturbation Theory ($\chi PT$) from Scherer's _Introduction to Chiral Perturbation Theory_. What I am currently having some trouble understanding are two things:   1. The quark condensate. What is this and why is it a sufficient condition for spontaneous breaking of chiral symmetry? What I do not really understand is where the operator $S_a=\bar{q} \lambda_a q$ comes from ($\lambda_a$ are the Gell-Mann matrices) and why the expectation value of this (which I gather is zero) gives us this thing called the quark condensate.   2. The formulation of the effective Lagrangian. There is some stuff in Scherer about the coset $G/H$ where in this case G is the full chiral group and $H$ is the vector subgroup which is left after spontaneous symmetry breaking, but I do not really follow how this discussion explains why the Lagrangian is given in terms of the SU(3) matrix $U=\exp{\frac{i}{F_0} \Phi} = \exp{\frac{i}{F_0} \phi_a\lambda_a}$ for (individual) Goldstone fields $\phi_a$? Why can't we write down the effective Lagrangian in terms of the actual degrees of freedom in the theory, i.e. the Goldstone fields? I've read something about them not transforming non-linearly (and the $U$ transforming linearly) but could not really follow so if someone could elaborate on this I would be very glad. A big thanks in advance for all help given! And another thing - if anyone has another tip for an introductory reference to $\chi PT$, I would be very grateful. Scherer works decently but it's always good to read about things from a different viewpoint.


What does it mean to "convert energy into time"?

In a recent article about creating electron-positron pairs by colliding photons in a laboratory, Andrei Seryi, director of the John Adams Institute at Oxford University, was quoted to said: > It's breathtaking to think that things we thought are not connected, can in > fact be converted to each other: matter and energy, particles and light. > Would we be able in the future to **convert energy into time** and vice > versa? Let's ignore for the moment that a physicist is amazed that $\gamma\gamma\rightarrow e^-e^+$ process is possible in a laboratory. But what does he mean by converting energy into time? Is this just a poetic sentence to emphasize his amazement, or is there some deeper meaning behind it? Only thing that comes up in my mind is that, because our currently best theory of gravity - general relativity - is describing the dynamics of spacetime, "creating time" might mean inducing some change in the metric tensor $g_{\mu\nu}$, so he might be talking about the prospect of converting ordinary Standard Model particles into gravitational waves and vice versa. However, he might be talking about something else. Am I missing something here?


Orientation in GaAs

I can't find the precise definition of what is the orientation of a GaAs lattice. Being the superposition of two fcc lattices (one of Ga, the other of As), I would think that it is the direction of the closest Ga atom to any As atom of the structure. Is it right?


What is the path integral exactly?

I asked a question here about path integrals and QFT. I just want to confirm something. Is the path integral in quantum field theory a mathematical tool only? I thought the path integral meant that particles literally do take all paths as stated by Victor Stenger and Stephen Hawking in their interpretation of what happens in reality? If particles are field excitations, as QFT says, does that mean Stenger and Hawking's interpretation is right and in reality particles do take all paths (sum over histories)?


Does heat kernel factorize on product spaces?

I have a doubt regarding whether the trace of the vector heat kernel on a product space factorizes into the corresponding heat kernel traces on each manifold in the product space. I know this holds for the spin-0 as well as spin-1/2 cases, but I am not sure about the spin-1 case. Can somebody please help me out in this regard?


Canonical form representation of a Linear Gaussian CPD

I'd like to know how a linear Gaussian conditional probability distribution can be represented to a canonical form. For example, let X and Y be two sets of continuous variables, with |X| = n and |Y| = m. Let p(Y | X) = _N_ (Y | _a_ \+ BX; C) where _a_ is a vector of dimension m, B is an m by n matrix, and C is an m by m matrix.


Educational applications of a small Michelson interferometer?

The community college where I teach has some nice old Michelson interferometers. There appear to be a bunch of versions of these that used to be sold (may still be sold?) which were all probably knock-offs of some manufacturer's original model. Ours are Cenco 71990-549, and are identical to the one in the photo here (but without the stepping motor). Here and here are manuals for a couple of other similar ones. (We no longer have the manuals for ours.) The arms are about 20 cm, the movable mirror can travel about 25 mm, and it has a micrometer that is accurate to about 2 um. What is a good educational application for these sweet old instruments? Here are the things I've tried, both by myself and using students as guinea pigs:   1. Crank the micrometer while counting fringes with a Na discharge tube. Determine the wavelength.   2. Like 1, but with laser light, projecting the fringes rather than squinting into the instrument.   3. Equalize the arms by finding the condition where fringes are visible with white light. Insert a sample of glass with an unknown index of refraction, and reacquire the condition of equal optical path length. Determine the unknown index of refraction.   4. The Na line is actually a close doublet, so as you move the micrometer you go back and forth from a condition of high-contrast fringes (the bull's eye patterns for the two wavelengths agree) to a condition of little or no contrast (the patterns disagree and counteract each other). If $\Delta d$ is the distance traveled between two successive positions of maximum contrast, then the splitting of the doublet is $\Delta \lambda\approx \lambda^2/2\Delta d$. Comments on how these went:   1. Incredibly tedious, and nearly impossible to do with high precision because you pass through the conditions of low contrast described in #4. The same result could be obtained with much higher precision in five minutes with a diffraction grating.   2. Not as bad as #1, but still doesn't use the instrument for an application to which it's uniquely well suited.   3. This was very cool. It was amazing how accurately the equal-length condition could be detected using this technique (to within 1 um). However, it doesn't seem to be a high-precision way to determine an unknown index of refraction, because you have to use a thin sample like a microscope slide, and your precision ends up limited by the determination of the thickness of the slide using Vernier calipers (e.g., 1.8 mm, so only 2 sig figs).   4. This is sublimely tricky, but I doubt that most students understand it. Measuring the spacing of a close doublet, while impressive to a spectroscopist, is of little intrinsic interest to a student in a freshman survey course. Other than the Michelson-Morley experiment (which obviously can't be done in a 3-hour student lab with a tabletop instrument), the classic historical application seems to have been the determination of the meter in terms of atomic standards. (See Hardy and Perrin, p. 584.) This doesn't seem practical for us. I've seen references to gas cells for use with this setup, presumably using the technique of #3. A back-of-the-envelope calculation seems to suggest that you would get a low-precision measurement, so I don't really see the point. Is it possible to use a technique sort of like #3 to determine the ratio of two wavelengths from the same source with high precision? This would have some intrinsic interest for hydrogen, whose wavelengths are in the ratios of small integers.


Maximum theoretical efficiency of internal combustion engine

This Wikipedia article states that: > Most steel engines have a thermodynamic limit of 37%. Is it correct? If so then it is not clear where this number comes from. Is it derived somehow from Carnot's theorem? I tried to google it but found nothing usefull. If this limit of 37% is wrong then what is correct limit?


Bravais lattice with sublattices : why multiple bands?

I have a very naive question : given a tight-binding model (with nearest- neighbor hoping) on a lattice defined by a Bravais lattice with a number of sublattices (for instance the honeycomb lattice is a triangular lattice with two sublattices), why is there a band associated with each sublattice ? For instance when a lattice is defined by a Bravais lattice with a 2 sublattice basis, the tight-binding model will have 2 bands.. But why is that ? There must be something very simple that I am not getting here... is it just because that to define a band structure, you need some translation invariance ?


How and why can random matrices answer physical problems?

_Random matrix theory_ pops up regularly in the context of dynamical systems. I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive example or a basic introduction to the topic? I would also appreciate hints on relevant literature.


Detail of deriving Berry Curvature From Berry Connection

The Berry curvature of the $n^{\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\vec{R}$ can be derived in part by writing the following two lines: $B^n(\vec{R}) \equiv \nabla \times A^n(\vec{R}) = -Im \left[ \left< \nabla n(\vec{R}) | \nabla n (\vec{R}) \right> \right]$ where $A^n(\vec{R})\equiv -Im \left[ \left< n(\vec{R})| \partial_{\vec{R}} n(\vec{R}) \right> \right]$ is the Berry connection. I don't know exactly how perform this step in the calculation in detail: $\nabla \times A^n(\vec{R}) = -Im \left[ \left< \nabla n(\vec{R}) | \nabla n (\vec{R}) \right> \right]$ Help would be greatly appreciated. Thank you very much.


Spoon released out of a satelite moving with orbital velocity

![enter image description here](http://i.stack.imgur.com/VwmZU.png)Suppose we are in a satelite revolving around the earth with a velocity equal to orbital velocity. Now if we release a spoon from the satelite then will it reach the earth or will it just revolve around the earth like the satelite with a velocity equal to orbital velocity?


Explanation of the classical coupling of the Higgs Field to Electromagnetism

I'm interested in learning about the classical coupling of the Higgs Field to Electromagnetism. There are numerous sources explaining the Higgs mechanism quantum mechanically, i.e. How does the Higgs mechanism work? In addition to the question about classical coupling I'm also interested in how the Mexican hat describes charged particles?


Neutrinos arrived before the photons (supernova)

A while back I read about the super Kamiokande detector detected a large neutrino flux and then several hours later a supernova was seen. Anyone know of this with sources? I don't recall the source at the moment.


How do electrons get the energy to jump from one orbital to the next when in stationary orbits the electron does not radiate energy

My Question is how do electrons get energy/lose energy to jump up or down an orbital as in a stationary orbit they do not gain energy and their energy is finite?


Point charge moving towards a conducting plane

A point charge $q$ of mass $m$ is released from rest at a distance $d$ from an infinite grounded conducting plane. Show that the charge hits the plane after an amount of time given by: $ \Delta t= \frac{\pi d} {q} \sqrt{2\pi\epsilon_0md} $ I can't seem to start this one. I'm assuming I need to start by finding the force by the charge as a function of $z$, and setting it equal to $\frac{md^2z}{dt} $ giving a differential equation. The diff eq. seems hard to solve, but a hint I recieved was to multiply both sides by $\frac{dz}{dt}$, but I don't see how this simplifies the problem.


Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?

I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did next is correct: I took the normalized eigenvectors, placed them in matrix form, and did matrix multiplication with the basis set of solutions. Let me try to be more precise since I am not sure I am using the right language when mentioning the basis solutions. In the problem we are using the set of solutions of the particle in a box model as our basis. I can increase the number of basis elements in the calculation of the matrix of the Hamiltonian (which amounts to doing $<\psi_n|H \psi_k>$ over a specified range of $n$ and $k$) in order for some of my smallest eigenvalues to begin to converge. Once I have this $H$ matrix built, and that I see that my eigenvalues are converging to some degree, I take the eigenvectors of the $H$ matrix, format them to be in matrix form, and multiply them by the set of basis solutions. I hope that makes things clearer.


Experimental evidence for the relic neutrinos

What are the experimental (indirect) evidence for the cosmic neutrino background? Where can I read more about this? The discussion on the wikipedia page about the C$\nu$B seems to me to be more about the evidence of the number of generation of neutrinos, than about the cosmic neutrino background...


A question about Gell-Mann Low theorem?

Let $|\Psi_0\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\epsilon=H_0 + e^{-\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\epsilon \rightarrow 0^+ $ and $|t|\rightarrow\infty$. Let $U_{\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\epsilon\rightarrow 0^+$ of $|\Psi^{(\pm)}_\epsilon \rangle = \frac{ U_{\epsilon I} (0,\pm\infty) |\Psi_0 \rangle}{\langle \Psi_0 | U_{\epsilon I}(0,\pm\infty)|\Psi_0\rangle}$ exists, then $|\Psi^{(\pm)}_\epsilon \rangle$ are eigenstates of $H$. Here is my question: In interacting field theory, whether the infinite past state $|p_1p_2>_{in} = a^{\dagger}_{in}(p_1)a^{\dagger}_{in}(p_2)|0>$ is some kind of $|\Psi_0\rangle$ I think it may be wrong. Where is my fault? Thanks!


Needle Floating on a Water Surface

So, I'm trying to find the largest radius of a needle before it ends up sinking, and I know that it's a balance of needle weight vs surface tension. However, I'm not really sure how the surface tension force is defined. In a lot of resources, I see the surface tension force define as gamma, which is the force / length. Also, I am seeing surface tension defined in terms of curvature give by the Young-Laplace equation. Which one am I supposed to use? If I can use the gamma one, it would be straightforward, but I'm not clear on what length to use. Obviously, it's the portion of the circumference in contact with the needle, but this length can vary based upon a lot of factors. Aside from that, I'm thinking that I would have to set the contact angle equal to zero (for the maximum surface tension force), but does that mean that I use gamma*2*pi*r? Wouldn't that imply that the needle is completely submerged? Any help would be appreciated!


Is the gravitational potential of a planet in orbit always equal to minus the squared velocity?

Say a planet (mass $m$) is orbiting a star (mass $M$) in a perfect circle, so it is in circular motion. $F=ma$ and the gravitational force between two masses $F=\frac{GMm}{r^2}$ so $\frac{GMm}{r^2}=ma$ $\frac{GM}{r^2}=a$ And in circular motion $a=\frac{v^2}{r}$ so $\frac{GM}{r^2}=\frac{v^2}{r}$ $\frac{GM}{r}=v^2$ And gravitational potential $V=-\frac{GM}{r}$ So $v^2=-V$ Is there a (qualitative/less math/wordy) reason why this is the case? (or have I got this wrong?) and is this limited to the specific case of perfect circular motion?


While holding an object, no work done but costs energy (in response to a similar question)

I read the answer to Why does holding something up cost energy while no work is being done? and wanting to know more, I asked my teacher about it without telling him what I read here. Instead of referring to muscle cells and biophysics, he answered my question in terms of entropy. He told me that while my arm muscles are stretched when I hold the object, they are more ordered. When my arm is at rest and muscles are not contracted, the muscles are less ordered (more entropy). So his conclusion was that the energy is required to keep the system (my arm muscles) from going to a state of higher entropy. However, the answer in terms of muscle cells doing work on each other (i.e the answer to the hyper-linked question) made more sense to me. Could someone please give me some intuitive sense to my teacher's answer or explain the link between the two answers if there are any...


Calculating the uncertainty of $r$, when $1/r^2$ is used

I have tried to search through the forums and the Internet, however I haven't been able to find a source I was able to understand fully. I have a distance $r$, with an uncertainty of $\mathrm{±\ 0.003\ m}$, however in order to linearise my graph, I need to divide $1$ with $r$ squared ($1/r^2$), what would the final uncertainty for $r$ be? - Furthermore, what would the unit be? Would that still be $m$? or $m^2$?


Rijke tube: Why Sound is maximum when heated 1/4th length from bottom of tube

Why is it that the maximum sound coming from a Rijke tube happens only when heated wire gauze is placed at 1/4th the length of the tube from the bottom. According to Rayleigh: > If heat be periodically communicated to, and abstracted from, a mass of air > vibrating in a cylinder bounded by a piston, the effect produced will depend > upon the phase of the vibration at which the transfer of heat takes place. > If heat be given to the air at the moment of greatest condensation, or be > taken from it at the moment of greatest rarefaction, the vibration is > encouraged. On the other hand, if heat be given at the moment of greatest > rarefaction, or abstracted at the moment of greatest condensation, the > vibration is discouraged. Wouldn't the sound be maximal if heated gauze is placed at the bottom, where there exists an antinode?


The Synchrotron at CalTech

In the Feynman Lectures Vol. 1, it says that the Synchrotron at the California Institute of Technology is capable of producing electromagnetic Radiations with a frequency of $10^{24}$ cycles per second. Since I'm not familiar with _cycles_ , I'd like to know where this lies in the UV Spectrum.


Is it possible to observe interference from 2 independent optical lasers?

It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true? How fast should photo-detectors be for observing of the interference of beams from two of the "best available" lasers? What is the coherence length of the best- available laser? More specifically, does there exist any (optical single- wavelength) laser with coherence length exceeding 500 meters?


What if the LHC doesn't see SUSY?

A question in four parts.   1. What are the main problems which supersymmetry purports to solve?    2. What would constitute lack of evidence for SUSY at the proposed LHC energy scales (e.g. certain predicted superpartners are not in fact observed)?    3. Are there alternative theoretical approaches which would address the SUSY problem set and which would still be credible in such an LHC no-SUSY-scenario?   4. Where would LHC-disconfirmation of SUSY leave String Theory? I would like to think that these four points could be taken together as one question.


What is going on in the system? How are the formulas `mg sin(x)` and `mg cos(x)` derived?

When a load is resting on an inclined plane, there is force $mg \sin(x)$ that's vertical to the inclined plane and force $mg \cos(x)$ horizontal to the plane acting on it like this: ![enter image description here](http://i.stack.imgur.com/mfCsM.png) My textbook says the weight $mg$ has two components: $mg \sin(x)$ and $mg \cos(x)$. But when adding up $mg \sin(x)$ to $mg cos(x)$ it does not equal to $mg$. If `mg` and N are the only forces in this system, which means the net force should be 0, how should I explain the existence of $mg \sin(x)$ and $mg \cos(x)$? Does $mg \sin(x)$ and $mg \cos(x)$ have their own reaction force to cancel them out and turn the net force to be 0 too or is it wrong the say the system's net force is 0? And how are the formulas $mg \sin(x)$ and $mg \cos(x)$ that are used to describe these two forces derived?


Why do neutrons repel each other?

I can understand why 2 protons will repel each other, because they're both positive. But there isn't a neutral charge is there? So why do neutrons repel? (Do they, or have I been misinformed?) The reason why I'm asking this is because, I've just been learning about neutron stars and how the neutrons are forced (as in, they repel) together according to my teacher (he's a great teacher btw, though what I just said doesn't make it seem so). So I wondered, why they have to be forced by gravity and not just pushed?


If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?

> Are the more recently discovered symmetries in string theory such that the > theories based on mirroring geometries are absolutely the same from an > observable point of view? I have mirror symmetry in mind, i.e. by symmetries I mean the dualities encountered in string theory. * * * (v2) Say you take e.g. take the Schwarzschild metric from general relativity and consider its quivalent in some theory of quantum gravity theory. There, if you don't look close enough, i.e. if you look at averages, the quantum- Schwarzschild space would look classical, namely like the general relativity Schwarzschild space. Now the target space in string theory is one manifold and it leads to a quantized theory. But if there are now actually two mirroring manifolds where this theory can come from, the I guess the classical version looks like neighter of these geometries. The following question comes from this observation, that there are totally different (mirroring) options with which Calabi-Yau you can start (considered in the Lagrangian) if you go in the quantization direction and the conclusion that neighter of will be the limit, or there are more limits. > If we assume that the world is described by such a theory, do we besically > live in two indestingushable geometries at once? To put it differently, I ask if string theory can be considered a quantum gravity in the sense that there is a classical limit down to classical differential geometry, especially regarding the extra dimensions. And if so, to what geometry in these higher dimensions does the limit lead? I do know that at least some of the geometry must come from the string excitation representing a quantizations of a metric/graviton. But I suspect different metrics in the non-linear sigma model Lagrangian will also affect the classical picture in some essential way, directly or not.


Refraction of light in water

Here's a question in a paper based on refraction of light. I can't seem to solve it for some reason. > A man looks down at a fish length of 20 cm. His eye is 2m above the surface > of the water ($\mu = 4/3$) an the fish is 2m below the surface shown in the > figure. The ratio of angular width $\Delta\theta_1$ of the fish as seen by > the man in presence of water to the $\Delta\theta_2$ in the absence of water > is:   >  _(A) 6/5 $\hspace{3cm}$ (B) 5/6 $\hspace{3cm}$ (C)7/8 $\hspace{3cm}$ > (D)8/7_ ![fish](http://i.stack.imgur.com/5zvpD.jpg) * * * _Here's what I've done:-_ When there is no water present: $$\frac{\Delta\theta_{air}}{2} = \sin^{-1}(10/400)$$ $$\therefore \Delta\theta_{air} = 2\sin^{-1}(1/40)$$ ...from the triangle obtained by the eye, and half the fish. When the water is present: ![diag](http://i.stack.imgur.com/qCTOk.jpg)   ($P.S:$ the equation written below the diagram is wrong, I've just realised. I've written the correct version below) Let the light rays at angle $\phi$ with the normal refract at the water surface and converge at the eye. Here the angle of refraction is $\theta$. So we have: $$\sin\phi . \mu_{water} = \sin\theta . \mu_{air}$$ $$\therefore\frac{\sin\theta}{\sin\phi} = \frac{\mu_{water}}{\mu_{air}}$$ $$\therefore\frac{\sin\theta}{\sin\phi}=\frac{4}{3}$$ After which I am stuck and have no clue how to go about this. Can anyone help me? * * * _**Addendum_** : Will taking $\sin x = x$ help? $x$ is very small anyway, and $\sin x \approx x$ is turning out to be true ($x = 0.025$, $\sin(0.025) = 0.024997...$). _**Addendum2_** : Another thing I figured out is that the water will increase the observed angular width, so the ratio is definitely $>1$. So we can safely rule out (B) and (C) which are $<1$.


What happens to an astronaut who's floating in a spaceship (in space) when it begins to move?

I feel this is somehow a stupid question, but I don't know the true answer. What happens to an astronaut who's floating in a spaceship in space when it begins to move? Will the astronaut not move until he smashes onto a wall in the spaceship? Or will he move with the spaceship due to gravity? Or does it depend on the speed/acceleration of the spaceship; if it's too fast, the astronaut will smash onto a wall and if it's enough slow, he'll move with the spaceship? Thanks for helping me out!


Is the "consistent histories" interpretation of QM a "many worlds interpretation" in disguise?

The so called consistent histories interpretation is claimed to be a correction of the Copenhagen Interpretation. One of its aim, as much as I can see is to show that observers don't have any special role and a history is represented as a sequence of projection operators in the Heisenberg picture. It has been convincingly shown that due to environmental decoherence the density matrix will be almost diagonalized within a short time (expressed in terms of a certain basis of interest). It will smell just like a "collapse" of the wave function. However, if we want to see the whole picture, we see that superposition still exists in the entangled whole. My question is whether this interpretation is actually a "many worlds interpretation" in disguise? If not, then why not?


Is it possible to write a Density Matrix in the following form?

Is it possible to write an arbitrary density matrix $\hat{\rho}$ in the following form ? $$\hat{\rho} ~=~ \frac{1}{N} \sum_{\ell=1}^N \left|x_{\ell}\right\rangle \left\langle x_{\ell}\right|,$$ where $\left\\{\left|x_{\ell}\right\rangle\right\\}_{\ell = 1}^{N}$ are normalized states (but not necessarily orthogonal). If yes, how can one prove this ?


Poynting vector direction

Just a quick question if I may. The Poynting vector, or the energy flux density, is given by: $\mathbf{S} = \frac{1}{\mu_{0}}(\mathbf{E} \times \mathbf{B})$ So it's the cross product between the $\mathbf{E}$-field and $\mathbf{B}$-field. So depending on the direction of the fields, the Poynting vector will point in some direction. So lets say the $\mathbf{E}$-field has the direction $\mathbf{e}_{y}$ and the $\mathbf{B}$-field has the direction $\mathbf{e}_{z}$, then the resulting direction for $\mathbf{S}$ will be $\mathbf{e}_{z}$. So my question is, is that the direction of which the energy is flowing, or is there some fancy thing I need to know, like it's the opposite or something like that ? Thanks in advance :)


Beta function of pure $SU(N_\text{c})$ Yang-Mills theory

What is the dependence of the beta function of pure $SU(N_\text{c})$ Yang- Mills theory on the number of colors? I guess $$\mu\frac{dg_\text{YM}}{d\mu}=-\beta_0N_\text{c}g_\text{YM}^3-\beta_1N_\text{c}^2g_\text{YM}^5-\beta_2N_\text{c}^3g_\text{YM}^7-\cdots\,,$$ with the appropriate constants $\beta_0,\beta_1,...$, as has been computed in QCD (including quarks) at four loops  [arXiv:hep-ph/9701390]. Is this correct? How could it be proved (to all orders)? Then the 't Hooft coupling $\lambda=g^2_\text{YM}N_\text{c}$ runs independently of $N_\text{c}$: $$\frac{\mu}{2}\frac{d\lambda}{d\mu}=-\beta_0\lambda^2-\beta_1\lambda^3-\beta_2\lambda^4-\cdots\,.$$ The goal is to assure that in the large $N_\text{c}$ limit the coupling runs independently of $N_\text{c}$, so that the confinement scale is held fixed. So the series in $\lambda$ could be truncated for $N_\text{c}\rightarrow\infty$ (there could be negative powers of $N_\text{c}$).


What happens to pipe outlet temperature here?

I have a fluid flowing through a pipe in the ocean and there is heat transfer from the ocean to the fluid in the pipe. I prepared a simulation and the results show that if I increase the mass flow rate of the fluid through the pipe then the heat transfer rate to the fluid increases and the outlet temperature increases. These results don't seem right to me. I would think if the heat transfer rate increases, then the outlet temperature should also increase. Does anyone have any suggestions as to why my intuition might be wrong? It is also quite possible the simulation could be wrong.


Describing quantum intereference with only currents and densities

I know about and believe to understand the general wave equation based Kirchhoff diffraction formula, which in the Fraunhofer limit leads to a farfield complex wave function by Fourier transforming the complex wave function of the incident beam in the aperture. I believe it should be possible to describe this phenomenon through only the density (wavefunction squared) and probability current (quantum mechanically). These two should be able to provide the phase and amplitude information necessary to interfere something. What I'm looking for is a "hydrodynamic" theory of diffraction and/or interference, which I think should be very well possible. Has this been done before? Am I barking up a wrong tree?


Does the universe follow a general physical and computational principle of evolution?

Evolution is a principle in biology, whereby organisms evolve their ability to replicate and multiply in numbers over successive generations. From a computational point of view, the organisms employ a common programming (DNA, RNA, etc.) and runtime environment (cell biology). This evolution has eventually evolved a more powerful ((in a restricted sense) computational environment, the hierarchy of biological brains, with the currently most powerful class instance being the human brain. A collection of human brains is now working on evolving a quantum computer. Perhaps one day the the quantum computers will produce a.... From an anthropic point of view, we are here to think about these things, because our universe creates environments such as our planet earth, which support the evolution of life. So my question is whether there is a general physical and computational principle at work here, demonstrated by the following incomplete, in parts almost surely incorrect, and highly speculative hierarchical chain of evolution:   1. Within the Multiverse, universes are instantiated following particular physical laws originated and adapted from their embedding universe   2. Each instantiated embedded universe follows computational rules (the particular physical laws of this universe) and create a finite number of embedded universes (the only candidate seems to be for singularities of black holes to correspond with a white-hole, inflationary, embedded universe), each of which evolve in the same general manner   3. Each universe produces a finite number of computational schemes to build organisms that follow certain classes of program patters. These organism evolve according to the theory of biological evolution.   4. Biological organisms evolve biological brains, a higher class computational scheme   5. biological brains build computational devices, which at some point attain the property of replication   6. The replicating computational devices build higher classes of computational devices, following the general principle of evolution...


How does one calculate where the "surface" of a gas-giant would be?

Okay, so Jupiter, Saturn, et. al are gas giants. I understand that they have large gassy atmospheres, which, due to the pressure would eventually become more and more dense as one approaches the center of the planet. Falling into Saturn or Jupiter, would we pass through it until we hit the nucleus? hits my model about right - gas eventually compressed by gravity would form, I suppose, a liquid core, followed by a solid surface. Assuming that Jupiter and Saturn were made of pure hydrogen (It's actually about 90%, but I'll simplify it a bit, how would one go about calculating the distance at which there is a "surface"? Put another way, at some distance, gravity should compress the gas into a liquid, and at a point even lower, I would assume that the liquid would compress into a gas. Is there an equation that would tell me what those points are? And, for bonus points, assuming that Saturn has a diameter of 72,367 miles / 116,464 km, at what distances would the transition points from solid to liquid and liquid to gas be? Full disclosure: I do realize that the same surface pressures would actually implode a human being trying to step on said surface. Despite desires to attempt this in reality, I am not actually going to "try this at home." This is a thought experiment more about how I calculate where gravity forces a compound into a different state of matter. I am curious more about how one models the effect of mass into gravity and compression. As a thought experiment, I suppose technically this is not a 'problem I face,' but I am curious. Thanks in advance :)


$d=\frac{1}{2}at^2$ not giving expected results

Say I have an object that starts falling from rest at 2 m/s2. Its total displacement and velocity at the end of every second should look something like this:               t    d    v     1    2    2     2    6    4     3    12   6     4    20   8      At the end of the first second, it should have accelerated to 2 m/s and moved 2 m. At the end of 4 seconds, it should moved 20 m. But when I use the equation for getting displacement, $d=\frac{1}{2}at^2$, I get 16 m. Shouldn't it be 20 m? What is going wrong here?


Rotational speed of a coil in a uniform magnetic field at equilibrium

I'm looking at the following problem from "Physics 3" by Halliday, Resnick and Krane (4th edition): > The armature of a motor has 97 turns each of area 190 cm² and rotates in a > uniform magnetic field of 0.33 T. A potential difference of 24 V is applied. > If no load is attached and friction is neglected, find the rotational speed > at equilibrium. My attempt is as follows: Initially, the potential difference is 24 V. Because of this potential difference, current starts flowing through the motor. Then, because of the 0.33 T magnetic field, the motor starts rotating. The rotational motion changes the magnetic flux through the motor's coil, which produces an induced EMF (electromotive force) that opposes the original EMF. If we call N the number of turns, A the area of the motor's coil, and θ the angle from the magnetic field vector B to the vector normal to the plane of the coil, then the magnetic flux through the coil is: $\Phi=NAB\cos{\theta}$ So, the induced EMF is: $\varepsilon_{ind}=-\frac{\mathrm{d} }{\mathrm{d} t}(NAB\cos{\theta})=NAB\frac{\mathrm{d} \theta }{\mathrm{d} t}\sin{\theta}$ If we call ω the angular velocity as a function of time ($\frac{\mathrm{d} \theta }{\mathrm{d} t}$), then the net EMF as a function of time is: $\varepsilon = 24 - \varepsilon_{ind} = 24- NAB\omega\sin\theta$ Is this correct so far? I'm not sure how to proceed from here. Anyway, the answer given in the back of the book ( **39.5 rad/s** or **6.3 rev/s** ) seems to suggest that the correct way of finding ω is by using $\varepsilon=NBA\omega$ and plugging in the values, that is, ε = 24 V, N = 97, B = 0.33 T and A = 0.0190 m². This gives the correct result of 6.3 revolutions per second. But I'm not sure how to obtain $\varepsilon=NAB\omega$ from the expression that I found above ($\varepsilon = 24 - NAB\omega\sin\theta$). * * * _**Edit**_ : Based on the suggestions given in the comments to the answer below, I understand that the "equilibrium" that this question is referring to is when the torque on the coil reaches the value zero; this happens when the total EMF ($\varepsilon = 24 - NAB\omega\sin\theta$) on the coil equals zero. That is: $\omega = \frac{24}{NAB\sin\theta}$ To reach the desired result, $\sin\theta$ should equal $1$; but I don't understand why that is the case here. Is this reasoning correct? How should I proceed from here?


when is the stationary phase approximation exact?

I am thinking about some topological field theories, and I am wondering when one can say that the stationary phase approximation (ie. a sum of the first- order variations about each vacuum) is exact. I am looking perhaps for conditions on how the space of vacua is embedded into the space of all field configurations. I suspect that when the action is a Morse function (and I suppose the space of field configurations is finite dimensional) that the exactness of the stationary phase approximation implies some very strict topological constraints on the configuration space... torsion-free and so on. Anyone have a good reference or some wisdom? Also I'd like to dedicate this question to the memory of theoreticalphysics.stackexchange


Physical explanation for capacitive circuit

In capacitive circuit, the voltage and current are out of phase. the current leads the voltage by 90 degrees. this can be explained by these two equations: $$V = V_{\max} \sin \omega t$$ $$I = I_{\max} \cos \omega t.$$ Now, my question is that what does this statement mean physically that "current leads the voltage by 90 degrees? It is clear mathematically that if $V = 0$ then $I$ is maximum but how do we explain this physically? hope you understood my question


Is kinetic energy in QM a state-property or is it distributed?

Suppose we have a quantum mechanical system, which is well described by its wave function in r-representation $\Psi$. We are interested in the properties of an observable, say the kinetic energy $T$. So we let the kinetic energy operator act on the wave function, multiply the result by Psi and integrate over spatial coordinates. Thus we obtain the expectation value $\langle T \rangle$. We note that this result is a property of the operator and also of the quantum state. Now in QM text books one will often find detailed discussions of the result, in order to provide physical meaning and context. For example, in discussing the kinetic energy of the ground state of a hydrogen atom, it is pointed out that the kinetic energy $T$ is intimately related --through the Schroedinger equation-- to the electrostatic potential $V$ and the total energy $E$. Both $V$ and $E$ are operators that are multiplicative in $r$-representation. So $T$ (which at its core is represented by the Laplace differential operator), when written as $E$-$V$, may be regarded as multiplicative. Thus we get (with energy in Rydberg energy units and distance in Bohr radius units): $$T = -1 + \dfrac{2}{r}$$ This result is then discussed in terms of "the classical turning point" $r=2$. The region $r > 2$ is forbidden in classical physics, since the kinetic energy is here negative. Yet in QM there is a non-zero probability that the electron is found at $r > 2$. I wonder how much "truth" there is the above formula for $T$. It is suggestive to regard it as a proper function of the distance to the nucleus $r$. However, as soon as one digs a little deeper, conflicts arise. For example, one may compute the Fourier transform of the wave function. Now since the initial wave function was spherically symmetric, so is the FT. Hence in $\Phi(k)$ there are only states with positive wave factor $k$. There is simply no room for negative kinetic energies! So immediately a paradox arises when we regard kinetic energy as a distribution in $r$- or $k$-representation. Furthermore, one may look at the higher moments of the kinetic energy operator. This leads to new problems. For example, the second moment $\langle TT\rangle$ can be computed as either $(\Psi, TT \Psi)$, or as $(T \Psi, T \Psi)$ [provided $T$ is self-adjoint when acting on these wave functions]. Of course it is satisfactory that these two integral representations lead to the same net result. However their integrands are obviously different! Therefore if the kinetic energy is distributed within the state, it can not be pin- pointed without some level of ambiguity. I would like to hear your comments on this issue !


Superficial Degree of Divergence for Feynman Diagrams

The superficial degree of divergence for a diagram is defined as the power of $k$ in the nominator minus the power of $k$ in the denominator. It is written to be equal to $4\times$ (number of loops)$ - $(number of internal Fermion lines)$ - 2 \times $(number of internal Boson lines). Why is it 4 and not 3? When you do a 4-dimensional integral in spherical coordinates, you get a factor of $k^3$ from your volume element. In particular, when you calculate the QED vertex, for instance, you get: $$ \int_{\mathbb{R}^4} \frac{d^4k}{(2\pi)^4} \frac{\gamma^\rho (\not\\!{p_1}-\not\\!{k}+m)\gamma^\mu(-\not\\!{p_2}-\not\\!{k}+m)\gamma_\rho}{k^2\left[\left(p_1-k\right)^2-m^2\right]\left[\left(p_2+k\right)^2-m^2\right]} \overset{k\to\infty}{\propto} \int_{\mathbb{R}^4} \frac{d^4k\, k^2}{k^6} \overset{\text{spherical}}{\underset{\text{coordinates}}{\propto}} \int_0^\infty\frac{dk}{k} $$ Thus I would say that the superficial degree of divergence of this diagram is $3\times1-2-2=-1$ and not $0$. What am I missing?


Calculating new temperature of an object when air temperature changes

I'm trying to calculate the new temperature of an object when the air temperature around it changes, given a period of time. Basically I get periodic readings from an air temperature sensor in a refrigerator. In some cases these readings are every 5 minutes, in others every 1 minute, so the time between readings is variable. For each reading I get, I'd like to also calculate the approximate temperature of food at its core; something like a chicken for example (I know that part is vague, but if there is a variable I can tweak then that is fine). The result should be a "damped" version of the actual air temperature, as obviously any objects will slowly change temperature to eventually meet the air temperature. Initially there used to be "food simulant" put around the sensor, so the temperature would automatically be damped, but this is no longer the case. I do not know much about thermodynamics. I'm not sure if I can just add a percentage of the temperature change to the previous damped value, or if I need a calculation based on the last few air temperature readings, or what. I guess I'm looking for a result a bit like:               10:00 2 degrees (air), 2 degrees (product)     10:05 2.5 degrees (air), 2.1 degrees (product)     10:10 2.5 degrees (air), 2.2 degrees (product)     10:20 2.7 degrees (air), 2.5 degrees (product)      I could do something really cheap like averaging the readings over the last 30 minutes but I don't think that will cut it! I'd appreciate any help - thanks very much.


Another faster-than-light question

Imagine we have something very heavy (i.e supermassive black hole) and some object that we can throw with 0.999999 speed of light (i.e proton). We are throwing our particle in the direction of hole. The black hole is so heavy that we can assume that in some moment acceleration of gravity would be say 0.0001 speed of light/s^2. So the question is what will be the speed of proton in few a seconds later, assuming we have such distances that it will not hit the black hole before.


How do I use MAUD for phase quantification of XRD data with AMCSD standards?

I'm a biologist with some raw powder XRD data of biominerals, need to quantify different phases, can't access commercial software or databases at the moment. Does anyone have experience with using the cif standards from the AMCSD with MAUD? It seems to display the list of phases fine, but when I pick one it doesn't load. Alternatively, suggestions of other free/trial software that works on Win 7, accepts the AMCSD and is not too impenetrable to noobs would be welcome. I have tried XPowder12 trial (which kept crashing), FullProf (which I couldn't figure out), and GSAS (which I couldn't even install). MAUD seemed like such a nice program until it didn't load the standards...


Seeing cosmic activity now, really means it happens millions/billions of years ago?

A Recent report about a cosmic burst 3.8 billion light years away. It is written as though it is happening now. However, my question is, if the event is 3.8 billion light years away, doesn't that mean we are continuously looking at history, or is it possible to detect activity in "realtime" despite the distance?


Planck satellite 2013 results impact on cosmological models

I've recently watched a "Through the Wormhole" episode from 2010 named "What Happened Before the Beginning?", where cyclic and ekpyrotic cosmological models are explained. In this episode the creators of the ekpyrotic model, Neil Turok and Paul Steinhardt, say that the results from the Planck satellite about gravitational waves will either support one theory or the other. After doing a little bit of research about the Planck satellite I've come across some 2013 results on the Wikipedia page. However, this information is far too complex for me to understand. How does this information affect the cyclic and ekpyrotic models? Are the results leaning towards one of the models?


Information Preservation and Burning Books

I recently read an article in the NY Times called A Black Hole Mystery Wrapped in a Firewall Paradox. I really liked the article, but reading one quote immediately made me think of asking Physics.SE a question: > From the material in the smoke and flames of a burning book, for example, > one could figure out whether it was the Bible or the Kama Sutra; the same > should be true of the fizz and pop of black holes, these physicists argued. So, Physics.SE, _theoretically_ , how would you figure out what book was burned from the smoke and flames? I do not intend for this to be a silly question. If the idea of "information loss" inside of a black hole is so difficult for some physicists to come to terms with, I figure those same physicists should have a perfectly good answer to my question.