question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
|---|---|---|
<p>Reading about cloud formation, I learned that to a droplet to form, the water vapour needs a <a href="http://en.wikipedia.org/wiki/Cloud_condensation_nuclei" rel="nofollow">Cloud Condensation Nuclei</a>, which is an aerosol with a size in the order of 0.0001mm. And if no CCN is found temperatures as low as -13°C are needed to form the droplets. Why this is happening? What does the contact of the solid with water do to make it condense?</p> | g10286 | [
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<p>From what I understand the expansion of the universe has no "center". If we're flying through space away from the "center of the big bang", there's basically no way to tell. Every two given points in space gets farther away from each other, and we can pick any point as center if we like.</p>
<p>I also understand that the speed of light is not relative to the speed of the source emitting the light. If I go on a train in $c/2$, turn on a flash light pointing forward, the light emitted from the flashlight will still travel at the speed of $c$.</p>
<p>Now here's my question: Why can't we set up a sphere with photodetectors with synchronized clocks on the inner walls, turn on a light in the center, record the exact time at which each photodetector detects the light, and compare the times to figure out if the sphere was traveling in some certain direction?</p>
<p>I mean if we turn on a lightbulb and light travels with the speed of $c$ in all directions at the same time, my intuition tells me that we should be able to figure out some form of "reference stand still".</p>
<p>(Tagging this with general relativity because I <em>suspect</em> that it's impossible to set up the experiment the way I like due to relativity.)</p> | g10287 | [
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<p>Could someone help explain the uses of AC <a href="http://en.wikipedia.org/wiki/Electromagnet" rel="nofollow">electromagnets</a>. Wherever I look it says that DC electromagnets create stronger magnetic fields. I understand why AC electromagnets could be used in transformers but why use them in motors for example? Power stations also use electromagnets in the generator to create the magnetic field, would these be AC or DC? Finally, which type of current would electromagnets in household motors use e.g. food processors, blenders etc?</p> | g10288 | [
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<p>This question came to me when I realized how the linear speed varies while listening to a Vinyl LP.</p>
<p>The linear speed variation has to be compensated with a variation in the resolution of the grooves, that is, since the linear speed decreases, the groove resolution also has to decrease in some measure. What is this measure of reduction, or else, how much does the linear speed reduce? And how does that influence the sound definition?</p> | g10289 | [
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<p>I <a href="http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1431&context=ecetr" rel="nofollow">read</a> a statement saying that the inner product between divergence-free current and a gradient field is zero.</p>
<p>Divergence-free surface current is $\nabla\cdot\vec{J}=0$, and $\vec{J}$ could be represented as $\vec{J}=\nabla\times(\psi\hat{n})$, where $\hat{n}$ is the normal vector of the surface. So the statement becomes: $\nabla\times(\psi\hat{n})\cdot \nabla \varphi=0$.</p>
<p>I think according to the identity: $$\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$$
we have
$$\nabla\times(\psi\hat{n})\cdot \nabla \varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)+\psi\hat{n}\cdot\nabla\times\nabla\varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi),$$
but what next?</p>
<p><strong>Update</strong>
Thank you Luboš Motl. I suppose I now understand why, but I have no enough points to reply below, so just update here my answer.</p>
<p>Target is to prove $\int_s \vec{J}\cdot\nabla\varphi ds=0$
The whole process is as follows:</p>
<p>First, $\vec{J}$cannot go across the surface edge, so $\vec{J}\cdot\hat{t}=0$,
where $\hat{l}$ is the surface edge direction and $\hat{t}=\hat{l}\times\hat{n}$ is the edge out direction</p>
<p>Second, according to the identity $\nabla\cdot(\vec{A}\times\vec{B})=\vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$, we have $\vec{J}\cdot\nabla\varphi=\nabla\times(\psi\hat{n})\cdot \nabla \varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)+\psi\hat{n}\cdot\nabla\times\nabla\varphi=\nabla\cdot(\psi\hat{n}\times\nabla\varphi)$</p>
<p>since $\nabla\times(f\vec{A})=\nabla{f}\times\vec{A}+f(\nabla\times A)$</p>
<p>$\psi\hat{n}\times\nabla\varphi=-\nabla\times(\varphi\psi\hat{n})+\varphi\nabla\times(\psi\hat{n})$
Then
$\nabla\cdot(\psi\hat{n}\times\nabla\varphi)=\nabla\cdot(-\nabla\times(\varphi\psi\hat{n})+\varphi\nabla\times(\psi\hat{n}))=\nabla\cdot(\varphi\nabla\times(\psi\hat{n}))$</p>
<p>finally,</p>
<p>$\int_s \vec{J}\cdot\nabla\varphi ds=\int_s\nabla\times(\psi\hat{n})\cdot \nabla \varphi ds
=\int_s \nabla\cdot(\varphi\nabla\times(\psi\hat{n}))ds=\oint_l \varphi\nabla\times(\psi\hat{n})\cdot\hat{t}dl=\oint_l \varphi\vec{J}\cdot\hat{t}dl=0$</p>
<p>I think here the important things are:</p>
<ol>
<li><p>Generally speaking, divergence-free current usually can be expressed as $\vec{J}=\nabla\times\vec{T}$, and $\vec{J}=\nabla\times(\psi\hat{n})$ is specially for surface current.</p></li>
<li><p>the $\hat{n}$ is only valid on the surface(there is no meaning of $\hat{n}$ for point in side of a body). the integral is on the surface rather than on the body. According to the original article, it is just talking about PEC and surface current.</p></li>
</ol> | g10290 | [
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<p>I read in a physics today paper,</p>
<p>The electron can have nonvanishing EDM only if nature violates symmetry under time reversal (T) and under the combined operations of charge conjugation (C), which replaces particle by antiparticle (as we all know), and parity inversion (P).</p>
<p>I checked <a href="http://physics.stackexchange.com/questions/34484/what-makes-electric-charge-special-wrt-cpt-theorem">here</a>. So it will be nice if you can show me qualitatively and quantitatively about the above mentioned point.</p>
<p>Note: Simplicity in explanation is always nice. (I don't have much idea of these big words)</p> | g329 | [
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<p>Our eyes contain 3 photoreceptor cells (cones) to perceive three wavelength ranges of light. Here is a visual representation of the wavelengths by these receptors (S, M and L).</p>
<p><img src="http://i.stack.imgur.com/5olj4.png" alt="Wavelength per photoreceptor"></p>
<p>So if we have light of 440 nm, it results in the color blue. If we have light of 540 nm, it results in the color green. If we see light of 650 nm, it results in the color red.</p>
<p>I think I understand our ability of the brain to mix the results of the signals of these receptors, producing colors like yellow. However, what I do not understand is how the color spectrum is displayed like this:</p>
<p><img src="http://i.stack.imgur.com/IUxg1.gif" alt="Visible spectrum representation by wavelength"></p>
<p>Given that spectrum, I would suggest that the color 'blue' is actually a mix between the receptors S and M. And the pure result of activating the S-receptor would result in the color 'purple' (I would describe the color in the left of the image as purple, right). Therefore, the receptor colors should instead be RGP (red, green, purple) instead of RGB (red, green, blue).</p>
<p>However, there is one problem with this which I cannot explain. How come that mixing red light with blue light also results in purple light? How is it possible that purple light can be achieved through mixing (additively) blue and red light, just as going to the shortest wavelength boundary of what we can see (from blue to ultraviolet via purple)?</p>
<p>So the actual problem here is:</p>
<ol>
<li>Purple is the color at the very shortest wavelength we can see.</li>
<li>Purple is an additive mix between what we see as red light and blue light.</li>
</ol>
<p>That just doesn't make any sense. I don't see how our brain can possibly perceive this as being the same color. Shouldn't both purple colors actually be different colors (thus we would have a new different color for that)?</p> | g10291 | [
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<p>Is the construction of an etalon / <a href="http://en.wikipedia.org/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer">Fabry-Pérot interferometer</a> within the reach of amateur telescope makers? Are there any resources pointing to such projects?</p> | g10292 | [
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<p>What do these units mean: the large velocity widths of emission lines (in AGN) are 2,000 - 10,000 km s^-1? I've looked for the answer but keep getting swamped in myriads of details. I want to know what the term <em>velocity</em> means in <em>large velocity widths</em> and also why <em>km s^-1</em>. Thanks</p> | g10293 | [
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<p>I would like to know if acceleration is an absolute quantity, and if so why?</p> | g10294 | [
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<p>Astronomers estimate that there are between 200 billion to 400 billion stars contained within the Milky Way and Andromeda Galaxy probably has 1 trillion stars. There may be around 500 billion galaxies in the observable universe. </p>
<p>So, my question is, statistically speaking because of the number of stars, there is should be lots of chances of stars going supernova; where at times they glow brighter than a whole galaxy. Why then do we not see many supernovas, for example like the 1987A? Why have we not been able to see one in our own galaxy since the SNR G1. Should there not be more supernovas in surrounding galaxies and even our own one as it has around 400 billion stars. </p>
<p>I appreciate that there are different types of stars and various life span, but our galaxy being almost as old as the universe, surely there should be stars dying all the time.</p>
<p>Thanks</p> | g10295 | [
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<p>It is generally believed that $10^{-35}$ seconds after the Big Bang, the symmetry of a GUT was broken and after $10^{-12}$ seconds the electroweak force was broken:</p>
<p>\begin{equation}
\mathrm{SU(2)} \times \mathrm{U(1)} \rightarrow \mathrm{U(1)}
\end{equation}</p>
<p>This symmetry breaking is a result of the universe cooling down and undergoing a phase transition. I'm aware that the temperature of the universe it about $2.7$ Kelvin, so the temperature of the universe cannot decrease much more, but I was wondering if there is a chance that another phase transition might happen again in the future?</p> | g10296 | [
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<p>Diffraction theory is scalar. How you deal with beam propagation in fourier optics that is sensitive to the to the polarization?</p>
<p>If I have linearly polarized gaussian beam incident on glass surface, how is the polarization included in the propagation code/theory ?</p>
<p>What I am ultimately interested is to have let say linearly polarized input gaussian beam to propagate through a hollow fiber. I will use split step to propagate if that matters. And how is it done if thats not good approach ?</p> | g10297 | [
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<p>I know that if some object acquires potential energy, it also gains mass- is it the same for kinetic energy?</p> | g10298 | [
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<p>I am trying to get the definition of capacitance nailed down. This is my work so far:
$$
\frac{\mathrm{d}U}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega_a}{ u(\mathbf{x},t)\mathrm{d}t} = \int_{\Omega_a}{ \partial_t u(\mathbf{x},t)\mathrm{d}t} = -\int_{\Omega_a}{ \mu_0^{-1}\nabla\cdot[\mathbf{E}\times\mathbf{B}] + \mathbf{E}\cdot\mathbf{J}\mathrm{d}t}\\
U(t) = \frac{1}{2}C[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]^2\\
\text{ Assuming C constant in time}\\
\frac{\mathrm{d}U}{\mathrm{d}t} = C[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]\frac{\mathrm{d}}{\mathrm{d} t}[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]\\
C\frac{\mathrm{d}}{\mathrm{d}t}[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)] = \frac{\frac{\mathrm{dU}}{\mathrm{d}t}}{[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]} = \frac{-\int_{\Omega_a}{ \mu_0^{-1}\nabla\cdot[\mathbf{E}\times\mathbf{B}] + \mathbf{E}\cdot\mathbf{J}\mathrm{d}t}}{[\phi(\mathbf{a},t)-\phi(\mathbf{b},t)]}
$$
So where can I go from here? I have never seen anyone talk about electrodynamics and capacitance at the same time. I guess there is some kind of dimensional analysis in play that allows certain approximations of the Maxwell equations?</p> | g10299 | [
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<p>Suppose I place a gyroscope in a theoretically perfectly quiescent, closed room. Let its output be given as a vector ${\bf v} = (v_x, v_y, v_z)$ indicating rate of rotation around three orthogonal axes, where the $x$-axis points towards true north and the $z$-axis points vertically. Given the data, how could I calculate the latitude of the test location? Safe to assume Earth is a perfect sphere.</p>
<p>I reckon that it equals $\frac{180}{\pi}\arctan(v_z/v_x)$ should do it, because at the equator (where the latitude is zero), $v_z=0$, and as you increase the angle (move N, say) more and more of the magnitude of the rotation vector is accounted for by the $z$-axis.</p>
<p>Am I thinking about this correctly? Can someone help me formalize the argument?</p> | g10300 | [
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<p>Neutrinos travel straight through earth at the speed of light. Therefore, it seems that they could be great for intercontinental communication. Of course, I assume a lot still needs to be learned about detecting, producing and controlling neutrinos before they can be used for the practical purpose of communication.</p>
<p>My question: In principal, could neutrinos be manipulated similarly to radio waves for the purpose of communication? I mean, modulation, filtering, etc. ?</p> | g10301 | [
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<p><a href="http://en.wikipedia.org/wiki/AdS/CFT_correspondence">AdS/CFT</a> seems like a really hot topic and I'd like to start reading about. I am looking for the best introduction at my level, i.e. I have a background in QFT, CFT and general relativity at the level of a master student in theoretical physics. What would you recommend me to start tackling the subject? I have been looking for resources and so far I have noticed:</p>
<p>-this synthetic introductory lectures by Horatiu Nastase: <a href="http://arxiv.org/abs/0712.0689">http://arxiv.org/abs/0712.0689</a></p>
<p>-the videos of lectures done by P. Vieira at Perimeter for Perimeter Scholar International students: <a href="http://www.perimeterscholars.org/341.html">http://www.perimeterscholars.org/341.html</a></p>
<p>-the subject starts entering the most recent textbooks on string theory. We have the Schwartz and Becker ( <a href="http://goo.gl/jh45U">http://goo.gl/jh45U</a> ) and also the Kiritsis ( <a href="http://goo.gl/ulEVw">http://goo.gl/ulEVw</a> )</p>
<p>I probably missed a lot of resources, as the literature on the subject is already quite huge. I would really appreciate some advice on that, as I already had the frustration of losing my time on not-so-good books when started to learn something new, so if I could (to the best) avoid that this time...</p> | g555 | [
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<p>In the weak interaction limit, behaviors of electrons can still be described by a 1-partitcle equation with a modified mass, where the change in mass can be understood as effects of other neighboring electrons as a single electron drags along. Since the cluster of electrons couples to gravity as usual, then the effective mass of a quasi-particle couples to gravity in the same way as that of a genuine electron. Moreover, the effective mass can be negative which is the case of holes which are “bubbles” in the electron liquid.</p>
<p>In graphene, there are massless Dirac electrons. Do they couple to the gravity?</p>
<p><strong>I think what I intended to ask should be "How to incorporate effects of gravity in a many-electron system?"</strong></p>
<p>(Originally I asked "do effective masses couple to gravity?")</p> | g10302 | [
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<p>I'm having real problems trying to solve the following:</p>
<p>"The particle is moving in the plane. The trajectory is given by $y=cx-bx^2$, where $c$ and $b$ are positive constants. The acceleration of the particle is always constant and given by $\underline{a}=-\underline{j}$. Find the speed of the particle at the origin O (0,0)."</p>
<p>I am aware that trajectory differentiates to velocity which differentiates to acceleration but I can't work out how to apply it in this case... I'm used to seeing trajectory given in terms of $r(t)=...$ and I can't deduce how to convert it into this form using the information given. I'm really hoping this is actually very simple and I'm missing one key idea!</p> | g10303 | [
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-0.016711276024580002,
0.0161270871758461,
0.03771992400288582,
0.012974613346159458,
0.0272... |
<p>So here's the statement:</p>
<p>A pulley of a mass $M$ is hanged using a spring (stiffness of the string being $k_1$), as shown in the image. What is the frequency of the pulley's oscillation?</p>
<p><img src="http://oi43.tinypic.com/255lxf5.jpg" width="300" height="360"/></p>
<p>So that's as far as I could get:</p>
<p>Linear motion equation:
$$T_2+T_1-mg=ma \tag{1}$$
Where $T_2$ is string tension on the spring side ($T_2=-kx$) and $T_1$ being string tension on the other side.</p>
<p>Now for the rotational equation:
$$R(T_2-T_1)=I\alpha \tag{2}$$</p>
<p>$$a=R\alpha\tag{3}$$</p>
<p>$$I=\frac{mR^2}{2}\tag{4}$$
Now I have a feeling that the problem should be solved using the equations:</p>
<p>$$x'' + w^2x=0$$</p>
<p>$$T=\frac{2\pi}{w}$$</p>
<p>In this case $x''=a$. From the equations (2), (3) and (4) we derive that</p>
<p>$$T_2-T_1=\frac{ma}{2}\tag{5}$$</p>
<p>And adding (5) and (1) we get</p>
<p>$$-2T_2+mg+\frac{3}{2}ma=0$$</p>
<p>or</p>
<p>$$2kx+mg+\frac{3}{2}ma=0$$</p>
<p>And there I'm stuck. Could anyone tell if at least I'm going to the right direction? Any help is appreciated!</p> | g10304 | [
0.03261049836874008,
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0.009888894855976105,
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0.022821826860308647,
-0.030449409037828445,
-... |
<p>In my introductory physics class, $V$ is the symbol for electric potential (joules per coulomb) and $U$ is the symbol for electric potential energy (joules). </p>
<p>Since the Schrodinger equation is the sum of Kinetic and Potential energies in the system, $V(r)$ must represent $U$... if so, is there any particular reason why $V(r)$ is used as opposed to $U$? </p> | g10305 | [
0.04353933781385422,
-0.013454560190439224,
-0.000603706284891814,
0.019048046320676804,
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0.050500866025686264,
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0.004790900275111198,
0.030472... |
<p>Every possible reaction in chemistry is to attain stability. In physics, the alignment of an electric dipole in an external electric field and in all other physical systems (at least those I study in high school) attains stability at the lowest energy state. But, why is it so? </p> | g10306 | [
-0.02894679084420204,
0.02683786302804947,
-0.020509472116827965,
0.031766705214977264,
0.07102907449007034,
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-0.019316943362355232,
-0.03070911578834057,
-0.... |
<p>I need a good and simple reference for studying about ADM mass. Can someone introduce me one?</p> | g330 | [
0.01444956660270691,
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0.016237277537584305,
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0.05166635289788246,
0.01720697060227394,
0.11891508847475052,
-0.036... |
<p>I know this is probably an easy question, but it's been a while since I've studied physics and I've started reading some circuit analysis textbooks.</p>
<p>I'm finding hard to understand the relationship between between the verbal definition of quantites and the mathematical definitions. For instance, in the Sadiku's book "Fundamentals of electic circuits", I've got the following verbal definiton for voltaje (literal)</p>
<p>"Voltage (or potential difference) is the energy required to move a unit
charge through an element, measured in volts (V)."</p>
<p>And then, it says that this mathematically "means"</p>
<p>$$
v_{ab} \triangleq \frac{dw}{dq}
$$</p>
<p>I can't understand well the relationship between this two "definitions" could someone explain further the relationship?</p> | g10307 | [
0.00557169783860445,
0.0031030362006276846,
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0.02960074320435524,
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0.... |
<p>We have the formula for the Lorentz force $$\textbf{F} = q \space(\textbf{E} + \textbf{v} \times \textbf {B})$$</p>
<p>This is a simple formula you learn in high school, but I'm interested to self-study electromagnetism and I found out a different notation for this formula:</p>
<p>$$ \textbf{F}(\textbf{r}, \dot{\textbf{r}} , t,q) = q[\textbf{E}(\textbf{r}, t) + \dot{\textbf{r}} \times \textbf{B}( \textbf{r}, t)]$$</p>
<p>I want to know where I can study this type of notation, as I've never encountered it during high school. I'm guessing the LHS needs to include every unit on the RHS, but I don't understand for example why we have $\textbf{E}(r,t)$, what is the significance of the position vector and time, and how can you know?</p> | g10308 | [
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0.0002175635745516047,
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0.08052153140306473,
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0.04294568672776222,
0.04687103256583214,
0.02... |
<p>I'm not really sure if this is the place to ask this, but anyway here is my question:</p>
<p>Let's say I have the Kepler orbital elements of the ISS, for example, (<a href="http://www.spaceflight.nasa.gov/realdata/sightings/SSapplications/Post/JavaSSOP/orbit/ISS/SVPOST.html" rel="nofollow">NASA stuff</a>). Now I want to compute the coordinates relative to the earth at a specific time so that it can be displayed, like in a sky map (I am actually experimenting with Google Sky Map).</p>
<p>I found many websites discussing Kepler orbital elements, but I have only found 2 pdf's that talk about this conversion:</p>
<p><a href="http://www.phas.ubc.ca/~newhouse/p210/orbits/cometreport.pdf" rel="nofollow">http://www.phas.ubc.ca/~newhouse/p210/orbits/cometreport.pdf</a> and I can't post a 3rd link yet.</p>
<p>I do not fully understand which values I need to calculate, and which are given by NASA. </p>
<p>I am also assuming that the data given by NASA is valid for the vector time provided, meaning that time obviously has to come into the calculations somewhere.</p>
<p>Thanks, I would appreciate some clarification.</p> | g10309 | [
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0.02061477303504944,
0.005773728713393211,
-0.0585007444024086,
-0.020714029669761658,
0.014989642426371574,
-0.02550758607685566,
0.01433321088552475,
-0.032771751284599304,
0.033802200108766556,
0.02995694987475872,
0.05524680018424988,
0.0761425793170929,
-0.003696... |
<p>I cannot seem to find any peer-reviewed (or other) reference to an integer-spin Stern-Gerlach experiment. It shouldn't be too hard to do: just find you friendly neighbourhood Deuterium ion and shoot it through a Stern-Gerlach magnet.</p>
<p>Can one devise a photonic Stern-Gerlach experiment, i.e. spatial seperation of polarization states? One should also see only two states in this case, because the spin-0 photon state is "reserved" for EM-interactions (this might be too simple a statement, but this is how I understand it currently).</p>
<p><em>EDIT</em> it seems some of you are misunderstanding the question: I am inquiring about a Stern-Gerlach-like experiment, where spin states have been split, and by extension the perpendicular nature of non-commuting measurements. So only the concept of the S-G experiment as extensively described in introductory QM textbooks such as Sakurai.</p> | g10310 | [
-0.05227605253458023,
-0.00046783749712631106,
0.002874228172004223,
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-0.035606615245342255,
0.029740115627646446,
0.0003052491520065814,
0.04185652360320091,
-0... |
<p>I have four questions about black holes and universe formations.</p>
<ol>
<li><p>Do new universes form on the other side of black holes?</p></li>
<li><p>Was our own universe formed by this process?</p></li>
<li><p>Was our big bang a black hole seen from the other side?</p></li>
<li><p>Are there solid reasons why this might not be the case?</p></li>
</ol> | g10311 | [
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0.0008092709467746317,
0.020544614642858505,
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0.029617462307214737,
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0.026208819821476936,
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0.... |
<p>I am trying to show the invariance of the following Yang Mills Lagrangian:
$$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$
under the following gauge transformation ($\theta$ being a rotation in color space and $g$ related to the structure constant):
$$L \rightarrow -\frac{1}{4} \left( F^a_{\mu\nu} +\epsilon ^a_{jk} \theta^j F^k_{\mu \nu}\right)\left(F_a^{\mu\nu} + e_a^{jk}\theta_j F_k^{\mu\nu}\right) + \left( J_a^\mu + \epsilon_a^{jk} \theta_j J_k^\mu \right) \left( A_\mu^a + \epsilon^a_{jk}\theta^jA_\mu^k -\frac{1}{g} \partial^\mu \theta^a \right),$$</p>
<p>where each term is now transformed accordingly.</p>
<p>I was able to simplify the above to and obtain:
$$L \rightarrow -\frac{1}{4} \left( F^a_{\mu\nu}F_a^{\mu\nu} + \epsilon^a_{jk} \theta^j F^k_{\mu\nu} \epsilon_a^{j' k'} \theta_{j'} F_{k'}^{\mu\nu}\right) + J_a^\mu A_\mu^a - J_a^\mu \frac{1}{g} \partial^\mu \theta^a + \epsilon_a^{jk} \theta_j J_k^\mu \epsilon^a_{j'k'} \theta^{j'}A_\mu^{k'} - \epsilon_a^{jk} \theta_{j} J_k^{\mu}\frac{1}{g} \partial^\mu \theta^a.$$</p>
<p>How could I possibly reduce it to a form similar to the original, untransformed Lagrangian? There are about 4 terms I can't get rid of, though it has been suggested to me that I use the equation of motion of YM, which I have handy but can't seem to use them appropriately. Any help would be greatly appreciated. Also note that I may end up with a boundary term which would vanish when varying the action, thus possibly giving me say 3 terms instead of the original 2 (which is fine, though I can't identify them yet).</p> | g10312 | [
0.015996908769011497,
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-0.019191861152648926,
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0.03520442172884941,
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-0.005035621114075184,
0.022680925205349922,
0.026716792955994606,
0.005341540556401014,
0.0... |
<p>If an air cylinder is pushing two platens apart with a force of $100\: \mathrm{lbs}$, do the platens need to push back at $100\: \mathrm{lbs}$ or $50\: \mathrm{lbs}$ each to keep the cylinder from moving? Assume no friction and both platens are not fixed.</p> | g10313 | [
0.04542779549956322,
0.06412049382925034,
0.024043120443820953,
0.04772479832172394,
0.05745479464530945,
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0.02572585456073284,
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-0.00703771598637104,
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0.031571466475725174,
-0.10487248748540878,
-0.03690986... |
<p>If time is treated as a fourth dimension of spacetime, what is relation between length and time units?</p>
<p>Or in other words, how can I convert time units to length units, for instance seconds to meters?</p> | g10314 | [
0.04143262282013893,
0.007606495637446642,
-0.016734205186367035,
-0.0548880361020565,
0.00748436851426959,
0.030781909823417664,
-0.010879741981625557,
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0.018992263823747635,
-0.016459202393889427,
-0.027327867224812508,
-0.02030133828520775,
0.... |
<p>In <a href="http://profmattstrassler.com/2012/10/15/why-the-higgs-and-gravity-are-unrelated/#comment-22975" rel="nofollow">this comment in a blog</a> kudzu computes the energy of a marshmallow with mass $M=25 grams$ by using $E=mc^2$:</p>
<p>$E=Mc^2 = 2.247\times 10^{+15} Joules$</p>
<p>I may be wrong but this seems like a huge energy for a marshmallow. Can anybody help me put this in context. </p> | g10315 | [
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0.05747896432876587,
0.008373413234949112,
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0.026666909456253052,
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0.... |
<p>I was looking through my bathroom window this night and I was wondering if any of the photons my retina is hit with are from 13 (40) billion light years away ?!</p>
<p>I was looking through it a few seconds, was that too less of a time?</p>
<p>Do I need a Hubble for this?</p> | g331 | [
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0.00007424302748404443,
-0.04045739769935608,
-0.00... |
<p>Why is one-point Green's function for scalar field equal to zero? Can one prove it using path integral formalism? </p> | g10316 | [
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-0.01281... |
<p>In inelastic collisions, the kinetic energy of the system is not conserved but the momentum is. </p>
<p>Kinetic energy is: $0.5 \times \text{mass} \times \text{velocity}^2$. Momentum is: $\text{mass}\times\text{velocity}$. </p>
<p>I think that, considering that mass is constant:</p>
<ul>
<li><p>if Ke must be different also the velocity of the centre of mass of the system <em>must be different</em>, after the collision. On the other hand: </p></li>
<li><p>if the momentum of the system is conserved, the velocity of the centre of mass of the system <em>cannot be different</em>. </p></li>
</ul>
<p>So, how can there be a change in kinetic energy of the system if there is no change in momentum? $mv = m_1v_1$</p> | g10317 | [
0.07042723149061203,
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... |
<p>I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations.
Here in my book, it is stated that </p>
<blockquote>
<p>Every unitary operator $\hat{\mathcal{U}}$ can be written in an exponential form as follows:
$$\mathrm{e}^{-i\alpha\hat{\mathcal{T}}}=\sum_{k=0}^{\infty}\dfrac{1}{k!}\left(-i\alpha\right)^{k}\hat{\mathcal{T}}^{k}
$$</p>
</blockquote>
<p>Provided that I have no knowledge of Lie Group/Algebra, my questions are:</p>
<ul>
<li>Why a unitary operator can be always represented by an exponential form?</li>
<li>What is the intuitive mathematical meaning of the exponential form/matrix?</li>
<li>What is the relation between the operator $\hat{\mathcal{U}}$ and the operator $\hat{\mathcal{T}}$?</li>
<li>What is its physical meaning?</li>
</ul> | g10318 | [
-0.03831776976585388,
-0.009313004091382027,
-0.03389288857579231,
-0.030651330947875977,
0.049173831939697266,
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0.01953495293855667,
0.04767506197094917,
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0.022458918392658234,
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0.02402121014893055,
0.003729776246473193,
0.0... |
<p>I have been trying to understand the difference between the Work function of a metal and the Local Work Function. </p>
<p>I did some experiments to find the Local Work function of Graphite using an STM and found values ~0.3-0.7 eV. However, the work function of Graphite is ~4eV. Literature values indicate that my experimental results are correct. </p>
<p>Some research leads me to these definitions on a website:</p>
<p>Work Function:
The work function corresponds to the minimum energy necessary to
extract an electron from the metal</p>
<p>Local Work function:
The local work function, which is defined as the energy required to take an electron from the Fermi level to a specified position outside the metal.</p>
<p>I frankly can't see a difference between these two definitions. Admittedly, my concepts in Solid State Physics are shaky. What is the difference between the two work functions? Can someone clarify, without assuming I know too much about Fermi levels.</p> | g10319 | [
0.04166283458471298,
0.025720840319991112,
0.007902010343968868,
-0.007428828161209822,
0.021944519132375717,
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0.03158995136618614,
-0.017137078568339348,
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0.010573466308414936,
-0.07348346710205078,
0.01366507913917303,
-0.0023181415162980556,
-0... |
<p><strong>Motivation:</strong></p>
<p>I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form</p>
<p>$$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$</p>
<p>where $'=\frac{d}{dr}$ and $\dot{} =\frac{d}{dt}$. (The expression for the function $f$ is long to I don't include it, but it contains terms like $\dot{u}^nu'^m$.)</p>
<p>I need to solve this numerically, and I have done this by defining the 'velocity' $Q = \frac{du}{dt}$ so that we can write the system as</p>
<p>$$\dot{Q} = f(t, Q, Q', u' ,u'')$$
$$\dot{u} = Q$$</p>
<p>which I then discretize on a grid (for the space) and evolve in time using a a staggered leapfrog:</p>
<p>$$Q_{n+1/2} = Q_{n-1/2} + f(t_n, Q_n, Q'_n, u'_n ,u''_n)\Delta t$$
$$u_{n+1} = u_n + Q_{n+1/2}\Delta t$$</p>
<p>This works fine for many applications, but I have some problems with getting this to work in all situations I'm interested in.</p>
<p><strong>Question</strong></p>
<p>I'm therefore looking for alternative methods to solve this equation. I welcome any suggestions. I have been thinking about Newton-Gauss-Seidel, but it's pretty computationally expensive so I'm hoping there are other methods I could try first. Also, if there is someone that has experience with numerically solving such equations it would be great to get some input on methods that are known to work well in practice.</p> | g10320 | [
0.03842184320092201,
0.016864078119397163,
0.0011740815825760365,
-0.018391238525509834,
0.0029810811392962933,
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0.013102088123559952,
-0.002870823722332716,
0.017865179106593132,
-0.02831915207207203,
-0.047193557024002075,
0.005207533948123455,
0.06717757880687714,
0... |
<p>Can we have elctromagnetic waves which are characterized by parallel electric and magnetic field $\vec{E} || \vec{B}$ ? I am not talking here about free space, but maybe some kind of materials or waveguides or something else.</p> | g10321 | [
0.06173500791192055,
0.08155152946710587,
-0.028808973729610443,
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0.0802062526345253,
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0.030951933935284615,
0.048814017325639725,
0.017289957031607628,
-0.020523294806480408,
0.04084117338061333,
-0.023054832592606544,
0.02... |
<p>Please forgive a string theory novice asking a basic question.</p>
<p>Over at <a href="http://physics.stackexchange.com/questions/70821/the-problem-of-a-relativistic-path-integral">this question</a> Luboš Motl gave an excellent answer, but he made a side comment that I've heard before and really would want to know more about:</p>
<blockquote>
<p>Quantum field theory is the class of minimal theories that obey both sets of principles [Ed: <em>SR and QM</em>]; string theory is a bit more general one (and the only other known, aside from QFT, that does solve the constraints consistently).</p>
</blockquote>
<p>What are the arguments that string theory is more general than QFT? I get that you can derive many different QFTs as low energy effective theories on different string backgrounds. But from my limited exposures to worldsheet perturbation theory and string field theory I can also see string theory as a very <em>special</em> kind of QFT. I also realize these approaches don't fully characterise string theory, but I don't know enough about their limitations to understand why the full definition of string theory (M theory?) surpasses QFT.</p>
<p>My naive guess would be that <em>no</em>, string theory can't be more general than QFT because surely there are many more QFTs which are asymptotically free/safe than could possibly come from string theory. Think ridiculously large gauge groups $SU(10)^{800}$ etc.. Since these theories don't need a UV completion into something else string theory can't be a more general framework than QFT. Logically you could also have theories which UV complete into something other than string theory (even if no such completion is presently known).</p>
<p>Alternately you could turn this around and say that string theory limits the kind of QFTs you can get in the low energy limit. This is saying that string theory is more predictive than QFT, i.e. <em>less</em> general. I always thought this was the goal the whole time! If it is the other way around and string theory really is <em>more</em> general than QFT, doesn't this mean that string theory is <em>less</em> predictive than, for instance, old school GUT model building?</p>
<p>So is the relationship between string theory and quantum field theory a strict inclusion $\mathrm{QFT} \subset \mathrm{ST}$ or more like a duality/equivalence $\mathrm{QFT} \simeq \mathrm{ST}$, or a more complicated Venn diagram?</p>
<p>Note that I am <em>not</em> asking about AdS/CFT as this only deals with special string backgrounds and their QFT duals. I'm asking about the general relationship between string theory and QFT.</p> | g10322 | [
0.040669579058885574,
-0.005890832748264074,
0.024547090753912926,
-0.007179380860179663,
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0.011032143607735634,
0.0... |
<p>We know that universe is expanding and galaxies are moving away from each other. Does this mean the galaxies are also expanding in itself and therefore I guess growing larger in volume? Depending on the answer of that, does the solar system also expands? </p> | g56 | [
0.008725378662347794,
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0.039475... |
<p>I don't know quantum mechanical model. So, I'm referring to just bohr's model of atom.</p>
<p>Any atom emits energy when it makes transition from higher excited state to lower excited state. Now, some times they say that this energy is light energy and some times heat energy. </p>
<p>I'm confused. What decides the emitted energy will be light or heat?</p>
<p>Do we have control over what kind of energy it emits? I mean can I make an atom emit light and not heat? or heat and not light? What are the factors influencing this?</p>
<p><strong>EDIT:</strong> To clarify the issue of what exactly do I mean by heat. Well, I myself am not quite sure. I just thought heat as just heat. This question araised from <a href="http://physics.stackexchange.com/questions/15162/what-is-changing-in-latest-light-bulb-technologies">my previous question</a> where I discovered that in a filament bulb 98% of energy is converted into heat energy and rest into light energy. But where as in CFL about 40% to light and rest into heat. I know these conversion is because of electron transistions. But whats controlling light & heat components?</p> | g10323 | [
-0.008872149512171745,
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0.04096589237451553,
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0.043803535401821136,
0.03906239569187164,
... |
<ol>
<li><p>Consider a charged particle (electron or proton) at rest. It is surrounded by its own electric field.</p></li>
<li><p>Now consider an electron moving with certain velocity (less than speed of light), Still is there electric field around it?</p>
<p>i. If it has electric field around it, why is it that when electrons are moving in a conductor (i.e.. current if flowing in a conductor) there is no electric field outside the conductor?</p>
<p>ii. Now, when a current is flowing in a conductor (I'm not sure what happens if the motion is not inside conductor) it produces magnetic field around it. I'm lost. What happened to the electric field? Is it still there? Are there both electric field & magnetic field? Why don't we discuss about it?</p></li>
<li><p>Hypothetically, If the electron is moving with speed more than that of light. What happens now? </p></li>
</ol>
<p>I'm totally lost. Kindly some one clarify these issues.</p> | g10324 | [
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0.0... |
<p>I read that:</p>
<p>If you take a rough surface and make it smooth, the coefficient of friction decreases. But if you make it <strong>super</strong> smooth, then the coefficient of friction increases. How come?</p> | g10325 | [
0.06482619047164917,
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0.02405252680182457,
0.04906946420669556,
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<p>I am working on a project involving a simulation of the motion of a projectile (in 3D) aimed at a moving target. The way projectile motion is analyzed in most introductory physics books is not accurate enough for this project. I would like to know what other influences on the motion of a projectile, including air resistance and spin, I need to take into account. What is a good book on this subject?</p> | g10326 | [
0.034743621945381165,
0.029658203944563866,
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0.050402700901031494,
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0.03339280188083649,
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0.07420754432678223,
-... |
<p>Is there a definition of a <a href="http://en.wikipedia.org/wiki/Black_hole" rel="nofollow">black hole</a> in a generic spacetime? In some books, for example Wald's, black holes are defined for asymptotically flat spacetime with strong asymptotic predictability, although the definition makes sense without the second condition. Is there a notion of a black hole in general spacetime, not necessarily asymptotically flat? Or is it the case that there is not a "natural" or agreed upon definition? </p> | g10327 | [
0.01304127462208271,
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0.025... |
<p>I read some methods but they're not accurate. They use the Archimedes principle and they assume uniform body density which of course is far from true. Others are silly like this one:</p>
<p><em>Take a knife then remove your head.<br/>
Place it on some scale<br/>
Take the reading<br/>
re-attach your head.</em></p>
<p>I'm looking for some ingenious way of doing this accurately without having to lie on your back and put your head on a scale which isn't a good idea.</p> | g10328 | [
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0.010409445501863956,
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0.06664754450321198,
0.00689741550013423,
-0.05012871325016022,
0.0680680125951767,
-0.0... |
<p>Just what the title states.
What is the reason that a Nuclear reactor has a characteristic dome/bell shape?</p> | g10329 | [
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0.04321613535284996,
0.001086162286810577,
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0.05... |
<blockquote>
<p>An open container is being filled with water from the top at a constant volumetric flow rate $300 \frac{cm^3}{s} $. At the same time, water is leaving the container trough a circular whole at the bottom, whose surface area is $0.6 cm^2$. What is the water level in a stationary state. </p>
</blockquote>
<p>First of all I do not understand this question. How is "stationary state" defined? Shouldn't the water level be increasing as time passes by? I don't want a solution to this problem but only a hint or an explanation of what this stationary state is.</p> | g10330 | [
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<p>Surface tension appears at the interface of two immiscible fluids if the cohesive force of attraction is more than adhesive force. What will be the physical effect if the <em>adhesive</em> force is more than <em>cohesive</em> force?</p> | g10331 | [
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<p>I'm working to craft a space-horror themed <a href="http://en.wikipedia.org/wiki/Role-playing_game" rel="nofollow">RPG</a> a la Dungeons & Dragons and I've run into a situation I can't resolve on my own.</p>
<p>At what rate is atmosphere lost to space through various sized apertures? I'm looking generally at these sizes(in diameter) 10' 5' 30" 15" 7" 3" 1"</p>
<p>Trying to determine how quickly the cabins/holds/etc will decompress. It is the deep space version of falling rocks :D</p> | g10332 | [
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<p>I'm solving an exercise about the Lagrange-Euler equations, that states the following:</p>
<blockquote>
<p><em>Let $\gamma (t) = \{ (t,q) : q = q(t), t_0 \leq t \leq t_1\}$ be a curve in $\mathbb{R} \times \mathbb{R}^2$. Further let $F(q,\dot{q},t)$ be the function from $\mathbb{R}^2 \times \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}$ for which the functional $\Phi = \int_{t_0}^{t_1} F(q,\dot{q},t) dt$ is the length of the curve.</em></p>
<p><em>(a) Which is the form of $\Phi$ in cartesian coordinates? Which is its form in polar coordinates?</em></p>
<p><em>(b) Give the Euler-Lagrange equations in both coordinate systems.</em></p>
<p><em>(c) Solve the differential equations in both coordinate systems and show that the solutions are the same.</em></p>
</blockquote>
<p>Now, my problem begins with giving the form of $\Phi$. I found that the element of length in cartesian coordinates is $ds = \sqrt{dx^2 + dy^2}$, so with
$$\int ds = \int \frac{ds}{dt} dt = \int \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2} dt,$$
We find that $\Phi = \int_{t_0}^{t_1} ||\dot{\gamma}(t)|| dt$.
Now, my plan is finding the element of length in polar coordinates, and plugging in the respective expressions in terms of $\gamma$. The problem is that I don't see how to find the element of length in polar coordinates. I looked it up on Wikipedia, and found $ds^2 = dr^2 + r^2 d\theta^2$. Now, for $dr^2$ I would plug in $||\dot{\gamma} (t)||^2$, for $r^2$ I'd set $||\gamma (t)||^2$, and for $d\theta$ I have no idea.</p>
<p>Can you help me, especially with the derivation of the polar line element and the form of $\Phi$ in polar coordinates?</p> | g10333 | [
0.04530135169625282,
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-0.... |
<p><img src="http://i.stack.imgur.com/WcibA.png" alt="enter image description here"></p>
<p>In the answer section it is written that,The magnetic field at O due to the current in the straight segments $AA^\prime$ and $CC^\prime$ is zero because ds is parallel to along these paths. </p>
<blockquote>
<p>My querry is that "How ds is parallel to along these paths?"</p>
</blockquote>
<p>EDIT: <img src="http://i.stack.imgur.com/NfFgO.png" alt="enter image description here"></p> | g10334 | [
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-0.028182124719023705,
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0.03448763117194176,
-0.0003159225161653012,
-... |
<p><strong>Why don't most physics programs study the primary sources?</strong> In other words: Why don't they include Newton's <em>Principia</em>, Lagrange's <em>Analytical Mechanics</em>, etc., in the curricula?</p> | g10335 | [
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-0.0087978728... |
<p>What classical information is carried by $\alpha|0\rangle+\beta|1\rangle$ and $\alpha|00\rangle+\beta|11\rangle$? How to quantify it? To be specific, A GHZ state, $\frac{1}{\sqrt 2}[|000\rangle+|111\rangle]$ can deterministically teleport both the states. Both the states appears to be same from the teleportation point of view. Does they carry same information? </p> | g10336 | [
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0.0007663809810765088,
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0.017153186723589897,
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<p>Historically, Bekenstein estimated the entropy associated with a black hole in 1973, obtaining:
$$
S_B = \frac{\ln(2)k_Bc^3}{8\pi\hbar G}A.
$$
He already acknowledges in his article that his estimates are based on classical principles and that a quantum mechanical treatment will yield a different constant, though within a factor of order one the same.
A year later Hawking derived:
$$
S_H = \frac{k_Bc^3}{4G\hbar}A,
$$
i.e. $S_B = (\ln(2)/2\pi) S_H$, such that $S_B<S_H$. </p>
<p>I am wondering if there could have been examples, showing that $S_B$ was not correct. So, without knowing Hawking's results, can we see that $S_B$ cannot be correct, maybe by giving a certain counter example, or using the fact that $S_B<S_H$?</p> | g10337 | [
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0.0166... |
<p>I really want to understand about the Surface Plasmon of metal nanoparticles, if anyone can explain it to me i'll be grateful.</p>
<p>I have checked Wikipedia for it but didn't get anything clear.</p> | g10338 | [
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0.03040756657719612,
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-0.0472772... |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/1775/why-is-there-no-absolute-maximum-temperature">Why is there no absolute maximum temperature?</a> </p>
</blockquote>
<p>On the Kelvin scale, absolute zero represents the temperature at which there is no thermal motion. Consequently, speaking of -r Kelvin has no physical meaning for any positive real number r.
My question is this: is there a value k>0 that is an upper bound for the Kelvin scale in the sense that speaking of k+r Kelvin has no physical meaning for any positive real number r?
Presumably there is such a k, based on "finite" energy, but is it so incomprehensibly large that it is no longer possible to distinguish between one level of largeness from another, say in the sense of <a href="http://www.math.osu.edu/~friedman.8/pdf/finiteseq10_8_98.pdf" rel="nofollow">Friedman</a>.</p> | g237 | [
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0.02... |
<p>I was wondering : </p>
<p>does the weight on the planet earth is equal over the years ?</p>
<p>meaning : </p>
<p>all the people , ground , water ,gas.</p>
<p>does the weight stays the same over the years ?</p> | g10339 | [
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... |
<p>Let me admit beforehand that this is quite possibly a very stupid question. I was also uncertain of where to post this question, as it doesn't fit cleanly into either physics or math stackexchange.</p>
<p>In dimensional analysis, it does not make sense to, for instance, add together two numbers with different units together. Nor does it make sense to exponentiate two numbers with different units (or for that matter, with units at all) together; these expressions make no sense:</p>
<p>$$(5 m)^{7 s}$$</p>
<p>$$(14 A)^{3 A}$$</p>
<p>Now my question is plainly this: <em>why</em> do they not make sense? Why does only multiplying together numbers with units make sense, and not, for instance, exponentiating them together? I understand that raising a number with a unit to the power of another number with a unit is quite unintuitive - however, that's not really a good reason, is it?</p> | g820 | [
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0.020762108266353607,
-0... |
<blockquote>
<p><strong>Q:</strong> Calculate the voltage that V1 has to provide</p>
<p><strong>Given:</strong> The following circuit. The voltage through LAMP1 is 5.0V. A (U,I) chart (not shown). LAMP1 and LAMP2 are identical, non-ohmic lightbulbs.</p>
</blockquote>
<p><img src="http://i.stack.imgur.com/3tFZy.png" alt="circuit"></p>
<p>My attempt at this problem:</p>
<p>LAMP1 and LAMP2 are two identical light bulbs. I know that $U(LAMP1) = 5.0 V$. According to the (U,I) chart, the value of $5.0$ V corresponds with $0.63$ A.</p>
<p>If $U(LAMP1) = 5.0$V, $U(R1) = 5.0$V too. That value corresponds with $I(R1) = 0.5$A.</p>
<p>$I(LAMP1 + R1) = 0.63 + 0.5 = 1.13$A.</p>
<p>Because LAMP1 and LAMP2 are identical, they are both $P = 5.0 \times 0.63 = 3.15$W lamps.</p>
<p>The voltage through LAMP2 has to be $U(LAMP2) = P \div I$. So $U(LAMP2) = 3.15 \div 1.13 = 2.78$V</p>
<p>Hence the voltage V1 has to provide is $5.0 + 2.78 = 7.8$V</p>
<p>Is this correct? Or am I using the wrong method(s)?</p> | g10340 | [
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0.028871655464172363,
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0.030936703085899353,
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0... |
<p>I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a scale transformation (renormalization group transformation in the QFT case and dilation of the mesh/ruler when computing the perimeter of a fractal). Is there a more profound link between the two? I have't read much about the subject but it seems that for any statistical model at criticality the fractal dimension of the clusters becomes a function of the full dimension of the field. Is it a general rule?</p> | g10341 | [
0.0217166505753994,
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0.016257667914032936,
-0.042543426156044006,
-0.004679129924625158,
0.03294054791331291,
-0.03871811181306839,
0.01743752509355545,
0.0038190071936696768,
-0.013260038569569588,
-0.00457408232614398,
-0.008929645642638206,
0.05413789674639702,
-0.0... |
<p>I have been told always that
$F$ is directly proportional to acceleration.</p>
<p>My question is that for finding such a relationship there should be source that produces desired force and in which the force can be adjusted i.e. Twice, thrice and more.
But the problem is initially before the discovery of laws of motion how can one say that a force is twice the other, how can he even judge the relationship between two forces without knowing the quantitative definition of force?</p> | g332 | [
0.053371816873550415,
0.03562362119555473,
-0.017572497949004173,
0.021442435681819916,
0.08178970962762833,
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0.020232614129781723,
0.037563394755125046,
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-0.044650644063949585,
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-0.055472392588853836,
0.029857106506824493,
0.... |
<p>Or to put the question another way - what is the result of a proton-positron collision, or an up quark-charm antiquark collision, etc.? As far as I know, annihilation happens only between particles of opposite charge and same mass, but perhaps I am wrong?</p>
<p>And if the types of annihilation mentioned above cannot happend, what are the reasons?</p>
<p>Thank you.</p> | g10342 | [
0.06102197617292404,
0.016766589134931564,
0.0020963563583791256,
-0.02085658349096775,
0.07681180536746979,
0.06787236779928207,
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-0.013252215459942818,
0.0032338874880224466,
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0.024420345202088356,
-0.0... |
<p>I'm trying to determine the electric field at a point P (located on the +Z axis) due to a square of side length [L] and centered at the x-y plane origin. The square has a constant surface density [s]. I'm thinking that I should go about splitting up the square into four smaller squares (one in each quadrant) and then calculate the field from each on the point P. Is this the correct way of solving this type of problem, or should I be splitting the big square up into infinitesimal strips and calculating the field that way?</p> | g10343 | [
0.05243489518761635,
0.027175314724445343,
-0.011316267773509026,
-0.0724252238869667,
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0.04030932858586311,
0.031046058982610703,
0.020617306232452393,
0.0028965710662305355,
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0.016571490094065666,
-0.... |
<p><img src="http://i.stack.imgur.com/Whguh.gif" alt="double slit ray diagram"></p>
<p>This is from Young Double slit experiment. But How to prove the the two $\theta$ are equal, I meant, how $\angle EAD= \angle PEC$? I see from the both triangle have $90^0$ but what about others?</p> | g10344 | [
0.05195264518260956,
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0.0799151062965393,
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0.05344880744814873,
0.009108190424740314,
0.04788950830698013,
0.004224273841828108,
0.017649341374635696,
0.06731797754764557,
0.02081274800002575,
0.000740... |
<p>I am curious to see what people think of the abiotic theory of oil deposit formation versus the traditional theory.</p>
<p>I have long wondered how enough organic material became trapped underground to degrade and form oil deposits. Especially when you consider the shear amount of oil that we have found underground over the decades. And considering how much oil we use. If you think about it, when most critters die the tend to decompose mostly or fully above ground. So I have trouble believing that such large amounts of oil, especially in such large concentrations, have formed from dying plants and animals over the centuries.</p>
<p>Recently I heard about the theory of abiotic oil. My understanding of the theory is that oil deposits are actually chemicals and material that is formed at the mantle boundary that seeps up further into the crust where it is consumed by micro-organisms that process it and turn it into what we know as oil. My understanding may be a bit off I realize.</p>
<p>But some evidence of this has been presented in the form of old, capped oil wells that after a few decades, when re-examined, seem to be full once more and are now run at full pumping capacity once more. There are other things I have read as evidence. It seems to me that this would be a better explanation for the large amounts of oil deep underground.</p>
<p>Does anyone know enough about these subjects to say that one theory holds more water than the other? Has there been any new evidence pointing to one or the other theory more?</p> | g10345 | [
0.05775518715381622,
0.009111235849559307,
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0.01514111366122961,
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0.0012372501660138369,
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0.0406939797103405,
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/2481/would-you-be-weightless-at-the-center-of-the-earth">Would you be weightless at the center of the Earth?</a> </p>
</blockquote>
<p>Supposing there is a cavity at the center of the Earth, what is the gravity there? What will be its direction and intensity? Will a body be attracted toward the center of mass of the Earth?</p> | g50 | [
0.028803257271647453,
0.029651565477252007,
0.012138968333601952,
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0.03273182362318039,
0.04220503568649292,
0.03825242817401886,
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0.037668392062187195,
-0.... |
<p>Radio-telescopes (e.g. the <a href="http://en.wikipedia.org/wiki/Very_Large_Array" rel="nofollow">Very Large Array (VLA)</a>) can simulate one gigantic dish by using separate smaller dishes.</p>
<p><strong>Q:</strong> Could such an array of <em>optical</em> telescopes potentially see an exoplanet at say 20 LY away?</p>
<p>Assumptions:</p>
<ul>
<li>The array is space-based, at an ideal location such as the Earth-Sun
L2 Lagrangian Point</li>
<li>Assume the exoplanet of interest is defined as a rockey planet up to
5 times the diameter of earth</li>
</ul> | g10346 | [
-0.05693665146827698,
0.07041337341070175,
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0.04287267476320267,
0.015923889353871346,
0.0556044764816761,
0.09525176137685776,
0.04372426122426987,
-0.0... |
<p>How can light be called <em>electromagnetic</em> if it doesn't appear to be <em>electric</em> nor <em>magnetic?</em></p>
<p>If I go out to the sunlight, magnets aren't affected (or don't seem to be). And there is no transfer of electric charge/electrons (as there is in AC/DC current in space). </p>
<p>In particular, the <a href="http://en.wikipedia.org/wiki/Photon">photons</a> (which light is supposed to be composed of) have no electric charge (nor do they have <a href="http://en.wikipedia.org/wiki/Magnetic_monopole">magnetic charge</a>). </p>
<p>I'm looking for an explanation that can be appreciated by the average non-physicist Joe.</p> | g485 | [
0.005005809478461742,
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0.06790772080421448,
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0.0... |
<p>I have a very basic question on the <a href="http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect">Aharonov-Bohm</a> effect.</p>
<p>Why is the curve integral $\oint_\Gamma {A}\cdot d{r}$ non-zero ?
$\Gamma$ is the "difference" of both paths $P_1$ and $P_2$.
If the magnetic field is limited to the interior of the solenoid $\operatorname{curl} {A}=0$ along the integral path $\Gamma$, so I can conclude that I can write ${A}=\nabla f$. A closed curve integral of a gradient function is zero. </p>
<p>I guess it is related with a possible singularity of $A$ in the very center of the solenoid. </p>
<p>Nevertheless if I travel around a point-like source of a gravitational field and compute the integral $\oint_\Gamma {F}\cdot d{r}$ where $F=-\nabla V(r)$ the closed curve integral over an conservative force field is certainly zero, whereas $V(r)$ even has a singularity and $F$ consequently too.
I would be very grateful for an explanation.</p> | g10347 | [
0.03458500653505325,
0.0163673497736454,
0.003675861516967416,
0.015115384943783283,
0.07097096741199493,
0.05196753516793251,
0.06962025165557861,
0.059099867939949036,
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0.012087167240679264,
0.02639162167906761,
-0.014621... |
<p>Sorkin is well known for his <a href="http://arxiv.org/pdf/Gr-qc/9904062.pdf" rel="nofollow">causal growth dynamics</a>. I think it is a sensible question to ask if his growth dyanamics, which I think can be seen as a poset map (though his work is much more complicated than that), represents a galois connection.</p> | g10348 | [
0.023073837161064148,
0.0706649199128151,
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0.014070753008127213,
0.006057514809072018,
-0.012892705388367176,
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0.10254884511232376,
0.0... |
<p>It is common to find models built on a compact spacetime. In mathematics, compactness is a very nice property $-$ and lot of powerful results depend on it.
But </p>
<ul>
<li>how <em>safe</em> is assuming compactness of spacetime in physics? </li>
</ul>
<p>Minkowski space is not compact, but, for instance, the treatment given to gauge theories in terms of bundles, assumes a compact basis $X$ for the principal bundle $G\hookrightarrow P \to X$ and the vector bundles containing matter $P \times_G \mathfrak{g} \to X$. This basis $X$ is thought of as Minkowski space-time (for sake of concreteness, assume an Euclidean spacetime $\mathbb{R}^4$, so one compactifies to $X=S^4$). Why can or cannot compactes be assumed? </p>
<p>This one is perhaps model-dependent:</p>
<ul>
<li>is there an existing experimental validation of the compactness of spacetime?</li>
</ul> | g10349 | [
0.0063783773221075535,
0.009956243447959423,
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0.022452158853411674,
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0.0034089297987520695,
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0.00072400615317747,
0.04767485335469246,
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-0.024666111916303635,
0.01... |
<p>What is meant with the fact that supersymmetry with $\mathcal{N}=4$ in three (2+1) dimensions is equivalent to supersymmetry with $\mathcal{N}=2$ in ordinary four (3+1) dimensions?</p>
<p>Which way does one go from one formulation to the other?</p> | g10350 | [
0.006560230161994696,
0.03530716150999069,
0.00793495960533619,
0.010446470230817795,
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0.035228170454502106,
0.032448429614305496,
0.0095455851405859,
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0.009796873666346073,
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-0.022054174914956093,
0.004174004774540663,
0.01588... |
<p>Consider photons and gluons have 0 mass and 0 charge. In many respects they're already understood as the absence of a particle by mathematical models. Couldn't this be interpreted to mean they operate by phenomenon similar to the charge carrying "holes" responsible for transmitting electro-magnetic waves (at the speed of light) in wires?</p>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Semiconductor#Holes:_electron_absence_as_a_charge_carrier" rel="nofollow">http://en.wikipedia.org/wiki/Semiconductor#Holes:_electron_absence_as_a_charge_carrier</a></li>
<li><a href="http://en.wikipedia.org/wiki/Standard_Model" rel="nofollow">http://en.wikipedia.org/wiki/Standard_Model</a> (note photons, and "gluons" which are thought to make up protons and neutrons, have 0 mass and 0 charge)</li>
</ul>
<p>The implications of this model would seem to be:</p>
<ul>
<li>There is a relativistic aether that is similar to the concept of a theoretical inviscid in fluid dynamics (see en.wikipedia.org/wiki/Inviscid_flow).</li>
<li>Differences in "pressure" at this fundamental level may be responsible for gravity, as well as providing possible explanations for a number of other phenomenon currently challenging physics.</li>
</ul> | g10351 | [
0.03888228163123131,
0.014076976105570793,
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0.06317268311977386,
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0.026251792907714844,
0.012681649997830391,
0.0009468338103033602,
-0.005811132024973631,
0.03845319524407387,
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0.03622359037399292,
0.03... |
<p>I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across.</p>
<p>$$ \hat{A}|\Psi\rangle = a|\Psi\rangle $$</p>
<p>The above equation implies to me that a single eigenket gives a single eigenvalue of $\hat{A}$.</p>
<p>However Ket Vectors that are composed of superpositions have multiple possible eigenvalues. Which leads me to believe that that equation is only valid for eigenkets which are Basis States.</p>
<p>However in the Schrödinger equation we have an Observable (Hamiltonian) acting on Wave Functions in Position Space which are composed of an infinite number of Basis States.</p>
<p>In this usage is it somehow assumed that every Basis State in the Position Basis corresponds to a single Energy Eigenstate? (I wouldn't think this would be the case. But what is the point/result of applying the Hamiltonian to any given Wave Function then?)</p>
<p>Further confusion arises from this because if the Energy is exactly known then shouldn't there be some sort of maximal uncertainty in time?</p>
<p>As a final question is there any kind of useful interpretation of multiplying the eigenket by it's eigenvalue as appears in the above Observable Equation? In all treatments I've seen this multiplication is simply ignored and the eigenvalue itself is the only focus. </p> | g10352 | [
-0.005361638031899929,
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0.0011661568423733115,
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0.0314338393509388,
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0.10886845737695694,
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-0.04216751083731651,
0.013826482929289341,
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0.024427585303783417,
0.04... |
<p>I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in mathematics called lattice theory which deals with ordering. </p>
<p>I am wondering whether the lattice theory in mathematics could actually help me in construct such "general algorithm" which can deal with any kinds physical lattice. If so, I will delve into this area. Please help me, thanks.</p>
<p>If you know a more specific mathematical area that deals with general physical lattice ( I have some names in my mind like, mathematical crystallography, lattice graph theory). Please hint me so that I could move forward. Thank you.</p> | g10353 | [
0.0142515879124403,
0.0448504313826561,
0.026254529133439064,
0.02439947985112667,
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0.035160768777132034,
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0.016528427600860596,
0.057198312133550644,
0.00825691968202591,
0.013051624409854412,
-0.0346... |
<p>I'm working with a mass density gradient with length $L$ and I'm trying to solve the heat equation in 1-D (mass diffusion equation, $\partial_t\rho(t,x)=D\Delta\rho(t,x)$), but I'm not sure which boundary conditions should I use and what would they physically mean.</p>
<p>The starting mass density profile ($\rho(0,x)=f(x)$) is a step-function, where the lower half has density $\rho_1$ and the upper half has $\rho_2$.</p>
<p>$f(x)=\begin{array}{ll}
\rho_1 & 0<x<L/2 \\
\rho_2 & L/2<x<L \\
\end{array}$</p>
<p>Considering the boundary conditions, I find more difficult to interpret them. The mass inside the sample's volume is kept constant during the experiment as the sample is isolated from surroundings. </p>
<p>For this purpose, would Neumann Boundary Conditions be appropriate?</p>
<p>$$\partial_x \rho(t,0)=0=\partial_x \rho(t,L) $$</p>
<p>I'm unsure because all the examples I found do not treat the heat equation/diffusion equation as function of mass density ($\rho$). Would Neumann BC suggest No Mass Transfer between the sample's volume and the surroundings?</p> | g10354 | [
0.04491908475756645,
-0.045815207064151764,
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0.0030050945933908224,
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0.053060710430145264,
0.0655517727136612,
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0.003062131814658642,
-0.03797733411192894,
-0.01905951462686062,
0.009814062155783176,
0.04... |
<p>Can one show that in quantum field theory <em>at least some example</em> massive particles propagate with speed less than speed of light, while massless travel at speed of light? Well, motion is a different thing in QM than in classical mechanics, and question might be formulated differently.</p> | g10355 | [
0.02900185063481331,
0.02307741902768612,
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0.04244426265358925,
0.05923847109079361,
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0.025164108723402023,
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-0.007928000763058662,
0.029686201363801956,
0.0176656823605299,
0.011823690496385098,
-0.01... |
<p>And if so, can we observe a difference in the electron scattering cross section with transversely polarized VS longitudinally polarized protons?</p>
<p>P.S. Let me make my question more precise. Consider the charge shape of the proton. In the rest frame of a proton with spin in +z direction, what's the spatial dependence of the expectation of the electric charge operator $j^0(x)$, at some particular renormalization scale? Can this question be sufficiently answered by the currently available data from polarized proton scattering experiments?</p> | g10356 | [
-0.003028743201866746,
-0.009232851676642895,
-0.007034014444798231,
0.016330765560269356,
0.07688605040311813,
-0.004257408902049065,
-0.04861856997013092,
0.03861921280622482,
-0.008889454416930676,
-0.009257001802325249,
0.008223782293498516,
0.0228261835873127,
-0.0352487750351429,
-0.... |
<p>Why commercial wind generators usually have just 2-3 blades?</p>
<p>Having more blades would allow to increase power OR decrease diameter.
Decreased diameter would also reduce stress due to different wind speed on different height...</p>
<p>But despite that commercial generators have few blades...</p> | g565 | [
0.03040171228349209,
0.1176338866353035,
0.00953624490648508,
0.037294115871191025,
0.0008486140286549926,
0.0009242013329640031,
0.025191161781549454,
0.031425245106220245,
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-0.01818198710680008,
-0.01995699666440487,
-0.018317347392439842,
-0.026174122467637062,
0.04... |
<p>I just saw a great clip on the <a href="http://espn.go.com/action/skateboarding/story/_/id/8098465/mischo-erban-breaks-world-record-fastest-skateboard-speed" rel="nofollow">fastest skateboarder</a> to date. He wants to reach 160 mph from a dropped board, but says his speed is limited by the roads currently available. What is the shortest run required for a "dropped skateboard" rider to reach 160 mph? Is it even possible?</p> | g10357 | [
0.0011000145459547639,
0.07440410554409027,
-0.017158305272459984,
0.05591268092393875,
-0.007826064713299274,
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0.02677679993212223,
0.007783909793943167,
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-0.0029019031208008528,
-0.04712410643696785,
-0.021476048976182938,
-0.03899194300174713,
... |
<p>Must all symmetries have consequences?</p>
<p>We know that transnational invariance, for example, leads to momentum conservation, etc, cf. <a href="http://en.wikipedia.org/wiki/Noether%27s_theorem" rel="nofollow">Noether's Theorem</a>.</p>
<p>Is it possible for a theory or a model to have a symmetry of some kind with no physical consequences at all for that symmetry?</p> | g10358 | [
0.008038288913667202,
0.014015480875968933,
0.009876268915832043,
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0.022962000221014023,
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0.014602684415876865,
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-0.051772162318229675,
0.02... |
<p>Imagine I have an infinitely strong container of volume $v_1$, and I fill it with some monoatomic fluid like liquid hydrogen. I then proceed to compress the walls of the container to reduce its volume by some fraction to $v_2$. </p>
<p>How much force/energy is required to reduce the volume of the container to ~99%, ~50%, then perhaps ~0.1% of its original volume? How might one characterize the various states of matter (presumably plasma) inside the container at different ratios of $\frac{v_2}{v_1}$? </p> | g10359 | [
-0.001546935411170125,
0.002643590560182929,
-0.020696274936199188,
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0.03495095670223236,
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0.025624098256230354,
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0.004296068102121353,
0.0018157997401431203,
-0.044537801295518875,
0.0005672716069966555,
... |
<p>Consider a simplest case of a heat exchanger - two parallel pipes of flowing liquids (say, hot and cold) that have physical contact along some part of their length. Hot water of a certain temperature goes from A to B. Cold water can go either from C to D or from D to C. Assume that heat exchange between liquids occurs only where pipes contact (at XY part). What is the favorable direction (meaning "the most heat is transferred) of a coolant flow relative to hot flow - in the same direction (C->D) or in the reverse direction (D->C)? How the coolant and hot flows' temperatures are distributed along the pipe contact?</p>
<p><img src="http://i.stack.imgur.com/Yr9sq.png" alt="enter image description here"></p> | g10360 | [
0.05920162796974182,
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0.005267592612653971,
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-0.01065212581306696,
0.010498793795704842,
0.09017008543014526,
0.011234869249165058,
0.0162... |
<p>Why do waves on the sea shore move towards the shore even when the tide is going out?</p> | g10361 | [
0.06126747652888298,
0.005351676139980555,
0.005443711765110493,
0.011893689632415771,
0.052882127463817596,
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<p>Hydrogen fusion requires two hydrogen nuclei to get close enough (typically a few fm) to fuse. Much of the problem of creating a fusion reactor is overcoming the Coulomb repulsion between a pair of nuclei - the millions of degrees for Maxwellian distributions, the Bremstrahlung losses for inertial confinement. </p>
<p>If we could align the paths of two neutral Hydrogen <b>atoms</b> (of whichever isotopes), what would the repulsion look like between them as they approach collision? Obviously at long range there is negligable force as both are neutral. But as they approach each other, what happens to the electron distribution? </p>
<p>Intuitively, I expect a bonding cloud to form between the nuclei, and antibonding clouds beyond them. This would presumably attract at first until reaching the usual Hydrogen covalent bond length, after which the internuclear repulsion would increasingly dominate.</p>
<p>But how does that compare to bare ionic collision? How much lower is the potential barrier? </p>
<p>Obviously if it was significantly lower and we could somehow engineer the collision to achieve fusion, the cross section would be larger than ionic fusion, but how much?</p>
<p>Or would the barrier be just as high over the final few femtometers?</p> | g10362 | [
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<p>A particle in one dimension moves under the influence of a potential $V(x)= ax^6$, where $a$ is a real constant. For large $n$, what is the form of the dependence of the energy $E$ on $n$?</p> | g10363 | [
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<p>Following up on this answer, is the point at which waves break on the sea shore a guide to the depth of the sea at that point? Could it indicate eg hidden rocks?</p>
<p><a href="http://physics.stackexchange.com/questions/13841/explain-the-direction-of-waves-on-sea-shore/13844#13844">Explain the direction of waves on sea shore</a></p> | g10364 | [
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<p>I have thought about this and looked for answers for a long time now, but I do not have any name or label for this problem, which is the reason for the long title of this question.</p>
<p>I have come across this many times, but I have never found any rationale for this, i.e. a clear interpretation or explanation, so here are a few examples of what I mean. I suspect that they have different answers.</p>
<p>I) In e.g. Shapiro and Teukolsy's book <em>Black Holes, White Dwarfs and Neutron Stars</em>, (1983), the Salpeter birthrate function for stars is defined as</p>
<p>$\psi_s\mathrm{d}\left(\frac{M}{M_\odot}\right) = 2\times10^{-12}\left(\frac{M}{M_\odot}\right)^{-2.35}\mathrm{d}\left(\frac{M}{M_\odot}\right)\mathrm{\ stars\ pc}^{-3}\mathrm{\ yr}^{-1}$</p>
<p>Questions: why does it matter that those differentials are included? Why make the statement look messier by multiplying both sides of the equation with something that obviously cancels out? If it is just to show what $\psi_s$ depends on, then it makes mre sense to me to define the function like so:</p>
<p>$\psi_s(M) = 2\times10^{-12}\left(\frac{M}{M_\odot}\right)^{-2.35}\mathrm{\ stars\ pc}^{-3}\mathrm{\ yr}^{-1}$</p>
<p>What is (or may be) the answer?</p>
<p>II) When defining the equation of continuity for brownian motion particles in 1D space, I have a lecture note that reasons in the following way:</p>
<p>Given the setup </p>
<pre><code> | n(x,t) |
| |
J(x,t) ----> ----> J(x + dx, t)
| |
| |
---------------------->
x x+dx
</code></pre>
<p>Where $n(x,t)$ is the number density of particles between positions $x$ and $x + \mathrm{d}x$ and $J(x, t)$ is the flux of these particles at given position $x$ and time $t$. The particle number is conserved, and they are assumed identical and non-interacting with each other.</p>
<p>In the notes that I have, the following equation is readily displayed as though it was crystal clear why it was written inthis way:</p>
<p>$n(x, t + \mathrm{d}t)\mathrm{d}x - n(x, t)\mathrm{d}x = - (J(x + \mathrm{d}x, t)\mathrm{d}t - J(x, t)\mathrm{d}t)$</p>
<p>or (multiplying the minus into the parenthesis)</p>
<p>$n(x, t + \mathrm{d}t)\mathrm{d}x - n(x, t)\mathrm{d}x = J(x, t)\mathrm{d}t - J(x + \mathrm{d}x, t)\mathrm{d}t$</p>
<p>Question: I can not myself justify why it makes sense to multiply with the diferentials as they are here. What is the significance, or rather the line of thought that lies behind the way this equation turns out?</p>
<p>I am unsure of my own answer which is only to look at the dimensions of the two functions themselves and then conclude what unit the factor should have on each side in order for the dimensions to make sense.</p>
<p>Just to complete the derivation, rewriting the equation by replacing the functions $n(x, t + \mathrm{d}t)$ and $J(x + \mathrm{d}x, t)$ with their Taylor expansion (to first order) gives</p>
<p>$n(x, t + \mathrm{d}t) = n(x, t) + \frac{\partial n}{\partial t}\mathrm{d}t$,</p>
<p>$J(x + \mathrm{d}x, t) = J(x, t) + \frac{\partial J}{\partial x}\mathrm{d}x$,</p>
<p>which, when inserted into the original equation gives</p>
<p>$\frac{\partial n}{\partial t}\mathrm{d}t\mathrm{d}x = -\frac{\partial J}{\partial x}\mathrm{d}x\mathrm{d}t$</p>
<p>so producing, again, a shared factor ($\mathrm{d}t\mathrm{d}x$) on both sides of the equation which can just be divided away (right?), and we end with the coninuity equation</p>
<p>$\frac{\partial n}{\partial t} = -\frac{\partial J}{\partial x}$</p>
<p>I hope this question makes sense or at least allows you to suggest a label for this kind of problem that I can use to find more information, Or better, help by answering the wuestion here.</p>
<p>Cheers.</p> | g10365 | [
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<p>I want to know the other forms of <a href="http://en.wikipedia.org/wiki/Beta_function_%28physics%29" rel="nofollow">beta-function</a> that make manifest certain properties of <a href="http://en.wikipedia.org/wiki/Renormalization_group" rel="nofollow">renormalization group</a>, for instance dependence on poles/residue and more. If possible can you state a reference, or perhaps a link, or sketch the method.</p> | g10366 | [
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<p>First of all, I would like to apologize in advance if I make stupid mistakes. I am a mathematician and I am trying to apply the Boltzmann distribution to places where I am not sure if it is applicable (albeit I have no choice). </p>
<p>The situation is: I have a system which consists in a discrete line of $M$ positions in which $N$ elements are distributed with a separation of at least $D$ positions between them. The state of each element should be its position in the line. Finally, (here's the fun part) each position in the line has an associated potential (so putting an element in the position $i$ means spending $\epsilon_{i}$ units of Energy). The usual approach to this problem as far as I have seen is just assigning $p_{i} = e^{-\epsilon_{i}/kT}$, where $p_{i}$ is the probability that there is an element in the position $i$. </p>
<p>I don't understand this approach and I am trying to derive that new one, but I am stuck when trying to force the particles to be separated. Any insight or reference would be very much appreciated.</p>
<p>Edit: If it is needed, we can also say that the particles might have a velocity (i.e. they can oscillate), but they should not be able to pass through each other.</p> | g10367 | [
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... |
<p>I want to learn how to code in Java and am in need of some cool computational physics projects that would provide motivation to do so. I was wondering if you guys could list some medium-level computational physics projects (from any field) that you think would provide a well-rounded insight into programming.</p> | g10368 | [
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<p>I've seen in a few gravity simulation games (ie. bouncing balls) the equation:</p>
<pre><code>force = G * m1 * m2 / distance^2
</code></pre>
<p>shortened to this by removing the gravitational constant:</p>
<pre><code>force = m1 * m2 / distance^2
</code></pre>
<p>I accept that it works fine and saves some calculations, but I'm wondering <em>why</em> it still works? Is the value just too small to matter? What's the physics behind this?</p> | g10369 | [
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<p>I have to do this exercise:</p>
<blockquote>
<p>9. The current $I(t)$ at time $t$ flowing in an electric circuit obeys the differential equation</p>
<p>$$I''(t) + RI'(t) + I(t) = \sin \omega t$$</p>
<p>where $R$ and $\omega$ are positive constants. The solution can be expressed in the form $I(t) = F(t) + A\sin(\omega t + a)$, where $F(t)\to 0$ as $t\to +\infty$, and $A$ and $\alpha$ are constants depending on $R$ and $\omega$, with $A > 0$. If there is a value of $\omega$ which makes $A$ as large as possible, then $\omega/(2\pi)$ is called a <em>resonance frequency</em> of the circuit.</p>
<p>(a) Find all resonance frequencies when $R = 1$.<br>
(b) Find those values of $R$ for which the circuit will have a resonance frequency.</p>
</blockquote>
<p>I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega t)+B\cos(\omega t)$.</p>
<p>I got the first and second derivates
$$I_{p}'=A\omega\cos(\omega t)-B\omega\sin(\omega t)$$
$$I_{p}''=-A\omega^{2}\sin(\omega t)-B\omega^{2}\cos(\omega t)$$
then, when I substitute in the equation
$$I''(t)+I'(t)+I(t)=\sin(\omega t)$$
I got the system on $(A,B)$:
$$\left\{\begin{array}{l}
-A\omega^{2}-B\omega+A=1 \\
-B\omega^{2}+A\omega+B=0
\end{array}\right.$$
so
$$A=\frac{(1-\omega^{2})}{(1-\omega^{2})^{2}+\omega^{2}}$$
I calculated $A$ because it follows $\sin(\omega t)$ and with this I can find all resonance frequencies.</p>
<p>Well, to find them, I gotta find the values of $\omega$ which makes $A$ as large as possible. To $A\to+\infty$, so, its denominator must goes to 0:
$$(1-\omega^{2})^{2}+\omega^{2}\to0$$
But that's not possible. </p>
<p>The answer is $\displaystyle\frac{1}{2\pi\sqrt{2}}$. Comparing with the exercise, I know that $\omega=1/\sqrt{2}$.</p>
<p>If the denominator was $(1-\omega^{2})^{2}-\omega^{2}$, I got the result, but I don't know what's wrong here.</p>
<p><strong>EDIT:</strong> I found after
$$I_{p}(t)=\frac{1}{\sqrt{(1-\omega^{2})^{2}+\omega^{2}}}\sin(\omega t-\alpha)$$
where $\alpha=\arctan(\omega/(1-\omega^{2})$. So,
$$A=\frac{1}{\sqrt{(1-\omega^{2})^{2}+\omega^{2}}}$$
and it's large just if its denominator is small... but how?!</p> | g10370 | [
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<p>In Griffiths' <em>Introduction to Electrodynamics</em>, Problem 1.28 (the triangular prism question) is especially challenging for me. I do not know how the limits of x are <strong>0</strong> to (<strong>1-y</strong>).
My concern is the latter limit, <strong>1-y</strong>. Can you explain systemically how this is arrived at?
On the face of it, I'm inclined to say the limits of x are <strong>0</strong> and <strong>1</strong>. I can see it's the diagonal across the x- and y-axes that's presenting the extra challenge, but I do not know how to go about establishing that indeed the limit is <strong>1-y</strong>.</p> | g10371 | [
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<p>I've been looking for questions about dark matter, and I've read some very interesting answers. However, I desire too look into it deeply. </p>
<p>This is not actually a question. I'm asking the community to recommend interesting references to understanding dark matter and dark energy. </p>
<p>I accept all sort of references: notes, books, scientific papers etc.</p>
<p>Let us assume some background on classical physics, thermodynamics and basics about quantum theory.</p> | g438 | [
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-0.... |
<p>I have problems fully understanding the concept of work, so please forgive me if this is simple. If I take a box of mass $ m $, and lift it a distance $ d $ vertically, <strong>why is the work I have done equal to $ gmd $, where $ g $ is the force gravity exerts on the box?</strong> I understand that work is equal to force times distance--so I'm not asking about the definition of work--but if I exert an upward force equal in magnitude to gravity's, won't the box remain motionless, i.e., net zero force, in which case the velocity is constant, and displacement and work done will be equal to zero?</p>
<p>Edit: To be clear, <strong>what I'm asking <em>is not a duplicate</em> of "Why does holding something up cost energy while no work is being done?", because I'm not asking about work done on an object with zero displacement, nor is it a duplicate of "What exactly is F in W=∫baFdx?", because I'm not asking about the distinction between the work done by an individual force and net force.</strong></p> | g315 | [
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<p>Can you suggest some references for rigorous treatment of thermodynamics? I want things like reversibility, equilibrium to be clearly defined in terms of the basic assumptions of the framework.</p> | g10372 | [
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