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<p>Say you have a hollow sphere with a uniformly distributed charge on the surface. Why is the electric field everywhere inside the sphere zero?</p>
<p>For the centre, its easy to add the vectors from the surface charges and show they sum to zero because of symmetry. </p>
<p>But can you show me how the field cancels out to zero for points other than the centre <strong>using vector addition?</strong></p>
<p>Thanks!</p>
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<p>Since a quantum mechanical system, even an isolated system containing one particle, can be described by a density matrix, with entropy for the system given by $\langle S\rangle=-k \rho\ln(\rho)$, is not entropy therefore a property of the system like mass or energy?</p>
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<p>If the universe is expanding, it would make sense that the spaces between particles are getting bigger. If this is so, then the particles which make up atoms are also affected. Does that imply the spaces between the components of an atom will become large for the subatomic forces to hold? Are atoms getting weaker?</p>
| 56 |
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<p>Consider a travelling wave produced by vibrating one end of a rope while the other end is made to freely move along a vertical line. Mathematically, the equation of the traveling wave that also represents the equation of motion of each point on the rope is given as follows.</p>
<p>$$y(x,t)=A\sin(kx-\omega t)$$</p>
<p>In the book I read, a certain particle on the rope moves up and down perpendicular to the propagation vector as wave does not transfer material but energy.</p>
<p>I don't agree with the statement that "a certain particle on the rope moves up and down perpendicular to the propagation vector". It is because I think to maintain the same point on the rope to move up and down, the arc length of $y(x,t)$ must be equal to that of $y(x,t+\Delta t)$ for $0\leq x \leq a$ and any $\Delta t$. However, according to the Notes below, their arc length are not generally equal.</p>
<p><img src="http://i.stack.imgur.com/8B30K.png" alt="enter image description here"></p>
<p>From the figure above, how can $P$ and $Q$ be the same point of the rope while the red and blue curves have different length?</p>
<p>So my question are: <strong>Does each point of the rope move up and down?</strong></p>
<h2>Notes</h2>
<p>From $x=0$ to $x=a$, generally the arc length of $y=\sin x$ is not equal to that of $y=\sin(x-x_0)$.</p>
<p>Let $S(x_0)$ be the arc length of $y=\sin(x-x_0)$ for the given interval. </p>
<p>\begin{align}
S(x_0) &= \int_0^a\sqrt{1+\left(\frac{dy}{dx}\right)^2}\, \textrm{d}x\\
&= \int_0^a\sqrt{1+\cos^2(x-x_0)}\, \textrm{d}x\\
\end{align}</p>
<p>If $S(0)=S(x_0)=\text{const}$ for any $x_0$ then $\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}=0$. Thus I have to check whether or not $\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}=0$.</p>
<p>\begin{align}
\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}
&= \frac{\textrm{d}}{\textrm{d}x_0}\int_0^a\sqrt{1+\cos^2(x-x_0)}\, \textrm{d}x\\
&= \int_0^a\frac{\textrm{d}\left(\sqrt{1+\cos^2(x-x_0)}\right)}{\textrm{d}x_0}\, \textrm{d}x\\
&= \int_0^a\frac{\cos(x-x_0)\sin(x-x_0)}{\sqrt{1+\cos^2(x-x_0)}}\, \textrm{d}x\\
&= \int_{-x_0}^{a-x_0}\frac{\cos y\sin y}{\sqrt{1+\cos^2y}}\, \textrm{d}y\\
&= \sqrt{1+\cos^2 x_0}-\sqrt{1+\cos^2(a-x_0)}\\
&\not = 0
\end{align}</p>
<p>It implies that generally they are not equal.</p>
| 10,011 |
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<p>I am given that a wave function at time 0 is given as
$$\psi(x,t=0)=\sin^6(\frac{\pi x}{2L})\cos(\frac{\pi x}{2L})$$
I am asked to find this wave function as a function of time. In order to do this, I feel like I need to find the combinations of eigenfunctions that make this function and then I can find how they each evolvewith time, so that
$$\psi(x,t)=\sum_{n=1}^{\infty}c_n\psi_n(x)e^{-iE_nt/\hbar}$$</p>
<p>Am I correct in saying this? I feel that this makes sense, and then I can find $c_n$ as $$c_n=\int_0^{2L}\psi(x,t=0)\sin(\frac{n\pi
x}{2L})dx$$</p>
| 10,012 |
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<p>When a system rejects heat it can be assumed that work is done by the system. So as per law the sign convention must be negative. But my physics textbook says the reverse. So is the question.</p>
| 10,013 |
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<p>This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is:</p>
<p>Consider a field Lagrangian with only a kinetic term,</p>
<p>$$L = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi $$</p>
<p>Consider the very simple transformation $\phi \rightarrow \phi + \alpha$ ($\alpha$ constant), and so I understand here that $\alpha$ plays the role of $\delta\phi$. I determine the Noether current as
$$\frac{\partial L}{\partial[\partial_{\mu}\phi]}\delta\phi$$</p>
<p>and the result is
$$\alpha\partial_\mu\phi$$</p>
<p>But in Peskin & Schroeder (just above eq 2.14), the result they give is:</p>
<p>$$\partial_\mu\phi$$</p>
<p>And it doesn't seem to be an erratum. I don't care that "localized" Lagrangian very much (hey, wait before closing, please), but a <em>very general</em> question arises:</p>
<p>Is $\alpha$ dropped simply because $\partial_\mu\phi$ is too a conserved quantity (and so under "conserved current" one understands the general concept, momentum, energy or whatever, regardless of its value), or am I missing some other very basic detail that is assumed to be known by the reader?</p>
<hr>
<p>Later edit: I have eventually understood this question and more, by reading the beginning of chapter 22 of <a href="http://web.physics.ucsb.edu/~mark/qft.html" rel="nofollow">Srednicki</a>. I am finding that book (well, the free preprint for the moment) crystal clear, it seems excellent.</p>
| 10,014 |
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<p>Tesla patented a device for gathering energy from light, using the photoelectric effect. (<a href="http://www.google.com/patents/about?id=YitoAAAAEBAJ" rel="nofollow">US 685,957 - Apparatus for the Utilization of Radiant Energy</a>):</p>
<p><img src="http://i.stack.imgur.com/cSGJ1.png" alt="Telsa's photoelectric effect generator"></p>
<p>Basically just a sheet of "highly polished or amalgamated" metal (top) connected to a capacitor (the ⊔⊓ shapes) connected to the Earth. (The other stuff on the right of the image is a load intermittently driven when the charge on the capacitor rises high enough.)</p>
<p>The patent says it would be able to collect energy from the sun. Would this actually work? Why or why not? Does it need to be in a vacuum? The patent seems to say that it works better in a vacuum but that it is not required. Would the plate stay charged after electrons are thrown off, or would they be attracted and drift back to it and neutralize it? How does the capacitor to earth make this better than a plate hanging in space? It allows more charge to be stored for the same work function voltage of the metal?</p>
<p>The patent demonstrates that he doesn't understand the cause of the photoelectric effect ("sources of such radiant energy throw off with great velocity minute particles of matter which are strongly electrified, and therefore capable of charging an electrical conductor"), but he understands the results.</p>
<p>(Incidentally, I have tried building this with a piece of aluminum foil, electrolytic cap, and wire strong enough to stick a few inches into the ground. I didn't see any voltage with a multimeter in bright sunlight.)</p>
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<p>Consider a conducting wire of 1M and 1000KM. Now if we connect a battery and a bulb to both these wires. Bulb glows instantaneously its because (my guess:) electric filed travels from positive terminal of battery to negative terminal through wire almost instantaneously.</p>
<p>Now, My question is there any speed for this electric field? If yes, what is it? Can I induce any delay in in glowing a bulb by similar expermient?</p>
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] |
<p>I'm looking for something that will generate scatter plots comparing different properties of isotopes. Ideally I'd like some web page that lets me select axis and click go but a CSV file with lost of properties would work.</p>
| 10,017 |
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<p>I have recently become curious about modeling the repulsion of everyday objects in contact with one another. By repulsion I mean as you attempt to walk through a wall, the pain in your nose suddenly alerts you to the fact that it won't be possible. I've come up with this exercise:</p>
<p>What is the mutual pressure exerted by two dipole sheets of finite extent, each area $A$, separated by a distance $D$? Assume $D \gg$ d where the dipole moment $p = ed$ and $e = 1.6\times10^{-19}$. There are $N$ dipoles per unit area. Make an order of magnitude estimate for $N$ to represent a density one would find in ordinary matter. Make a numerical plot for $10^{-10} < D < 10^{-3}$ meters. </p>
<p>This is a crude electrostatic model of two electrically neutral materials brought close together. The oppositely oriented dipole sheets are generated by the electronic cloud repulsion and deformation as the two interfaces approach. It is meant to serve as a quantitative estimate of the <em>maximum</em> distance at which Coulomb repulsion could possibly stop one body from passing through another. Interface physics is pretty rich, but I wonder how this model does in predicting such pressures.</p>
<p>If anyone has related references please post them, especially any experimental studies.</p>
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<p>I have a general question concerning mean-field approaches for condensed matter classical of quantum statistical mechanic systems. Does determining the mean-field by a variational approach <em>always</em> imply that the self-consistency is satisfied ? Moreover are there some cases where it is physically justified to look for saddle points and where a variational approach is misleading (say the energy is unbounded from below with respect to the mean-field parameters for instance) ?</p>
<p>For example, consider the simple case of the ferromagnetic Ising model. There, one introduces the magnetization ($m$) as the mean-field parameter with $m=\left<s_i^z\right>$ where the expectation value is taken with respect to the mean-field Hamiltonian that depends on $m$ (and $s_i^z$ is the spin variable $s_i^z=\pm 1$). In this particular case, when one finds a solution for $m=\left<s_i^z\right>$, this is equivalent to finding an extremum for the energy or the free-energy. Thus, my question is simply : instead of solving for the self-consistency can one instead look for the global energy (or free-energy) minimum with respect to the mean-field parameters (and does this approach always makes sense physically) ? Here I give the example of the Ising model, but my question also applies to any model (fermionic, bosonic, spin models, etc.).</p>
<p>I am more or less looking for counterexamples here, if there exist any. </p>
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] |
<p>The <a href="http://en.wikipedia.org/wiki/No-hair_theorem">no hair theorem</a> says that a black hole can be characterized by a small number of parameters that are visible from distance - mass, angular momentum and electric charge.</p>
<p>For me it is puzzling why local quantities are not included, i.e. quantum numbers different from electrical charge. Lack of such parameters means breaking of the conservations laws (for a black hole made of baryons, Hawking radiation then is 50% baryonic and 50% anti-baryonic).</p>
<p>The question is:</p>
<ul>
<li>If lack of baryonic number as a black hole parameter is a well established relation?</li>
</ul>
<p>OR</p>
<ul>
<li>It is (or may be) only an artifact of lack of unification between QFT and GR?</li>
</ul>
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<p>First I will have to explain my question. Look at the image below. This shows doppler shift when an object is moving horizontally to the direction of the wave. Keep the word 'horizontally' in mind. Now this happens because:</p>
<p>I will quote Jim from his answer for <a href="http://physics.stackexchange.com/questions/97881/redshifting-of-light-from-a-moving-light-source">Redshifting of light from a moving light source</a></p>
<blockquote>
<p>We all know that light is a wave, when you turn on your headlights and drive in reverse, the light is doppler shifted because of the motion of source. When not moving, each cycle of the light wave is emitted from the same position; it has a specific set of wavelengths. The distance between one crest of a wave and the next crest is equal to the speed of light, c, times the period of the light (which is determined by the oscillations in your headlights and won't change when you are in motion). When you drive backwards, the distance between one crest and the next becomes the period times c plus the period times your backwards velocity (approximately); the second crest is not emitted at the same location as the first, so it extends the wavelength. From your perspective, the emitted wave would not be red-shifted at all, but from a stationary observer's perspective it is.</p>
</blockquote>
<p><img src="http://i.stack.imgur.com/2FcVL.png" alt="enter image description here"></p>
<p>So now my question is, imagine a car which has a torch attached to one of its windows. The torch is switched on and the car begins to move. When the car moves, its movement is in the opposite axis from the propagation of the wave. So each crest will be released from a different location while the first crest is already on its way in a straight line. I will try to represent this graphically.</p>
<p><img src="http://i.stack.imgur.com/qakE0.png" alt="enter image description here"></p>
<p>The representation is very estimate. It just shows how would the light bend as each crest is released from a different location. Please explain this to me. Will the light actually bend? Why or Why not?</p>
<p><strong>Edit</strong></p>
<p>What I have concluded from the answers is that first a photon is emitted and then it continues as a wave and is in no way attached to other photons. Is this right? If I got this then I got the answer for my question.</p>
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<p>For evaluating the electric field of some charge distribution one can use $$\phi(r):= \frac{1}{4 \pi \varepsilon_0}\int_{\mathbb{R}^3} \frac{\rho(r')}{||r-r'||_2} dr'.$$</p>
<p>My question is: What symmetry do we need to have that we can write in spherical coordinates $$||r-r'||_2 = \sqrt{||r||_2^2+||r'||_2^2-2||r||_2||r'||_2\cos(\theta')}~?$$ </p>
<p>This is of course not the most general way to express this distance, as the $\phi$ dependence is missing. So, under what conditions can the distance be expressed like this? </p>
<p>Notice that $\theta'$ is the respective angle in spherical coordinates, so it's NOT the angle between $r$ and $r'$.</p>
<p>So in particular, your answer should clarify, why we can evaluate for example the electric potential of a sphere by integrating: $$\frac{1}{4 \pi \varepsilon_0} \int_0^{2\pi}\int_0^\pi \int_0^{\infty} \frac{\rho(r')||r'^2|| sin(\theta')}{\sqrt{||r||_2^2+||r'||_2^2-2||r||_2||r'||_2\cos(\theta')}}d||r'||_2 d\theta'd\phi',$$ but need to refer to a more general equation in this example, where the $\phi$ angle is used too: <a href="http://aforrester.bol.ucla.edu/comprobs/F06prob14.pdf" rel="nofollow">excercise 14b)</a></p>
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<p>Average velocity, as I have heard, cannot be found simply by finding the average of two numbers. I have a question on average velocity, but am simply unable to proceed:</p>
<blockquote>
<p><em>A particle moving in a straight line covers half the distance with speed of 3m/s. The other half of the distance is covered in two equal time intervals with speed 4.5m/s and 7.5m/s respectively. The average speed of the particle during this motion is?</em></p>
</blockquote>
<p>What I know: The average speed is the total distance divided by the total time
How exactly should I approach the question?</p>
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<p>I'm trying to figure out how many atoms are decaying spontaneously in a span of 2 seconds. Let's say that the quantum yield is 0.45, and that the lifetime "τ" (tau) is 10 microseconds. </p>
<p>Then I found that the radiative lifetime is 22.2 microseconds. However, at this point I'm stuck. I don't know a relationship in order to get the amount of spontaneously decayed atoms.</p>
<p>What relationship is there in order to get spontaneous decayed atoms with this information? Also, this is homework, so just a hint would be appreciated.</p>
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<p>Quantum mechanics has a peculiar feature, entanglement entropy, allowing the total entropy of a system to be less than the sum of the entropies of the individual subsystems comprising it. Can the entropy of a subsystem exceed the maximum entropy of the system in quantum mechanics?</p>
<p>What I have in mind is eternal inflation. The de Sitter radius is only a few orders of magnitude larger than the Planck length. If the maximum entropy is given by the area of the boundary of the causal patch, the maximum entropy can't be all that large. Suppose a bubble nucleation of the metastable vacuum into another phase with an exponentially tiny cosmological constant happens. After reheating inside the bubble, the entropy of the bubble increases significantly until it exceeds the maximum entropy of the causal patch.</p>
<p>If this is described by entanglement entropy within the bubble itself, when restricted to a subsystem of the bubble, we get a mixed state. In other words, the number of many worlds increases exponentially until it exceeds the exponential of the maximum causal patch entropy. Obviously, the causal patch itself can't possibly have that many many-worlds. So, what is the best way of interpreting these many-worlds for this example?</p>
<p>Thanks a lot!</p>
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<p>I`m looking for a nice introductary reference that explains how the turbulence coefficient or any kind of turbulence parameterization (in view of applications to atmospheric turbulence for example) can be derived from the gravity - fluid dynamics correspondance, such that even I can get it. I mean, if something like this exists ...</p>
<p>I`m basically quite familiar with the hydrodynamic part (NS equation, etc) of this correspondance whereas about the other side I feel a bit more shaky ...</p>
<p>I`m finally looking for a citable reference, but any "reasonable" source (slides of a talk, video, ect) that explains how a turbulenc coefficient / parameterization can be obtained would be welcome and appreciated.</p>
<p>Edit</p>
<p>To clarify what I mean, relevant papers for the topic are for example <a href="http://arxiv.org/abs/1104.5502" rel="nofollow">here</a>, <a href="http://arxiv.org/abs/1101.2451" rel="nofollow">here</a>, and jep <a href="http://arxiv.org/abs/1107.5780" rel="nofollow">this</a> one linked to by Mitchell. </p>
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<p>I've been reading through various materials on relativistic quantum mechanics, but I find the lack of simple examples disturbing.</p>
<p>I'm acquainted with the general form the solutions to the Dirac equations have, but I'm having trouble just practically getting any specific example solution.</p>
<p>Since the motion of a free Dirac particle is entirely determined with four-momentum $p^\mu$ and the spin (polarization) four-vector $s^\mu$, how exactly does one find the wave-function corresponding to some given $p^\mu,s^\mu$?</p>
<p>E.g. $p^\mu=m\{\sqrt{2},0,0,1\}, s^\mu=\{1,0,0,\sqrt{2}\}$</p>
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<p>I decided to dig a tunnel inside the Earth. In equatorial plane. It should be designed in such a way that it follows the Coriolis effect. That means if, say a stone is dropped from rest into the tunnel, then it travels in the tunnel without touching the walls of the tunnel until it reaches the tunnel's lip on the other side somewhere near the surface of the Earth. So the question itself is an equation of the curve of the tunnel. I suspect that there is no nice closed form for the curve. But who knows? Nevertheless, it would be good to know how deep the tunnel goes and where it rises again to the surface and the travel time. </p>
<p>You could say that it would be impossible to do this on Earth, but you actually could do this on the Moon.</p>
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<p>When I was student I was told that time is defined by the requirement that the physical laws are simple. For example, in classical mechanics time can be defined by the requirment that the velocity of an isolated body is constant. One could generalize this approach by assuming that <em>space-time</em> is defined by the requirement that the physical laws are simple. This can be easily expressed in mathematical language as follows.</p>
<p>In general relativity the universe is represented by the triple $(M, g, T)$ where $M$ is a four-dimensional differentiable manifold, and $g$ and $T$ are a Lorentzian metric and a tensor field on $M$ satisfying Einstein's equation.</p>
<p>According to the above approach, we could say that the "true" physical reality of the universe is represented by the pair $(M, T)$, while the metric $g$ emerges as an appeareance from the requirement that evolution appears simple, i.e., that $g$ and $T$ satisfy Einstein's equation.</p>
<p>From the mathematical point of view, this approach poses for example the following problems: does any tensor field $T$ admit a metric $g$ such that Einstein's equation is satisfied? To what extent a tensor field $T$ univocally determines the metric $g$?</p>
<p>Does this approach make sense, at least from the mathematical point of view? </p>
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<p>I get confused when I see expressions like "the universe is $x$ years old" or "$10^{-2}$ seconds after the big bang" since it seems to me that relativity shows such statements don't have meaning. Is it assumed or experimentally verified or proved that space-time is equipped with a projection to the real numbers whose differential is non-zero on tangent vectors with a non-zero time component? If assumed, why is this a reasonable assumption? and if proved, what are the initial axioms (causality)? Looking at a similar question, perhaps the point is that all geodesic given by an initial point and negative time-pointing tangent vector must converge to some given point in finite (backwards) time? And to get completely cranky, are there good scientific reasons to assume no closed geodesics?</p>
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<p>In an electron gun, the heating filament heats the cathode, releasing electrons by thermionic emission. I've read that <em>"electrons are negatively charged particles and the positively charged cylindrical anode develops a strong electric field that exerts a force on the electrons, accelerating them along the tube"</em>. However, I don't think that this explanation is very clear, and I was wondering specifically how the "strong electric field" inside the cylindrical anode is able to accelerate the electrons?</p>
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<p>I'm guessing that when the LHC ramps up to 4000 GeV this means they are increasing the current in the superconducting magnets as RF fields accelerate the beams. Where does this current go when they ramp down? Is it dissipated as heat? Is it fed back into the grid?</p>
<p>P.S.- by 'ramp' I mean the normal increasing/decreasing of the current in the magnet at the beginning and ending of a fill. It builds gradually (ramps) and decreases gradually. The whole time the magnet is still superconducting.</p>
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<p>I'm reading an article on liquid crystal lasers and they're saying periodicity determines lasing wavelength and cavity geometry determines emission wavelength. What is the difference?</p>
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<p>Can someone summarize as to what are the problems and/or the open questions with the <a href="http://www.google.com/search?as_q=hayden+preskill" rel="nofollow">Hayden-Preskill circuit</a>? (in the context of understanding black-holes or as a computer science question)It gives a framework to see how the black-hole can thermalize in a $log (entropy)$ time scale. What next? </p>
<p>To start off, </p>
<p>The sort of problem with the Hayden-Preskill circuit is that it works via random unitary transformations on disjoint pairs of qubits and hence doesn't have an Hamiltonian interpretation. I believe one has always believed that matrix models (of matrix size $\sqrt{entropy}$) can saturate this logarithmic thermalization bound via Hamiltonian evolution. </p>
<ul>
<li><p>Has this above belief about the matrix models been proven yet? </p></li>
<li><p>Is there an intuitive explanation as to why a theory with finite dimensional matrices should behave like an infinite dimensional system? </p></li>
</ul>
| 322 |
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<p>The statistical thermodynamics definition of entropy: $S = kN \ln (W)$ troubles me a lot with the problem of dimenstions. where $S$ is entropy; $k$, the Boltzmann constant; $N$ the number of particles in the system and W the number of microstates corresponding to a given macrostate of the system. </p>
<p>For the equation to be dimensionally correct, $W$ must be a dimensionless number.</p>
<p>But if $W$ were to be a dimensionless number, then rewriting the equation in the form: $S = \ln [(W)^kN]$, we find the quantity in square brackets doesn't make sense - since a pure number is raised to a power that has dimensions! More over, since the argument of a logerthmic quantity must be a dimensionless number, the RHS will have no dimensions while LHS has dimensions leading to a paradox.</p>
<p>Again, statistical thermodynamics is full of equations that give elaborations of the quantity $W$. These elaborations contain logerithmic terms with arguments having dimentions (mostly, properties such as $U$ or $E$ and $V$, OR the corresponding molar quantities such as $U/N$ etc.) which not only defies normal mathematics rules but also gives rise to the perennial problem of Gibbs paradox.</p>
<p>Quantum mechanics, information theory etc are broughin to account for Gibbs paradox - which arises in the first place due to a confusing mathematical expression for entropy.</p>
<p>While there is no confusion in equilibrium (classical) thermodynamics about the fact that S is an extensive property, statistical thermodynamics results/equations lead to arguments wheteher $S$ is an intensive quantity or an extensive quantity - all because of the statistical thermodynamics definition of entropy through the equation $S = kN \ln (W)$ - that is confusing with its inherent problem of dimensions.</p>
<p>Any clarifications regarding the dimensional analysis of the defining equation of entropy in statistical thermodynamics/statistical mechanics is requested.</p>
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<p>Regarding the cosmological selection hypothesis and testable predictions, Lee Smolin asserted the following:</p>
<blockquote>
<p>"Smolin: I did make two predictions which were eminently checkable by
astrophysical and cosmological observations, and both of them could
easily have been falsified by observations over the last 20 years, and
both have been confirmed by observations so far.</p>
<p>One of them concerns the masses of neutron stars and the prediction is
there can't be a neutron star heavier than about twice the mass of the
sun. This continues to be confirmed by the best measurements of the
masses of neutron stars."</p>
</blockquote>
<p>What is he referring to? As far as I know, the neutron star mass limit is a prediction of GR, and doesn't suggest that any fine-tuning is involved in it. Is this correct?</p>
<p><a href="http://www.space.com/21335-black-holes-time-universe-creation.html" rel="nofollow">http://www.space.com/21335-black-holes-time-universe-creation.html</a></p>
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<p>How might one consider a conductor to be an 'obstructor'? </p>
<p>Might the strength of the 'skin effect' of a conductor be in direct relation to conductance?</p>
<p>And how does this relate to insulators, as these material are literal obstructors to energy transfer.. Might electricity flow at full strength along the surface instead of through the insulator such as along the inside of wire insulation? </p>
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] |
<p>Is there any example of a transmission of energy in a medium that does not show wave nature? </p>
| 10,036 |
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<p>While i was in high-school i learn't the <strong>Doppler's Effect</strong> which if i remember correctly is:</p>
<ul>
<li>The Apparent change in the frequency of sound caused due the relative motion between the sound and the observer.</li>
</ul>
<p>This phenomenon seems obvious, but what i would like to know is, what use does <strong>Doppler Effect</strong> have in real life. Why is it useful?</p>
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<p>I wanted to know a few things about elliptical motion, that I do not understand so far.</p>
<p>Let's imagine that we have an attractive gravitational two body problem where both bodies carry out elliptical motion around the center of mass. In this case, the center of mass is in one focus of the ellipse, right?
Further, we are able to determine the closest distance between the center of mass and the 1st planet(perigee) and the largest distance(apogee). further we have the semi-major axis, that will enter our third kepler-law, but is different from apogee and perigee.</p>
<p>but is there also a physical meaning behind the semi-major axis? and is there any meaning behind the center of the ellipse, cause until now, we only discussed the one focus of the ellipse?</p>
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<p>An object is pulled up a slope at constant speed. I'm trying to find the Tension on the rope.</p>
<p>$$m = 65kg \Rightarrow W=-637$$
$$\mbox{constant speed} \Rightarrow a = 0$$
$$\mbox{angle of slope} = 45^\circ$$
$$T=?$$</p>
<p>I use Newton's first Law to determine that there is no acceleration because of constant speed:
$$\sum F=0$$</p>
<p>Online I found that I should use this:
$$W \sin\theta=T$$</p>
<p>I don't understand how they came up with that equation. I tried using mine like this but the answer is way too much to be correct. My equation: $\sum F_y=-W+\sin\theta*T_y=0$</p>
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<p>Can space-time singularities be treated as mathematical knots occurring in dimensions greater than four? I just drew an analogy with knots in one-dimensional strings. When a rubber-band is looped over again and again, it ultimately forms a clumped-up ball like structure, having 3 dimensions. Similarly, the fabric of space-time may also get looped over and over again to form singularities having dimensions greater than four.</p>
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<p>I am doing this problem:</p>
<blockquote>
<p><em>The upward normal force exerted by the floor is 620 N on an elevator passenger who weighs 650 N.</em> </p>
<p><em>What is the magnitude of the acceleration?</em></p>
</blockquote>
<p>This is how I solved it:</p>
<p>$$\sum F = F_1+F_2=N+(-W)=ma$$</p>
<p>I am saying here that the sum of all forces is equal to the $Normal \ Force + Weight$ and that all of this should equal $ma$. </p>
<p>I then re-arrange my equation:</p>
<p>$$(N-W)/m=a$$</p>
<p>Now here is where I am making a huge assumption, and where I am having problems. I decided to solve for $m$ by imagining that $m$ is the mass of the person. So in a way I am saying that the whole system in this problem is really the person. Is this a logical way to solve this kind of problem or will I run into problems by thinking this way?</p>
<p>If it is not clear what I did here is in equation format:</p>
<p>$$(N-W)/(W/g)=a$$</p>
<p>In other words: <strong>What is the proper way of thinking about the mass when applying $F=ma$?</strong> At the moment I am thinking of it as the mass of the objects that affect the problem. So if it was two people in the problem I would add up their masses and use that.</p>
| 10,041 |
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<p>I would like to determine the number of energy states two free, distinguishable particles in a box of length $L$ have. I would then like to determine the number of states two free, indistinguishable particles, with spin $3/2$ each, have in that box at the elementary level. Finally, determine the number of states in case these two particles with spin $3/2$ each are distinguishable.
I am familiar with the following formula for the energy states
$$E_n=\frac{(\hbar)^2 \pi^2n^2}{(2mL^2)}$$
but am not sure how to proceed. If the particles are distinguishable, does that entail that one is a fermion whereas the other is a boson? I am not sure.
Furthermore, if the two particles have spin $3/2$ each, that means they are both fermions, right? If that is correct, then, due to Pauli's exclusion principle, the two could not be at the same state. I also know that for spin $3/2$ there could be $4$ particles per energy level. But again I am not sure how to coherently process the given data and would appreciate some guidance.</p>
| 10,042 |
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<p>The phase density function is usually denoted as <a href="http://en.wikipedia.org/wiki/Boltzmann_equation" rel="nofollow">$f(\mathbf{x},\mathbf{v},t)$</a> which gives probable number of particles moving with velocity $ \mathbf{v}$ at position $\mathbf{x}$ at time t. Also we assume that position and velocities of particles are not correlated i.e. $\mathbf{x} $ and $\mathbf{v}$ are independent. </p>
<p>We use $f(\mathbf{x},\mathbf{v},t)$ as a basis Lattice Boltzmann functionto solve fluid dynamics equations.
$$ f_i(\mathbf {x} +\Delta \mathbf {x}, t + \Delta t) - f_i(\mathbf {x}, t ) = -\frac{\Delta t}{\tau} (f_i - f^{eq}_i)
$$</p>
<p>The entire point of solving fluid dynamics equation is(If I am correct) to know velocity at a given point and time. How can we use $f(\mathbf{x},\mathbf{v},t)$ to proceed with lattice Boltzmann equation when $f(\mathbf{x},\mathbf{v},t)$ has multiple values of velocities at given point $\mathbf{x}$ ?</p>
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<p>I am creating a particle in cell simulation that models an electron plasma in a cylindrical container. Part of this process is assigning charge density to grid points based on the position of each particle. My question is this: for the examples of these simulations that I have seen, the charge density is given by the amount of charge on a grid point divided by the volume of a unit cell, or: $\rho=\frac{q}{d\tau}$</p>
<p>which for cylindrical coordinates is: $\rho=\frac{q}{r\bigtriangleup_r \bigtriangleup_z \bigtriangleup_{\theta}}$, where the deltas represent the grid spacing that we have specified. For r and z, these are well defined.</p>
<p>however, we are taking the system to be symmetric about $\theta$, meaning that this simulation is basically in two dimensions, r and z, and that $\bigtriangleup_{\theta}=0$. How would I resolve this, because it looks to me like this would blow up?</p>
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<p>In general, a sphere with conductivity $\kappa$, heat capacity per unit volume $C$ and radius $R$ obeys the differential equation at time t:</p>
<p>$$C\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial r^2} + \frac{2\kappa}{r}\frac{\partial T}{\partial r}$$</p>
<p>Part (a): A sphere with a cavity of radius a generates heat at a rate $Q$. Heat is lost from the outer surface of the sphere to the surroundings, with surrounding temperature $T_s$, given by Newton’s Law of Cooling with constant $\alpha$ per unit area. Find the temperature $T_a$ at the surface of the cavity at thermal equilibrium.</p>
<p>Part (b): A second sphere without a cavity generates heat uniformly at $q$ per unit volume. Like the first sphere, heat is lost to the surroundings at its surface. Find temperature at the center of the sphere $T_0$ at equilibrium.</p>
<p><strong>Attempt</strong></p>
<p>Part (a)</p>
<p>Heat generated in cavity = Heat loss at surface
$$Q = \alpha (T_s - T_R)(4\pi R^2)$$
$$T_R = T_s - \frac{Q}{\alpha (4\pi R^2)}$$
We will use this as a boundary condition to solve for constants in the differential equation later on.</p>
<p>At steady state, $C\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial r^2} + \frac{2\kappa}{r}\frac{\partial T}{\partial r} = 0$ and solving, we get:
$$T = A + \frac{b}{r}$$
We need one more equation to solve for constants $A$ and $B$.</p>
<p>Using $\int \vec J \cdot d\vec S = k \int \nabla \vec T \cdot d\vec S$:
$$Q = -\kappa \frac{\partial T}{\partial r}(4\pi r^2)$$
$$Q = 4\pi \kappa b$$
$$b = \frac{Q}{4\pi \kappa}$$
Solving for $A$ gives $A = T_s - \frac{Q}{4\pi \alpha R^2} - \frac{Q}{4\pi \kappa R}$:.</p>
<p>Together, temperature at surface of cavity is
$$T_a = T_s - \frac{Q}{4\pi \alpha R^2} + \frac{Q}{4\pi \kappa}\left( \frac{1}{a} - \frac{1}{R} \right)$$</p>
<p>Part (b)
<strong>Attempt</strong></p>
<p>I have set up the DE, the main trouble is how to solve it.</p>
<p>$$ \kappa \nabla^2T + q = 0 $$
$$\kappa \frac{\partial^2 T}{\partial r^2} + \frac{2\kappa}{r} \frac{\partial T}{\partial r} + q = 0$$</p>
<p>Homogeneous solution reads $T = A + \frac{b}{r} $. Won't this screw things up at $r=0$?</p>
<p>For inhomogeneous solution, I try $\beta r^2$. I found it to be $-\frac{q}{6\kappa} r^2$. Won't this screw things up at $r = \infty$?</p>
| 10,045 |
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<p>According to the third law of motion, you van't have an mass move in a particular direction unless there is a proportional opposite mass/acceleration ratio in the opposite direction.</p>
<p>No-one has been able to provide a convincing argument otherwise, but the best one to date is Shawyer's <a href="http://en.wikipedia.org/wiki/EmDrive" rel="nofollow">EM Drive</a>. He claims some fancy relativistic effects are what allows his engine to work, but I have read some papers which claim he is a fraud.</p>
<p>My question is, <i>why</i> is it impossible to move a mass in a given direction without a proportional change in the opposite direction?</p>
<p>I'm not talking about a perpetual motion machine, or anything. Sure, the device would need to consume at least the amount of energy proportional to the energy required to accelerate the mass.</p>
<p>Here's a highly hypothetical example:
Say we either can project a gravity well in front of our vehicle, and/or project a gravity hill behind. In empty space, the effects of the gravity will be near-negligible by the time they reach any other object, however close to the vehicle they will be more significant. The end result would be the vehicle would move in the given direction, and nothing else around would really move at all.</p>
<p>An even cruder example would be to shine a bright torch out the back of your vehicle. Even though the photons have no mass, wouldn't the vehicle move forward?</p>
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<p>Someone sent me this link to a talk by <a href="http://www.ks.uiuc.edu/~kschulte/" rel="nofollow">Prof. Klaus Schulten</a> from the University of Illinois: (my emphasis)</p>
<blockquote>
<p><strong>Quantum Computing and Animal Navigation</strong></p>
<p>Quantum computing is all the rage nowadays. But this type of computing may have been discovered and used by living cells billion of years ago. Nowadays migratory birds use a protein, Cryptochrome, which absorbs weak blue light to produce <strong>two quantum-entangled electrons in the protein</strong>, which by monitoring the earth's magnetic field, allows birds to navigate even in bad weather and wind conditions. The lecture tells the story of this discovery, starting with chemical test tube experiments and ending in the demonstration that the navigational compass is in the eyes and can be affected by radio antennas. The story involves theoretical physicists who got their first paper rejected as "garbage", million dollar laser experiments by physical chemists to measure the entangled electrons, and ornithologists who try to 'interrogate' the birds themselves. <strong>This work opens up the awesome possibility that room-temperature quantum mechanics may be crucial in many biological systems.</strong></p>
</blockquote>
<p>Now here's my question: What's the big deal with entangled electrons? I mean, if I do not neglect electron-electron interaction, then pretty much all electrons in a condensed matter system are entangled, are they not? Electrons in the same angular momentum multiplet are entangled via Hund's rule, electrons on neighboring sites in a tight-binding (or, in the interacting case, Hubbard) model can all be entangled due to an antiferromagnetic exchange coupling, etc. etc.</p>
<p>Sure, for a quantum computer I'd like to have physically separated electrons maintain their entanglement, and I'd like to have fine-grained control over which of the electrons are entangled in which way etc, but for chemical processes in molecules such as these earth-magnetic-field receptors, is it not a bit sensationalist to liken such a process to quantum computing?</p>
| 10,047 |
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<p>I am reading Ludwig's paper "<a href="http://arxiv.org/abs/1010.0936" rel="nofollow">Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors</a>", and in this paper, although I am clear how they get the descent equation which introduced the relationship between anomaly and the existence of topological insultor, I am confused about the theta-term they mentioned on section VA5, they said that the integral of anomaly polynomial $\Omega_{2n+2}$ corresponds to the $\theta$ term. Could anyone help and explain to me what is this theta-term referring to and where can I find information about it?</p>
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<p>I've heard that the strong force doesn't decrease in strength with increasing distance, and that's why quarks must be confined within hadrons. But could there be, say, a single quark out there, so that the universe would be colour neutral without it, but with it, it's not? Would such a quark have a "strong field" that would extend through the entire universe?</p>
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<p>In relativity, if two events are simultaneous in a specified frame, they cannot be simultaneous in any other frame. </p>
<p>My question is this: given any two events, is there always a frame in which these two events are simultaneous? For example, if I drop a blue ball on one side of a tennis court, and my friend drops a red ball on the opposite side of the court one day later -- from my frame one day later -- is there a frame in which the blue and red balls hit the ground simultaneously? </p>
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<p>This is a conceptual question as much as an empirical one, but the question is: Does the lift of a wing change when the slats (or any other leading edge device) are deployed? I am stipulating that the angle of attack $(\alpha)$, freestream dynamic pressure $(q_\infty=\frac{\gamma}{2}p_\infty {M_\infty}^2)$, and wing area $(S)$ remain constant. We know that slats can allow the wing to operate at higher angles of attack, but do they change the lift of the wing at any given angle of attack? I am fairly confident that the answer to this question is no, but I was looking for a more rigorous (and possibly mathematical) explanation.</p>
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<p>Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$.</p>
<p>I've been taught that a density matrix is constructed as $\rho = \left|\psi\rangle\langle\psi\right|$, so in this basis: $$\rho = \psi_0(x_1,x_2) \psi_0^*(x_1',x_2') \left|x_1,x_2\rangle\langle x_1',x_2'\right|$$ The reduced density matrix of the second oscillator is then: $$\rho_2(x_2,x_2') = \psi_0(x_1,x_2) \psi_0^*(x_1,x_2') \left|x_2\rangle\langle x_2'\right|$$</p>
<p>However, in a paper I've come across this reduced density matrix is written: $$\rho_2(x_2,x_2') = \int_{-\infty}^{\infty} dx_1 \psi_0(x_1,x_2) \psi_0^*(x_1,x_2')$$</p>
<p>I'm not familiar with the Trace as an integral, though I can sort of see how it would work for continuous variables. Clearly there's also a difference in notation, since this last formula has no bras or kets.</p>
<p>I was wondering if someone could explain the difference, or maybe give some sort of overview.</p>
| 10,052 |
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<p>We have been discussing whether or not bigger cars are more dangerous to pedestrians. I will reduce it to a simpler question. </p>
<p>Imagine two cars, which are exactly identical in shape, but have different masses. One is 1000kg and the other 2000kg. One person is hit by the light car, and another is hit by the heavy car. The speed at collision is 50kmph, they both weigh 100kg, and they are identical in shape ;) </p>
<p>If you look at the energy itself it suggests that the heavier car will inflict more damage. But I am thinking that what will really matter for the guy's survival is how "fast" he will be accelerated (what g-force his organs must resist). I am not convinced that the 1000kg will make a big difference in how hard he is accelerated. </p>
<p>Could the 1000kg vs. 2000kg in some circumstances be the difference between life and death? </p>
<p>If the answer is that there is only little difference, it suggests that big cars with "soft" hoods or hoods designed to let the pedestrian slide over the car, could easily be less dangerous than much smaller cars with a design that did not prioritize this. </p>
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<p>Currently I'm taking an introductory course on thermodynamics. I've got a problem with understanding what is the meaning of pressure of a solid body. The question arose when I looked at phase diagram in P, T coordinates. </p>
<p>When we're talking about pressure of gas or liquid, it can be defined thus: it is the force per unit area with which the molecules of the substance hit the surface. This force is due to the molecular motion.</p>
<p>But in the case of solids, molecules don't "move", they just oscillate about their fixed positions. Therefore, they don't hit (in usual meaning) a testing surface and we have problems with pressure's definition. How then the pressure can be defined?</p>
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<p>Simply stated the <a href="http://arxiv.org/abs/1105.4714" rel="nofollow">casimir effect</a> can be observed bringing two plates together microscopically and in vaccum</p>
<p>Is this analogous to the observation of gravity?</p>
<p>Question: What is casimir effect?</p>
<p>Canonical, answers only please no wiki link one liners please</p>
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<p>I am reading a paper and saw the author wrote something like, </p>
<p>Because of no-slip wall assumption, so velocity vector $\vec{v}$ is $0$ on the wall, and also the variation of this velocity on the wall is $\delta \vec{v} = 0$. And he also said $\delta \vec{v}_t = 0$.</p>
<p>But it seems his subtext is $\delta v_n$ is not $0$. (btw, n is normal, and t is tangent)</p>
<p>Hmm, this seems quite obvious to the author, but not to me.</p>
<p>Could someone elaborate on it? I'd like to know the physical meaning of this, so to better understand it. Why it is so obvious?</p>
<p>Thanks a lot.</p>
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<p>Twin Paradox
Can anyone clarify and or correct the following for me?
A space ship is flying at speed v equal to 0.8 times the speed of light. Within the ship are three stations, a transmitter at station A, station B directly across the ship from station A and station C directly forward of station A. The distance AB is noted as d and the distance AC equals AB.
A photon is transmitted from A towards B at the speed of light c. At the same instant another photon is transmitted from A towards C. The transit time t for the photons is measured inside the space craft as d divided by c in both cases.
A stationary observer notes the transit of the photon from A to B and measuring the transit calculates the distance to be Tab multiplied by the speed of light c and also calculates the distance travelled by the space craft as v * Tab and concludes that :
d = Tab * (c^2-v^2)^0.5
And as d = t*c t = Tab * (1- (v/c)^2)^0.5
For v/c = 0.8, (1- (v/c)^2)^0.5 = 0.6 t = 0.6 * Tab</p>
<p>This is the Lorentz equation for time dilation on which the twin paradox is based.
Now consider what the observer sees of the photon travelling from A to C. Firstly Lorentz would contend that there is a shortening of lengths in the direction of travel and as a consequence the observer sees the distance A to C as d multiplied by (1- (v/c)^2)^0.5 hence the observed distance of travel is v*Tab + d*(1- (v/c)^2)^0.5 and:
Tac = (v*Tab + d*(1- (v/c)^2)^0.5)/c
Tac = 1.93333*t
If this sum is done for a photon travelling from C to A the numbers are even harder to understand
Tca = (-v*Tab + d*(1- (v/c)^2)^0.5)/c
Tac = -0.73333*t
As the time of arrival within the space ship is the same in all three cases then should not the observed time of arrival also be the same for all three cases not three different times and particularly the later one cannot be negative.
Please will someone resolve my paradox.</p>
| 10,057 |
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<p>In elementary QM, an electron is typically viewed as a cloud around a proton. The idea is that it's position can only be determined once a measurement is made. The probability that the electron will be found in a certain region is determined by it's wave function. </p>
<p>For background see the question
<a href="http://physics.stackexchange.com/questions/92565/is-it-that-electron-of-an-atom-can-be-found-anywhere-in-the-space">Is it that electron of an atom can be found anywhere in the space?</a></p>
<p>Assume a hydrogen atom in the ground state.</p>
<p>Now, there is chance, however small, that the electron will be found very far away from the proton. So far away, that it cannot reasonably considered under the proton's influence.</p>
<p>In other quantum situations, a particle can tunnel though a potential barrier if the barrier is finite. This allows a particle to escape as in beta decay. This is not exactly that situation, but the further away electron is found, the less influence the proton field will have on it. Is there a point where the electron becomes unbound? And is it the measurement process that causes this to happen? Or if we observe enough non-interacting, isolated hydrogen atoms, will we observe that some of the protons no longer have a bound electron if we wait long enough?</p>
| 10,058 |
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<p>We know that for constant pressure thermodynamic processes, $dH=dQ_p$. My question is, does it implies that only reversible work is possible in this processes so that $dw=0$ because $dv$ is zero? In addition, does $Q_v$ necessarily be reversible heat transfer in this case? What if the heat transfer is irreversible?</p>
<p>Similar question for $dU=dQ_v$, does the process need to be reversible?</p>
<p>Another question is, do the above relations have anything to do with whether or not the system is an ideal gas? I have heard from a lecture that $du≠dQ_v$ in Joule's free expansion for non-ideal gas. Consider reversibility and whether it's ideal gas there are four combinations of situations (ideal gas reversible,ideal gas irreversible, non-ideal gas reversible, non-ideal gas irreversible). I get confused with how these factors affect the thermodynamic relations.</p>
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<p>I have to compute the square of the Dirac operator, $D=\gamma^a e^\mu_a D_\mu$ , in curved space time ($D_\mu\Psi=\partial_\mu \Psi + A_\mu ^{ab}\Sigma_{ab}$ is the covariant derivative of the spinor field and $\Sigma_{ab}$ the Lorentz generators involving gamma matrices). Dirac equation for the massless fermion is $\gamma^a e^\mu_a D_\mu \Psi=0$. In particular I have to show that Dirac spinors obey the following equation: $$(-D_\mu D^\mu + \frac{1}{4}R)\Psi=0 \qquad (1)$$ where R is (I guess) the Ricci scalar.
Appling to the Dirac eq, the operator $\gamma^\nu D_\nu$ and decomposing the product $\gamma^\mu \gamma^\nu$ in symmetric and antisymmetric part I found: $$D_\mu D^\mu\Psi + \frac{1}{4}[\gamma^\mu,\gamma^\nu][D_\mu, D_\nu]\Psi=0$$Now I have troubles to show that this last object is related with the Ricci scalar. Can somebody help me or suggest me the right way to solve Eq. $(1)$?</p>
| 10,060 |
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<p>Are the speeds of the different wavelengths of visible light different or varying in a medium such as air? If so, please inform by how much?</p>
<p>Also, even if the wavelength speeds vary minimally, please inform.</p>
| 10,061 |
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] |
<p><a href="http://en.wikipedia.org/wiki/Einstein_field_equations">Einstein's field equation</a>:</p>
<p>$$G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} - g_{\mu\nu}\Lambda$$</p>
<p>I'm trying to understand each of the terms in this equation intuitively, but I'm struggling.</p>
<p>Basically, I want to understand how these equations allow me to predict the path of a particle, given the mass and energy distribution of a system.</p>
<p>I have some idea that $G_{\mu\nu}$ represents the curvature of spacetime, and that $T_{\mu\nu}$ represents the distribution of energy in the system, but it's not clear how.</p>
<p>Correct me if I'm wrong about any of this; I'm just starting.</p>
| 10,062 |
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] |
<p>Why Our Electricity Flowing in the Sequence of 11kv,33,66,132kv,why not Flow in 10 kv,11,12,13,14 that Sequence.</p>
| 10,063 |
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<p>I've been reading a fair amount about quantum chaos, and <a href="http://en.wikipedia.org/wiki/Random_matrix" rel="nofollow">random matrix theory</a> comes up a lot. I get that they're looking at the distribution of eigenvalues from an ensemble of random matrices, but I still don't know what the ensemble of those matrices looks like.</p>
<p>For example, take the <a href="http://en.wikipedia.org/wiki/Random_matrix#Gaussian_ensembles" rel="nofollow">Gaussian orthogonal ensemble</a>. If I understand correctly, the matrices are real and symmetric, and when generating such an ensemble, the probability of a given $n\times n$ matrix $H$ being generated is proportional to $e^{-\frac{n}{4}trH^2}$. How would I generate I generate such an ensemble (on a computer)? What is the distribution of the individual random elements of the matrix? Gaussian?</p>
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<p>I have found equation for moment of inertia $(J)$. I'm calculating $J$ for hemisphere, with rotational axis $Z$.</p>
<p>$$ J = \iiint\limits_V r^2 \cdot \rho \cdot dV $$</p>
<p>But if $\rho$ is constant (homogenous), I can do:
$$ J = \rho \cdot \iiint\limits_V r^2 \cdot dV $$</p>
<p>Which is:
$$ J = \rho \cdot V $$
$$ J = m $$</p>
<p>Am I right?</p>
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<p>Most people can ride 10 km on their bike. However, running 10 km is a lot harder to do. Why?</p>
<p>According to the law of conservation of energy, bicycling should be more intensive because you have to move a higher mass, requiring more kinetic energy to reach a certain speed. But the opposite is true. </p>
<p>So, to fulfill this law, running must generate more heat. Why does it?</p>
<p>Some things I can think of as (partial) answers:</p>
<ul>
<li>You use more muscles to run.</li>
<li>While running, you have more friction with the ground; continuously pouncing it dissipates energy to it.</li>
<li>While you move your body at a slow speed, you need to move your arms and legs alternately at higher and lower speeds.</li>
</ul>
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<p>OK so we've all heard of this from Carl Sagan, Lawrence Krauss and others and we know the argumentation, I don't refute that. There are other examples, for instance I once calculated (this was before I had taken any QM-course) that since there are more particles in a glass of water than there are glasses of water in the world, eventually, (given enough time, could be millions of years?) you will have a particle (atom/molecule) from each individual every lived on Earth (since we all drink water etc.)</p>
<p>But like I said this was before I had heard of indistinguishable particles etc. So later on, I wasn't so sure if I wanted to tell everyone I met about this amazing revelation, because quantum mechanically, every proton (say) is indistinguishable from every other proton, so this whole story breaks down right?</p>
<p>Similarly with the "we're all star dust"-thing. Is it not wrong to state that the protons in my right hand might come from this, and my left from that?</p>
<p>Or, am I overcomplicating things?</p>
<p>To make things clear: </p>
<p>My question is; if it is right to say that "This atom is from that supernova and that atom is from the other etc."
At what level is it right to speak of distinguishable particles. For instance, I breath in, breath out...then someone else in the same room takes a breath, swallowing my atoms/particles. Is it right to say that we've shared the same atoms/protons?</p>
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<p>In many physical applications, the Heaviside step fuction is defined as $$H(x) = \left\{\begin{eqnarray}
1, \quad x>0 \\
0, \quad x<0
\end{eqnarray}\right.$$
The value $H(0)$ is left undefined. Is there a physically prefered value of $H(0)$ or does it depend on the problem at hand?</p>
<p>For example, due to relation $H'(x) = \delta(x)$, it would be nice to have $H(0)=\frac{1}{2}$ if we think of $\delta(x)$ as an even function. However, when considering signal processing, we would like our functions to be (left/right) continuous at $x=0$.</p>
<p>Is there a way out of this arbitrariness?</p>
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<p>To my understanding the work done on an object is defined mathematically as:
$$W = \vec{F}\cdot\vec{S}=|\vec{F}||\vec{S}|cos\theta$$
This, I understand. My problem is that I don't understand that if the angle $\theta$ is 90 degrees how can the work done by $\vec{F}$ on the object is zero. For example; say you have a particle and the direction of the displacement is directly to the right, and you also have a force vector acting on the particle that is straight up(like the normal force on a box that is standing on a flat surface). How is it possible that the force vector is not doing any work? Must the particle not take a different route because of the force vector acting upward on the particle, like if you add the vectors together?</p>
<p>There has to be something wrong with my reasoning, but what is it?</p>
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<p>In what ways can a lunar eclipse occur?</p>
<p>Also, on what percentage of the Earth are they usually viewable? </p>
<p>I am aware that there are multiple configurations that constitute a lunar eclipse (umbral, penumbral, partial) and would like more information about each and how they occur.</p>
| 10,070 |
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<p>The very highest energy photons, gamma-rays, are too energetic to be detected by standard optical methods. In fact they rarely actually make it to the surface of the Earth at all but interact with molecules in the Earth's atmosphere. The high energy gamma-ray telescopes, such as Veritas and Hess use air Cherenkov telescopes to observer these photons. How does an air Cherenkov telescope actually work to measure the incoming gamma ray?</p>
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<p>Well, there is a measure of how a planet could be considered like Earth, called Planetary habitability. Based on this measure, what are the prerequisites needed to consider a planet to be a habitable one?</p>
| 10,072 |
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<p>Why it is easy to start the vehicle on hot day than on cold days?
Since on winter days it is diffcult to start than on hot days I thought it is due to the low temperature which in turn affects the chemical reaction in the cell.Since energy is produced in the cell due to reaction.But I don't think it is the correct one..</p>
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<p>so here's the problem: A simple RC circuit where the capacitor has been charged, and two resistors are in parallel configuration. How does one find the power (as function of time) through any of the two resistors? </p>
<p><img src="http://i.stack.imgur.com/mOGjJ.jpg" alt="enter image description here"></p>
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<p><img src="http://i.stack.imgur.com/dxSE3.jpg" alt="enter image description here"></p>
<p>![enter image description here][2]</p>
<p>This question is truly annoying, and I have been stuck for an hour on part D, would greatly appreciate if anyone could shed a light on this problem.
Why ans for part c and d are different?</p>
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<p>I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field.</p>
<p>Given a periodic graph (actually a physical lattice or crystal structure), we want to examine some periodic coloring ( or ordered but not periodic coloring) of the graph. This coloring in physics may be considered as filling the lattice site with atoms, ions and so forth. Is there some already formally established mathematical area that deals with the this?</p>
<p>Further, we could construct functions from these kinds of coloring to a real number. ( in physics we associate average energy with the specific structure) and we want to minimize it by searching the periodic or ordered graph space. Is there such a branch in mathematics dealing with this also?</p>
<p>Furthermore, I wish to construct a program to analyze any arbitrary periodic colored graphs (that is to analyze different crystal structure. ) By analyze I mean search the coloring that would give the minimum energy. Is there existing well established knowledge for doing this?</p>
<p>An reduced problem would be is there some established mathematical theorems tackling the coloring of an arbitrary periodic graph, in any possible way?</p>
<p>Thank you very much.</p>
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<p>In the theory of constrained Hamiltonian systems, one differentiates between primary and secondary constraints, where the former are constraints derived directly from the Hamiltonian in question and the latter are only realized 'on-shell', i.e. once the equations of motion are satisfied. </p>
<p>Further one can differentiate between first-class and second-class constraints, where first-class constraints have a vanishing Poisson bracket with all other constraints, and second-class constraints don't.</p>
<p>It can now be shown, that first-class primary constraints, generate gauge transformation. </p>
<p>The Dirac conjecture states that one could remove the requirement to have a first-class primary constraint and therefore that <em>all first-class constraints</em> (no matter if primary or secondary) generate gauge transformations.</p>
<p>The conjecture is shown to not hold in some very specific examples but is used in the literature nontheless. </p>
<p><strong>My question is whether at least the inverse is true: can every gauge theory be formulated as a Hamiltonian problem with a first-class constraint?</strong> </p>
<p>The case of formulating a gauge theory as a hamiltonian with primary first-class constraint probably just means adding gauge fixing terms to the theory. In that case the question might be rather, if a closed-form gauge fixing term can be found for any gauge group. And if not, can gauge theories for which this fails be formulated as first-class secondary constraints?</p>
| 10,077 |
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<p>According to the definition of angular momentum: </p>
<blockquote>
<p>Angular momentum, moment of momentum, or rotational momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a <strong>particular axis</strong>.[This definition has been extracted from the wiki encyclopedia-<a href="https://en.wikipedia.org/wiki/Angular_momentum" rel="nofollow">Angular momentum</a>] </p>
</blockquote>
<p>Does the wheel in motion,undergoes rotation with any particular axis of rotation (like gyroscope rotating about a particular axis), thus can we define angular momentum for it? </p>
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<p>We just saw parity symmetry and we were told about the experiments to see the non parity symmetry of disintegration, in particular one involving the reaction:</p>
<p>$$^{60}Co\longrightarrow^{60}Ni+ e + \bar \nu$$</p>
<p>Now, we were asked to check that if we prepare the system so that the spin is parallel to the $z$ axis and make a mirror symmetry in the $x$ would flip the spin, going from $|jm\rangle$ to $|j-m\rangle$.</p>
<p>I can't prove.</p>
<p>We were given as a clue, to have in mind the fact that the transformation is:</p>
<p>$$\left(\matrix{
-1&0&0\\
0&1&0\\
0&0&1
}\right)=
\left(\matrix{
-1&0&0\\
0&-1&0\\
0&0&-1
}\right)
\left(\matrix{
1&0&0\\
0&-1&0\\
0&0&-1
}\right)$$</p>
<p>In the decomposition, the first matrix is the parity matrix already studied, while the second one is a rotation of $\pi$ around the x axis.</p>
<p>How can I check that the spin actually flips after the reflection?</p>
<p>Thanks in advance.</p>
<p>First edition: I thought I could prove it seeing that the wave function of the orbital angular momentum is $Y_l^m$, and that the spin operator is an angular momentum operator, so we can use the symmetry of those functions:</p>
<p>$$Y_l^m=\alpha e^{im\phi}P_l^m(\cos\theta )$$
And so we have that the mirror symmetry is $\phi\mapsto \phi+\pi$, and so $Y_l^m\mapsto Y_l^{-m}$. This doesn't convince me because spin is not about spacial properties of a particle.</p>
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<p>Is there a reason to believe that the axial resonances be heavier than the vector resonances in the composite higgs models? </p>
<p>For instance, in <a href="http://arxiv.org/abs/0808.2071">http://arxiv.org/abs/0808.2071</a>, to have zero tree level contributions to S parameter, the constrain on the wave function mixing parameter $\epsilon$, makes axial resonances lighter.
However, models constructed with CCWZ formalism, seem to favor heavier axial resonances as compared to the $\rho$'s. </p>
| 10,080 |
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<p>The induced emf in a coil in AC generator is given as:</p>
<p>$$\mathbb E = NAB\omega \sin \theta $$</p>
<p>$\omega = d\theta/dt$</p>
<p>Now, when the angle between the normal of plane and magnetic field is zero degrees, the induced emf is zero i.e.</p>
<p>$$\Bbb E = NAB\omega \sin 0 = 0$$</p>
<p>But we also define emf as the time rate of change of magnetic flux so, why do we get zero emf in the above case, magnetic flux is still changing with time?</p>
| 10,081 |
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<p>I have confused myself about the following variant of Maxwell's demon and I can't seem to find out where the energy went. </p>
<p>Consider this: You have a chain (one dimension) of spins (up/down) with a nearest-neighbor coupling. Energy is minimized if spins are aligned. Let us say the energy difference between alignment and not-alignment is E. The zero temperature state is either all up or all down. If we heat the state up to a temperature T, some of the spins will flip with a probability given by the Bolzmann-factor, depending on the ratio T/E. So far so good. </p>
<p>Now the finite temperature state has energy because the states aren't all aligned, but the distribution is thermal and it's no useful (free) energy. However, if you knew which spins are misaligned, you could selectively flip them. Let us say the system is such that you can flip them by shooting a photon with energy E at the spin. Eg, you shoot at the middle spin in a series of three. If it's up,up,up then the photon will be absorbed and you end up with up, down, up. If it's up, down, up, the photon stimulates emission and you get up, up, up plus two photons of energy E. If you have up, down, down, the photon doesn't change anything about the total energy. The same happens if you exchange all ups with downs.</p>
<p>Now the thing is this: If you do not know which spins you have to flip, your chances of gaining or losing energy by shooting photons at the chain are the same. You just convert one thermal state into the other. But if you knew which spins to flip, you could topple them over selectively and get energy out of the system. Essentially, you extract it from the thermal bath that did heat up the chain. That's possible (I think) because you are using information to reduce the entropy of the system.</p>
<p>My question is this: How do I see that the energy needed to measure the spin orientations in the chain is at least as large as the energy I can gain by flipping them selective once I have measured? It isn't clear to me why it should not be possible to measure them with some very low-energetic probe, eg measuring the local magnetic field with the Hall effect. </p>
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] |
<p>Let's say we have an incomplete spacetime A that is globally hyperbolic, does there necessary exist a globally hyperbolic completion?</p>
<p>My guess is no, in which case what further restrictions can be placed on A to ensure that it can always be extended to geodesically-complete globally hyperbolic spacetime?</p>
| 10,083 |
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<p>The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$
if the constraint is nice enough (i.e. if $0$ is a regular value of $\Phi$), can be described as a $k$-dimensional submanifold of $\mathbb R ^{3n}$, and this clearly has some advantages. Those systems are called “autonomous” (and if someone could throw some light on the terminology I'd also be grate).</p>
<p>Now, suppose that $\Phi$ depends on time, i.e. the constraints are:$$\Phi(x,t)=0.$$
In this case it isn't immediately obvious to me what would be the most natural way to describe the configuration space. For example, one might define $g^t(x)=\Phi(x,t)$, suppose that $0\in \mathbb R ^{3n-k}$ is still a regular value of $g^t$ and define the configuration space at time $t$ as $$M^t=(g^t)^ {-1}(0),$$
and describe the position of the system at time $t$ with $k+1$ parameters $(q_1,\dots,q_k,t)$. This works good if, for example, the manifolds $M^t$ are essentially the same: for example, if $M^t\subset \mathbb R ^3$ is a ring that rotates about the $z$ axis, the manifold is simply $S^1$. But is this description always possible? I mean, $M^t$ could possibly change in time so that the coordinates $(q_1,\dots,q_k,t)$ don't mean a thing for some $t$.</p>
<p>Another conceivable way, I suppose, would be to consider the ($k+1$-dimensional) manifold: $$M=\Phi ^{-1}(0)\subset \mathbb R ^{3n+1}\ni (x_1,...,x_n,t).$$
For example, in this <a href="http://physics.stackexchange.com/questions/94381/noether-theorem-and-energy-conservation-in-classical-mechanics/94384#94384">post</a>, I sketched a proof of the Noether's theorem for non-autonomous systems following a similar idea. The main problem that occurs to me is that, in this way, we make the statement of propositions like, for example, d'Alembert principle more complicated, because we can no more consider virtual displacements as tangent vectors to the configuration space.</p>
<p><strong>So, to summarize</strong>: given a constraint of the form $\Phi(x,t)=0$, what is the most natural way to describe the configuration space of such a system?</p>
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<p>Given a coil initially in the x-y plane, rotating at angular frequency $ \omega $ about the x-axis in a magnetic field in the z-direction. This uniform time varying magnetic field is given by $B_z (t)=B(0)cos(\omega t) $ I am required to show that there is a voltage of frequency $2\omega $ across the loop. Clearly when t=0 the flux is at a maximum, but I dont understand how to relate to the frequency?</p>
<p>If the frequency is just the inverse of the period then $f=\omega / 2\pi $ ?
Clearly I am not understanding something. How does the voltage affect the frequency?</p>
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<p>A radar transmitter (T) is fixed to a system $S_{2}$ which is moving to the right with speed v relative to system $S_{1}$. A timer in $S_{2}$, having a period $\tau_{0}$ (measured in $S_{2}$) causes transmitter T to emit radar pulses, which travel at the speed of light, and are received by R, a receiver fixed to $S_{1}$. If I want to calculate the signal period in different points in $S_{1}$ (point A or B) will those be different and why? </p>
<p>I thought the signal periods at both points are just be the dilated time with respect to the moving frame i.e $$\tau_ {A}=\tau_{B}=\tau_{0}/(1-v^{2}/c^{2})$$ (Assuming $\tau >> \tau_{0}$). But then It doesn't consider the distance in between them.</p>
<p><img src="http://i.stack.imgur.com/3CBET.png" alt=""></p>
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<p>I mean look how big is the Cosmos and hof few planets and suns are outthere, compared to the vast "nothingness" between them. </p>
<p>With "nothingness" I mean no standard particles like atoms, electrons etc.</p>
<p>Why is that the ratio is so hugely in favor of "nothingness"?</p>
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<p>While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian $H(p,q)$ by </p>
<p>$$\hat{H}(\hat{P},\hat{Q}) \equiv \int {dx\over2\pi}\,{dk\over2\pi}\,
e^{ix\hat{P} + ik\hat{Q}} \int dp\,dq\,e^{-ixp-ikq}\,H(p,q)\;$$</p>
<p>if we adopt the Weyl ordering.</p>
<p>How can I derive this equation?</p>
| 10,088 |
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<p>It is stated in many books that analytic closed solutions to the time-independent electronic Schrödinger equation,
$$\hat{H}\Psi = E\Psi, $$
exist for the one-electron problem (e.g. hydrogen atom, assuming separability of nuclear and electronic motion) but that such solutions do not exist for systems with more than one-electron and thus approximation methods are required to solve the equation.</p>
<p>Specifically, on going from a one-electron system to a two-electron system, with fixed nuclei, something changes that makes closed analytic solution of the equation no longer possible.</p>
<p>Clearly this is related to the inter-electronic interaction because closed analytic solutions are possible for systems of non-interacting particles.
Many resources suggest that the many-electron problem is "too difficult" to solve analytically but do not give any further details. This raises the question: is it the case that closed analytic solutions <em>cannot</em> exist, or that they could exist, but it is very difficult to find them? And, if they cannot exist, then how is this determined?</p>
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] |
<p>In <a href="http://arxiv.org/abs/quant-ph/0504102" rel="nofollow">arXiv:quant-ph/0504102v1</a>, A.J. Bracken says</p>
<blockquote>
<p>if we think of the phase space formulation of QM as more fundamental,
arising directly from a deformation of classical mechanics in phase
space [12] we can think of the formulation of QM in Hilbert space and
the associated introduction of complex numbers as a computational
device to make calculations easier.</p>
</blockquote>
<p>Reference [12] is Bayen F, Flato M, Fronsdal C, Lichnerowicz A, and
Sterheimer, D, Annals of Physics 111 (1978) 61-110, 111-151. This seems to answer the question of why complex variables. Can anyone explain this? Thanks.</p>
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<p>I plan to graduate with an honours (four year) degree in philosophy and a general (three year) degree in mathematics. Are there any physics graduate programs that might admit me if I were to apply?</p>
<p>For context: My plan-A is to work in academic philosophy. A physics education might not appear on the resume of most philosophy professors, but I suspect that knowing about physics could improve a philosopher's thinking about a number of questions. Accordingly, it seems worthwhile to study physics during either, a one or two year detour, several years of concurrent distance education, or the summer terms of my philosophy education. Self-study is an option; however I'd prefer to earn a credential in order to improve my future applications to graduate schools and to employers.</p>
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<p><img src="http://i.stack.imgur.com/rKWj0.jpg" alt="enter image description here"></p>
<p><img src="http://i.stack.imgur.com/w95Zy.jpg" alt="enter image description here"></p>
<p>What i don't understand is why The solution to this problem says that,$$\tan\theta=\frac{m_{l}}{m_{r}}$$, isn't mass a scalar quantity?</p>
<p>The rest is straightforward</p>
| 10,092 |
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/13325/what-is-the-minimum-amount-of-fissile-mass-required-to-acheive-criticality">What is the minimum amount of fissile mass required to acheive criticality?</a> </p>
</blockquote>
<p>I want to make a red laser the only problem is that the power is not sufficient so I had to resort to a nuclear reactor.</p>
| 323 |
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<p>I was reading <a href="http://physics.stackexchange.com/questions/51838/why-do-tuning-forks-have-two-prongs">Why tuning forks have two prongs?</a>. The top answer said the reason was to reduce oscillation through the hand holding the other prong.</p>
<p>So if having 2 prongs will reduce oscillation loss, surely a 3-pronged tuning fork would be even more efficient.</p>
<p>Why don't you see more 3-pronged tuning forks?</p>
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<p>Are there any logical relationship between specific heat capacity and thermal conductivity ?</p>
<p>I was wondering about this when I was reading an article on whether to choose cast iron or aluminium vessels for kitchen.</p>
<p>Aluminium has more thermal conductivity and specific heat than iron ( <a href="http://www.engineersedge.com/properties_of_metals.htm">source</a> ).</p>
<p>This must mean more energy is required to raise an unit of aluminium than iron yet aluminium conducts heat better than cast iron.</p>
<p>Does it mean that aluminium also retains heat better ?</p>
<p>How does mass of the vessel affect the heat retention?</p>
| 10,094 |
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/14056/how-does-gravitational-lensing-account-for-einsteins-cross">How does gravitational lensing account for Einstein’s Cross?</a> </p>
</blockquote>
<p><a href="http://en.wikipedia.org/wiki/Einstein_Cross" rel="nofollow">Einstein's Cross</a> is a fascinating phenomena for which I have asked <a href="http://physics.stackexchange.com/questions/14056/how-does-gravitational-lensing-account-for-einsteins-cross/14063">explanation here</a>. However, I'm also interested to know if Einstein's cross was a specific prediction that people had obtained from Einstein's theories such as General Relativity or if the phenomena was merely named after Einstein. The wiki page on <a href="http://en.wikipedia.org/wiki/Einstein_ring" rel="nofollow">Einstein's Ring</a> explains how the gravitational lensing phenomena of rings was predicted by general relativity, however I have yet to discover evidence that Einstein explained why there are 4 dots instead of a ring/crescent. </p>
| 48 |
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<p>I read an article about phase transitions and I read about thermodynamic processes such as <a href="http://en.wikipedia.org/wiki/Adiabatic_process" rel="nofollow">Adiabatic process</a>, <a href="http://en.wikipedia.org/wiki/Isochoric_process" rel="nofollow">Isochoric process</a>, <a href="http://en.wikipedia.org/wiki/Isobaric_process" rel="nofollow">Isobaric process</a>, <a href="http://en.wikipedia.org/wiki/Isothermal_process" rel="nofollow">Isothermal process</a> </p>
<p>Do these processes affect phase transition in solid state metals? </p>
<p>The reason I ask is because when these thermodynamic processes are being defined or explained, a fluid is used to explain the concept behind them.</p>
<p>Take for example when a solid state metal is in use (say semiconductor chip) a lot of heat is evolved or given out, if the heat is too much, and the metal is considerable small, it might melt, thus going from solid of a liquid form.</p>
<p>How are these thermodynamic processes modeled? Is there any reference that you can provide?</p>
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<p>So this is the question given in my text book: </p>
<blockquote>
<p><em>A particle of mass m is at rest at the origin at time $t = 0$. It is subjected to a force $F (t) = F_0e^{–bt}$ in the $x$
direction. Its speed $v(t)$ is depicted by which of the following curves? (I am not posting the curve.)</em></p>
</blockquote>
<p>And this is the solution in my textbook :-$$F= F_0e^{-bt}$$$$\implies a = \frac{F}{M}=\frac{F_0}{M}e^{-bt}$$$$\implies \frac{dv}{dt}=\frac{F_0}{M}e^{-bt}$$$$\int dv=\int _0^t\frac{F_0}{M}e^{-bt}$$$$V=\frac{F}{Mb}e^{-bt}$$</p>
<p>So if I am right according to this at $t=0$ ,$V$ should be equal to $\frac{F}{Mb}$ but in my text book it is given at $t=\infty$ that $V=\frac{F}{Mb}$. So where am I making mistake can anyone point out.</p>
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<p>Usually in electrostatics we start by introducing the vector field $\mathbf{E}$ representing the electric field due to some charge distribution. Later when we study fields in materials we consider the electric displacement field $\mathbf{D}=\epsilon_0 \mathbf{E}+\mathbf{P}$ being $\mathbf{P}$ the polarization density.</p>
<p>When we want to generalize things to differential forms, $\mathbf{E}$ becomes a $1$-form and $\mathbf{D}$ becomes a $2$-form. Similarly occurs with $\mathbf{B}$ and $\mathbf{H}$.</p>
<p>Why is that? Why should we make those decisions when modeling <a href="http://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field" rel="nofollow">electromagnetism</a> with differential forms? </p>
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<p>In my introductory modern physics class we have examined time-independent solutions to the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" rel="nofollow">Schrödinger equation</a> in 1 dimension. We looked at a few cases without finite boundary, e.g., free particles and step potentials with $V_{\mathrm{max}} < E$. In each example, in class and in all introductory material I've found, it was stated without mathematical justification that the reciprocal complex exponential solution should be ignored as "unphysical". Particularly, though a more mathematically complete solution would be of the form
$$
A\mathrm{e}^{\mathrm{i}\alpha x} + B\mathrm{e}^{-\mathrm{i}\alpha x},
$$
I've only seen solutions in the unbounded region where $B=0$ under the interpretation that its term would be the result of some reflection, which can't occur if there is no reachable boundary. The usual statement runs, "the term in $B$ has a negative velocity and is unphysical..." I find this very unsatisfying. I'm not convinced it's proper to use the term "velocity" in this case (do they claim the time-independent wave equation "propagates"?), unless it's with respect to some aspect of probability current (which concept is not in the curriculum). An infinite boundary is an unphysical concept in itself. The solution isn't even square-integrable, as far as I can tell. I can't for the life of me find a mathematical justification, or more than a sentence of explanation (all relying, it seems, on implicit analogy to the physics of a string). It just comes across as arbitrary, or as an excuse to avoid a more complex treatment which I feel would be worthwhile.</p>
<p>Is there a justification for this assumption which can be written mathematically? It just feels like a trick to simplify an underdefined thought experiment.</p>
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<p>Biot-Savart law for a linear current distribution is:</p>
<p>$\displaystyle \vec{B}=\frac{\mu I}{4\pi}\int\frac{\vec{dl}\times \vec{r}}{r^{3}}$.</p>
<p>In the book that my professor uses says that if we have current flowing in a wire of very small cross section $S$, in a length $\vec{dl}$ of the wire we have $dV=Sdl$ so we can write that $\vec{J}dV=\vec{J}Sdl=\vec{I}dL=I\vec{dl}$.</p>
<p>Then we can obtain Biot-Savart law for volume distribution. How can that happen? In the linear distribution we have one integral and in the surface distribution we have triple integral! How is that possible? I don't mean why we use triple integral for volume distribution. I mean how is that possible to go from one to another.</p>
| 10,099 |
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<p><strong>Question:</strong> How hot is the water in the pot? More precisely speaking, how can I get a temperature of the water as a function of time a priori? </p>
<p><strong>Background & My attempt:</strong> Recently I started spend some time on cooking. And I'm curious about it. I have learned mathematics as a undergraduate student for four years, but I know a little about thermodynamics. (I listened to such a lecture once. So I've heard of $dU = TdS - pdV$, Entropy and Gibbs energy for example though I forgot almost everything; anyway I think I've never seen a formula depending on time.) So I conduct a small experiment first: I heat 100ml of water by IH correspond approximately to 700W and measure its temperature every 30 seconds. Here is the results.
<img src="http://i.stack.imgur.com/fRsWY.png" alt="experiment results"></p>
<p>It looks almost linear, but I think linear approximation is inappropriate; because if so, the water gets higher than $100^\circ\mathrm{C}$. So I guess it's some convex increasing function like $T(t) = 100 - \alpha e^{-t/\beta}$ for some positive constant $\alpha$ and $\beta$. <del>But it doesn't fit the data. </del> (It does fit the data. I just made a mistake in simple calculation. See <a href="http://physics.stackexchange.com/questions/60095/the-time-evoluation-of-the-temperature-of-a-water/60216#60216">my answer</a>.)</p>
<p>I think I ignored too many factors. So feel free to assume anything reasonable. I would greatly appreciate if you help me. Thank you.</p>
<p><strong>Additional question:</strong> I do a experiment and some calculation to deal with a problem pointed out in the comments of my answer: bad fitting at lower temperature. However, I cannot get a better solution. Fitting seems worse than before... Here is the results what I got. I heated 100ml water in pot with 9cm radius by IH correspond to 700W. (For calculation, I added linear interpolation values in the graph.) How can I get a better solution?
<img src="http://i.stack.imgur.com/OgELx.png" alt="logistic-curve">
(Light blue curve is a logistic approximation defined by $T = \dfrac{100}{1 + 1.62 e^{-0.0168 t}}$ as mentioned <a href="http://physics.stackexchange.com/questions/60095/the-time-evoluation-of-the-temperature-of-a-water/60103#comment121694_60103">here</a>.)</p>
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<p>Consider we have two atoms $a$ and $b$. They are entangled with each other in position and momentum, with some wavefuction describing them in position space that is $\Psi(x_a, x_b)$. This initialization of the entangled state is achieved as described in this paper: <a href="http://arxiv.org/abs/quant-ph/9907049" rel="nofollow">http://arxiv.org/abs/quant-ph/9907049</a></p>
<p>We can trap both atoms by putting them in a harmonic oscillator potential with lasers and get them to both entangle with methods described by the paper I referenced above and by using a nondegenerate optical parametric amplifier (NOPA) as the entanglement source.</p>
<p>From what I understand, we can think of each particle having its own local axis and separated by some arbitrary global distance. Let us also consider only one dimension, the $x$-dimension, to keep things simple. </p>
<p>Say we apply the position operator, $\hat x_a$ to particle $a$ and given that it was in a harmonic oscillator potential, $V$, we will obtain a single measurement $x_a$. If we had identically, <em>or as close as we can</em>, prepared systems and we applied the same operator, or measurement, multiple times for an ensemble of entangled pairs, and we put all of these measurements together, then we'd get the probability distribution or the "quantum state of position" of atom $a$. Essentially, we do quantum state tomography. Referring the figure I've attached, I've made up a scenario where the atom happens to be between $x$ = -4 and 4 with arbitrary units with respect to their local axis.</p>
<p><img src="http://i.stack.imgur.com/Z6gmY.png" alt="enter image description here"></p>
<p>Atoms $a$ and $b$ should roughly have the <strong>same distribution</strong> ... because each single measurement of position for each entangled pair in the ensemble should yeild $x_a = x_b$ with respect to their local axes ... because this is what it means to be entangled. (Need a check here)</p>
<p>Now, let us consider an two ensembles of atoms called $A$ and $B$. They are separated in Lab A and Lab B. They are entangled such that we will get the same distribution if position is measured at either laboratory.</p>
<p>We know that given a certain potential, we will obtain a characteristic distribution when we measure the ensemble. Basically, applying operator, $\hat x$, and potential, $V_0$, will produce a different distribution than $\hat x$, and potential, $V_1$, whatever those potentials maybe. (Need a check here) </p>
<p>We also know that we if we apply $\hat x$, and potential, $V_0 + V_1 = V_2$, it too will produce a different distribution, different than $\hat x$, and potential, $V_0$, or $\hat x$, and potential, $V_1$, alone. (Need a check here)</p>
<p>Scientist at Lab A has ensembles $A_1, A_2, A_3, ... $ and a scientist at Lab B has ensembles $B_1, B_2, B_3, ... $. Ensemble $A_1$ is entangled with $B_1$, $A_2$ with $B_2$, $A_3$ with $B_3$, and so on.</p>
<p>Scientist A chooses to apply $V_0$ to $A_1$ in their "local" area, <em>Note this is actually applying to $\Psi(x_a, x_b)$, because the ensemble is entangled.</em> Scientist A skips ensemble $A_2$ and chooses to apply $V_0$ to $A_3$, etc. Scientist B applies $V_1$ blindly to all the ensembles $B_1, B_2, B_3, ...$ <strong>at the same time agreed ahead of time with synchronized clocks</strong></p>
<p>The distribution scientist B should get is the lower right hand side of the following figure: (Note, I made up the distribution shapes. They are just shown to be different to make the case or explain the thought experiment.)</p>
<p><img src="http://i.stack.imgur.com/vwEz1.png" alt="enter image description here"></p>
<p><em>The logical deduction based on my previous understanding and setup, if its all consistent is that one could send a message using ensembles of entangled atoms, synchronized clocks, and simultaneous measurements at a specified time agreed ahead of time.</em></p>
<p>If scientist A wants to send a message to scientist B, then he or she would choose which ensembles to apply $V_0$ to and then scientist B would blindly apply $V_1$ regardless of what scientist A did. Scientist B <em>would know</em> what scientist A sent because they both were applying their respective potentials at the same time to the same shared or entangled system, $\Psi(x_a, x_b)$ resulting in characteristic distributions that can be assigned to "1" "0" "1".</p>
<p><strong><em>Does this proposed thought experiment have a major flaw? If so where?</em></strong> (First violation is obviously FTL information exchange. It seems to violate it, but where?)</p>
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<p>Well this got me stumbled, because I've been wondering what the "question" is. One of the example examns got the following question (strangely there are no supplied solution books):</p>
<blockquote>
<p>A $0.2 {\rm m^3}$ thermally insulated rigid container is divided into two
equal volumes by only a thin membrane. Initially, one of these
chambers is filled with air at a pressure of $700\, {\rm kPa}$ and $37 \, {\rm C}$ while
the other chamber is evacuated. </p>
<p>$C_p = 1.005 \frac{{\rm kJ}}{{\rm kg \cdot K}}\,$ and $\, C_v =0.721 \frac{{\rm kJ}}{{\rm kg \cdot K}}$</p>
</blockquote>
<p>Now the questions:</p>
<blockquote>
<p>a) Determine the change in internal energy of the air when the membrane is ruptured.<br>
b) Determine the final air pressure in the container</p>
</blockquote>
<p>Is this now a really silly question or am I missing something important? Cause isn't the internal energy an intrinsic property that has to be looked up/ experimentally determined?</p>
<p>And for the second problem, as there can be no energy transfer the internal energy also has to stay the same - so the temperature doesn't change and the pressure simply halves. ($PV = {\rm constant}$). Or am I missing something?</p>
| 10,102 |
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<p>I have been following the methods suggested by members of this site to calculate the solar irradiance outside of the earth's atmosphere, see <a href="http://physics.stackexchange.com/questions/116596/convert-units-for-spectral-irradiance">here</a>.</p>
<p>I now want to calculate the solar irradiance reaching the earth's surface. </p>
<p>I calculate the irradiance outside the atmosphere as:</p>
<p>$$ L_{\lambda} = \frac{2c^{2}h}{\lambda^{5}\left( \exp\left[hc/\lambda kT \right] - 1\right)}$$</p>
<p>where
$ h = 6.626\times 10^{-34}$</p>
<p>$ c = 3 \times 10^{8} $</p>
<p>$ T = 6000 $</p>
<p>$ k = 1.38066\times 10^{-23} $</p>
<p>$ \lambda = 0:20 \times 10^{-9}:3200 \times 10^{-9} $</p>
<p>I then convert the units as:
$$ L_{\lambda} =L_{\lambda} \times 10^{-9} $$</p>
<p>multiply by the square of the ratio of the solar radius of earth's orbital radius
$$ L_{\lambda} =L_{\lambda} \times 2.177 \times 10^{-5} $$</p>
<p>apply Lambert's cosine law
$$ L_{\lambda} =L_{\lambda} \times \pi$$</p>
<p>which results in the upper curve seen in the following figure, i.e. the energy curve for a black body at 6000K:</p>
<p><img src="http://i.stack.imgur.com/m0MDJ.png" alt="enter image description here"></p>
<p>I now wish to generate a second curve, one that shows the irradiance at the earth's surface. I know that the scattering and absorption processes that take place in the atmosphere not only reduce the intensity but also change the spectral distribution of the direct solar beam. </p>
<p>I want to show the spectral distribution of solar irradiance at seal level for a Zenith sun and a clear sky. So, the curve that I want to show is the spectral distribution as it would be if there were scattering but no absorption. For this I would also like to make the assumption that the solar elevation is more than 30 degrees. </p>
<p>Does anyone know how I could produce the curve explained above? </p>
| 10,103 |
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