Resistances in a circular loop

![enter image description here](http://i.stack.imgur.com/B7pBq.png) I have no idea what formula I'm supposed to use.I just want to know the concept that would allow me to get the answer. I know basic concepts of electricity like charge, currents, etc., but as far resistances go I know only about series and parallel. I tried looking for resistances in a circular loop but I didn't understand what was being explained. Also I didn't find any concept that helped me understand what was to be done, probably because I didn't understand them.I used some help from a different question and got the resistances but my answer wasnt correct.


Electrons of conductors Free?

Are electrons of any conductor really free ? I mean are they always already moving or do they move only when electrostatic field of some sort is applied across them. I suppose if they were always moving each and every conductor must have some varying magnetic field around it at all times which though negligible musy be detected. Is that really so ? Or are the so called free electrons not really free ? Also is there some other experiment which i do not know about by which we have already proven that they are always moving ? does it take into account the earths and other electric fields which may be causing that movement ? Addendum : the question is not duplicate of Are free electrons in a metal really free as that post talks about ease of electrons to interact with external fields, my question is about the nature of motion of electrons in absense of external field interference as in are electrons in free motion even when they are not under the influence of any electromagnetic field.


Has this boundary condition been used in fluid flow?

I would like to know whether anyone has seen a boundary condition used in a fluid flow problem, of the following type. Suppose viscous incompressible fluid is to the left of a plane $x_1=a$, so the outer normal to the fluid region is $n=(1,0,0)$. The condition specifies that the $x_1$ momentum flux is a nonnegative multiple of the velocity: $$ m\begin{bmatrix} u_1\cr u_2\cr u_3\end{bmatrix} = \begin{bmatrix}\rho u_1^2+p-2\mu u_{1,1} \cr \rho u_2u_1-\mu(u_{1,2}+u_{2,1}) \cr \rho u_3u_1-\mu(u_{1,3}+u_{3,1}) \end{bmatrix} $$ where $m$ could be a number $\ge0$ or even a nonnegative 3 by 3 matrix. (This is connected with my question at http://www.mathoverflow.net/q/163772/) Thanks for any references.


momentum conservation and gluons

The process is the following: $$e^-e^+ \rightarrow photon \rightarrow quark + antiquark$$ Regarding the momentum conservation law, how come we have a photon of spin 1 and at the end some meson with spin 0? Are gluons "behind this"? If this is correct, at which point are they radiated? From quarks? Or? Is this photon a virtual photon or not? I'm a bit confused here.


What exactly is a virtual displacement in classical mechanics?

I'm reading Goldstein's Classical Mechanics and he says the following: > _Avirtual (infinitesimal) displacement of a system refers to a change in the > configuration of the system as the result of any arbitrary infinitesimal > change of the coordinates $\delta \mathbf{r}_i$, *consistent with the forces > and constraints imposed on the system at the given instant $t$*. The > displacement is called virtual to distinguish it from an actual displacement > of the system ocurring in a time interval $dt$, during which the forces and > constraints may be changing._ Then he discusses virtual work and so on. Now, I can't grasp what this thing of virtual really is. By this text there's a diference between one infinitesimal change and one virtual change and I really don't get what this virtual really is. Also, this is based on infinitesimals. How can this be expressed rigourously without refering to infinitesimals? I tried looking on Spivak's Physics for Mathematicians where he considers these virtual displacements as tangent vectors to a certain manifold, but I'm not sure this is the most "standard" way to do it rigorously.


Rotating platform with pulleys and stacked blocks

This was a practice test question. > Consider the following figure. (If the frictional coefficients are unclear, > between A and B we have $k_1 = 0.3$, B and platform is $k_2 = 0.2$. If you > further doubts about the diagram ask me in the comments below and I will > clarify.) ![enter image description here](http://i.stack.imgur.com/PRMA5.jpg) > The blocks are resting on the rotating platform (bold) attached to the rod > which is rotating at constant angular velocity $\omega\ \mathrm{rad/s}$. The > masses of blocks and distances from the pulley are as given in the figure. > The distance of the pulley from the rod (axis) is negligible. Block B is on > verge of slipping. Find $\omega$. What I did : Let mass of $A = m_1$. Distance of $A$ from axis = $r_1$. Mass of $B = m_2$. Distance of $A$ from axis = $r_2$. If block $B$ is on verge of slipping $A$ is also on the verge of slipping. Questionable observation. But leads to correct answer. Anyway, we consider a rotating frame attached to the rod. So we give a centrifugal force to $A$ = $m_1 \omega^2 r_1$ and to $B$ equal to $m_2 \omega^2 r_2$. Tension in string $ = T$. We check two cases. Let $B$ slip inwards. Then $A$ slips outwards. So $A$ slips rightwards relative $B$. $A$ was on verge of slipping. Static friction on $A = k_1 m_1 g$. Static friction on $B = k_2 (m_1 + m_2) g$. Then equating forces and solving the system and putting values we get $ \omega = \sqrt{22}\ \mathrm{rad/s}$. The other case ($B$ goes outwards) leads to contradiction. Though the result is correct, I have a few questions. Why does the fact that $B$ is on the verge of slipping imply that $A$ is also on the verge of slipping. I mean, suppose we just eliminate the rotation and add an external force on $B$ instead. Let the magnitude of force reach $k_2 (m_1 + m_2) g$. Then why is it necessary that the force of static friction between $A$ and $B$ is $k_1 m_1 g$ ? Am I wrong, and is it just that the answer matched, without correct physics? A detailed explanation of a solution to this problem would be very helpful as I'm a beginner, and I intend to clarify these essential concepts.


How can a gas support tensile stresses?

In working through a rigorous derivation of the compressible Navier-Stokes equations, I find that the momentum flux in the X-direction should be driven not only by the normal pressure gradient $\frac{\partial p}{\partial x}$ and shear stress terms $\frac{\partial(\tau_{yx})}{\partial x}$ and $\frac{\partial(\tau_{zx})}{\partial x}$, but also by the gradient of the normal stress $\frac{\partial(\tau_{xx})}{\partial x}$. It's intuitively clear to me how adjacent lamina moving at different speeds can transfer momentum across their interface, and so the shear stress terms in the momentum equation are readily intelligible. The normal stress term, on the other hand, is far less intuitive because I cannot see how a freely-deforming fluid can support tensile stresses. Positive normal stresses (i.e. compression) are not that hard to understand, but it's proving exceedingly difficult to fully envisage an element of a fluid "pulling on" an adjacent element in a way even remotely analogous to the behavior of a solid under the same conditions. I am also unclear on the difference between "pressure" and "normal stress" in the fluid. How exactly are these terms different? I am interested primarily with gases not liquids.


Why is electric potential 0 in this case?

On a test, we had a question where there are 4 point charges at the vertices of a square. The 2 charges at the upper vertices have charges of `+q` and the 2 charges at the lower vertices have charges of `-q`. The magnitude of the charges are equal. According to the answer sheet, the electric potential is 0 along a horizontal line halfway between the 2 upper and 2 lower charges. Why is it 0? Shouldn't the test charge be attracted to either to top or the bottom depending on its charge?               +q        +q          ------------    <--- 0 electric potential          -q        -q


How absorption coefficient determines which material is used to make solar cells?

Does the knowledge of the material absorption coefficients aids engineers in determining which material to use in their solar cell designs? If yes, how?


How do crocodiles jump?

In a video (Here), I saw crocodiles jump vertically about three meters without using any solid surface. The wonderful thing is that when they start to jump, their vertical velocity is approximately zero, unlike fish who jump using initial velocity. It seems that crocodiles create an upward force that counteracts gravity, because when they are rising, their velocity seems to be constant. How is this possible? Could anyone explain this phenomenon using physics laws?


Feynman's $i \epsilon$ prescription in loop expansion

I have some questions about the $i\epsilon$ factor in Feynman diagrams. First, what is the physical meaning of $i\epsilon$ in loop amplitudes. Second, how does it ensures unitarity? And third, Dyson series assume that incoming and outgoing particles are free, this can be implemented by assuming that the interaction Hamiltonian switches off adiabatically, $e^{-\eta\,t}H_{I}(t)$. Is this $\eta$ related with the $i\epsilon$?


How to design a house to be cooled passively?

I live in Louisiana these days, in an area that is known for its numerous antebellum plantation homes (circa early 1800s). While touring one of these homes it was clear that almost everything about the house was designed around keeping cool in the summer. Some examples:   * 4 meter high ceilings to allow hot air to rise to the ceiling.   * Floor to ceiling windows to allow hot air at the top to escape and cool air to be drawn in at the bottom.    * Porches on the sunny sides of the house to prevent sunlight from entering the windows.   * Large central staircases to allow hot air to rise to the second floor, drawing cool air in on the bottom floor.   * Some have a cupola, a central observation room at the top of the house, again to allow hot air to escape at the top of the house and draw air in from the bottom. * * * My question is: Given our modern understanding of thermodynamics, how could one design a home today to be cooled passively? Could we do any better than the plantation owners of the 1800s? Lets define cooling as making the house more comfortable for humans. This means that it is not only important to reduce the temperature, but also to block sunlight and maintain airflow. Also, if possible, it would be very beneficial to extract moisture from the air.


Why the electric bulb turns on almost instantly when the switch is closed?

**The electron drift speed is estimated to be very low.How could there is current almost the instant a circuit is closed??**   ![enter image description here](http://i.stack.imgur.com/JOVuz.jpg) By the discussions it is known that **The information about beginning of the flow of current is transmitted through the propagation of electromagnetic waves(electric impulse)and not with the drift velocity of the electrons.**   But I want any one to explain how this process takes place.CURIE:)


How can I calculate the wave propagation speed in a copper wire?

First of all: I am a computer science student, so I don't have much knowledge of physics. So please keep your answers simple. I recently learned something about circuit design and its problems (differend kinds of hazards). To model the problems, we introduced a "dead time model" ("Totzeitmodel" in German, I have if it is "dead time model" in English)   We added some dead time to each element of the circuit, but we didn't add dead time to the wires of the circuit. I asked the prof. why we didn't add dead time to the wires. He responded that the signal is moving much faster and you can neglect the time that signals need to pass the wires. Now I would like to calculate the speed of the signal (is this the wave propagation speed?) for some very simple settings:   * assume we have a copper wire   * the wire is a perfect cylinder with diameter of 1mm   * the current is 2A   * the voltage is 12V Can you help me with this? Do you need something else to calculate the speed? * * * Notes: I found the wikipedia article Wave propagation speed and some questions on physics.stackexchange.com, but the questions and answers were either too complicated or didn't directly give numbers (like that one) A little side question: When I think about the electric signal, I imagine some elastic balls. When there is a signal at one end, you push the ball. It gets compressed and expands later, which compresses the next ball a bit and it expands, ... This way, the last ball gets moved and the signal arrives at the end. Do I have to get another thinking-model for simple circuits or will I be able to understand simple circuits with this model in mind?


Axisymmetric fluid flow

I'm having trouble with a boundary condition. In a fluid mechanics problem, I have flow at $z = \infty$ flowing into a solid plate at $z = 0$ and then flowing radially, and the problem is given as axisymmetric. I know that the velocity field has $v_r$ and $v_z$, and $v_\theta = 0$. How in the world would I express the symmetry boundary conditions without $\theta$?


Possible intergalactic celestial objects

We know there are rouge stars floating in intergalactic space, thought to be caused by galactic collision. What other other classes of celestial object could be found floating around in intergalactic space? Within galaxies there are the following types of object:   * Nebulae    * Stars   * Brown Dwarfs   * White Dwarfs   * Black Holes   * Neutron Stars   * Comets   * Asteroids   * Interstellar rogue planets   * Planetary Systems   * Star Systems   * Star Clusters   * Debris Disks Which of these can possibly exist in intergalactic space? Asteroids and comets are probably quite common, but is it known for entire planetary systems to exist in intergalactic space? Even star clusters? Out of all the possible types, could I also have a link / designation to each object thats been observed, floating in intergalactic space. For example, here's an intergalactic star HE_0437-5439


For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we get absorption lines?

For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we see absorption lines? Similarly, why do we see emission lines if the planet is hot on the surface and gets cooler as you move to the center? Note, for this question I am only thinking of the planet as a black body, not as something, for example, transiting a star and observing the spectra (transit spectra) of the light that shines through the planet's atmosphere. The source of the spectra is the planet itself.


How is the Hanbury-Brown and Twiss effect used to measure the size of stars?

I understand what an Hanbury Brown and Twiss (HBT) interferometer _does_ , but how can this be used to measure the apparent angular diameter of some object? What is the mathematical explaination?


Problem involving 1st law of thermo and ideal gas law

Problem: > $1.0 \text{ kg}$ of air at pressure $10^6 \text{ Pa}$ and temperature $398 > \text{ K}$ expands to a five times greater volume. The expansion occurs such > that in every instance the added heat is a quarter of the work done by the > gas. Calculate the final pressure. $1 \text{ kmol}$ has a mass of $29 \text{ > kg}$ and $C_V = \frac 52 R$. I solved the problem but I have some questions regarding the approaches I made. I interpreted heat being a quarter of the work done every instance as $\text{d}Q = \frac 14 \text{d}W$. By the 1st law of thermodynamics ($Q = W + \Delta U$), this yields $-\frac 34 \text{d}W = \text{d}U$. But for molar heat capacity we know that $\text{d}U = nC_V\text{d}T$, hence $-\frac 34 \text{d}W = nC_V\text{d}T$. By definition, $\text{d}W = p\text{d}V$ and therefore $-\frac 34 p\text{d}W = nC_V\text{d}T$. We treat the gas as ideal and thus we can substitute $\frac {nRT}V$ for $p$, arriving at $-\frac 34 \frac{nRT}V \text{d}V = nC_V\text{d}T$. Separation of variables yield $-\frac 34 \frac{nR}V \text{d}V = \frac {nC_V}T \text{d}T$. We take the definite integrales $ \displaystyle \int_{V_1}^{5V_1} \\!\\!\\!\\!\\! -\frac 34 \frac{nR}V \text{d}V = \int_{398}^{T_2} \frac {nC_V}T \text{d}T$ and can then finally solve for $T_2$ (it turns out to be approximately $245 \text{ K}$). At last, we can determine the final pressure from $\displaystyle \frac {p_1V_1}{T_1} = \frac {p_2V_2}{T_2}$. * * * **(1)** My main question is whether you guys know of a different way to solve this problem not involving having to solve differential equations? **(2)** I've seen the 1st law written as $\text{d}U = \text{d}Q + \text{d}W$ but if we use that expression we definitely won't get the right answer; what's up with the inconsistency? **(3)** I saw no other option than having to use the molar heat capacity equation $\text{d}U = nC_V\text{d}T$ but doesn't this assume constant volume? In our problem the volume is definitely changing, so is it not contradictory to use that very equation?


Unitary Confirmation

I am asked to show that an new defined operator: $$U_{\beta} = \exp(\displaystyle\frac{i\beta L_z}{\hbar})$$ is unitary, where $$L_z = -i\hbar\,\,(x\displaystyle\frac{\partial}{\partial y} - y \frac{\partial}{\partial x }).$$ I tried the following: $$ U_{\beta}^{\dagger} U_{\beta} = \exp \left( \frac{i\beta(-L_z^{\dagger}+L_z)}{\hbar} \right)$$ So I couldn't make the inside of exponential zero.


Does the Casimir effect give the correct value for Dark Energy?

My understanding is that the Casimir Effect is caused by vacuum energy. Quantum mechanics (QED) predicts vacuum energy, but gets the value grossly wrong, by a factor of $10^{120}$. On the other hand, from what I have read, Dark Energy is understood to be caused by vacuum energy. Has anyone checked whether the measured value for the Casimir Effect fits the value required for Dark Energy to make up 70% of the energy in the universe? If it did, that would be an excellent validation of Dark Energy.


Is it reasonable to assume that de Sitter temperature will stop growing when it reaches equilibrium with the temperature of CMB? Currently the de Sitter temperature (the temperature of the Universe's boundary) is $$T_{dS}=2.67\times10^{-30}\,\mathrm K$$ and rises. At the same time, the temperature of CMB is $$T_{cmb}=2.7\,\mathrm K$$ and falls. I wonder, does it mean that at the moment they will equate, the de Sitter temperature will stop growth, and so the expansion of the universe will also stop accelerating, thus reaching the thermodynamic equilibrium? In other words, whether the excessive temperature of CMB compared to the de Sitter temperature is the main source of the accelerated cosmic expansion?


interaction between mathematical structures

From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one...


Faradays Law: How Does Curl E Change Over Time In a Coil?

I think I am currently making a mistake regarding my interpretation of Faraday's Law. $\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$ Assuming we have an electric field $\vec{E}$ in the plane between 2 particles given by: $\vec{E}=(\frac{x}{x^2+y^2}+\frac{1-x}{(x-1)^2+y^2}, \frac{y}{x^2+y^2}-\frac{y}{(x-1)^2+y^2})$ Then $\vec{\nabla}\times\vec{E}=(0,0,\frac{y\cdot(2\cdot x-2)}{((x-1)^2+y^2)^2}+\frac{(2\cdot(1-x))\cdot y}{((x-1)^2+y^2)^2}) = (0,0,0)$ As expected, now, if there's an electric field through a coil due to potential difference, then why will the curl be non-zero only when the electric field is initiated and closed off from the coil (e.g. connected/disconnected from a power source)? Currently; I assume that the derivative of the magnetic field converges to 0 over time - because it can not grow indefinitely if we simply have a potential difference (coil connected to a power source). So my question is: How does the electric field (I am correct in that E indeed stands for the electric field, right?) change over time in the time we connect the coil (get potentials at both ends, and a current), until the coil's E curl has stabilized (becomes 0)? It'd be great if someone had an example of an $\vec{E}(\vec{r}, t)$ that changes with time so that $\vec{\nabla}\times\vec{E}$ converges to 0. Where $t_0$ is the connection time and $t_1$ the stabilization time, and $\vec{r}$ the position in space. Are my questions based on false premises? I feel really confused, I've tried my best to look for answers but I can not seem to find them.


Is there a Newton's third law for the em field?

There is a momentum associated with the em field that ensures the conservation of total momentum for a system of interacting charges. Can the same be done in an analagous way to ensure Newton's third law is also true?


Interpretation of Dirac equation states

In Pauli theory the components of two-component wavefunction were interpreted as probability amplitudes of finding the particle in particular spin state. This seems easy to understand. But when talking about Dirac equation, we have four-component wavefunction, two of which correspond to usual spin components of Pauli electron, and another two... How do I interpret positron-related components of Dirac electron? Are they probability amplitudes for the particle to appear to be positron? Or maybe to appear to _not_ be positron (taking Dirac sea picture into account)?


At what size will self-gravitation contribute more to stability than surface tension?

The governments of Earth have embarked on an experiment to place a massive ball of water in orbit. (umm... special water that doesn't freeze) Imagine this to be a fluid with a given density, $\rho$ ($kg/m^3$), surface tension, $\sigma$ ($J/m^2$), and formed in a sphere of radius $R$ ($m$). I think that the viscosity $\mu$ is not needed for this question, but correct me if I'm wrong. At what size will the restorative forces from gravity (after some small perturbation) become more significant than that from surface tension? Would the type of perturbation make a difference? Just for fun, here's a video of a ball of water stabilized by surface tension.


Can an invisible light source cast shadows?

Let's assume that we have a mechanism for producing EM radiation suspended in the air, and that that mechanism itself is invisible to the naked eye (e.g. a microscopic light bulb on a microscopic wire or a suspended molecular reaction giving off energy.) When off, the light source would not be visible. When turned on however, any light cast from the source to the environment around it would also reach the viewer's eye, and identify the source of light. Is it theoretically possible to cause the lit light source to cast shadows on the environment that are visible, but for the source itself to remain invisible? As an example, most 3D graphics programs can create invisible light sources which are only identifiable by the scene they illuminate. I imagine this may be possible with certain environments, such as an environment covered in phosphorescent paint and a black light source, where the light source is not seen but the environment which it illuminates is identifiable. But what about in the general case, when we limit the variables to the source of the light itself, stipulating it must work in a general environment?


How is a Rydberg Blockade Radius defined?

Rydberg blockade is a phenomena in 3 or more level systems of Rydberg dressed atoms.


Annihilation and creation operator - $\phi$ and $\pi$ for Klein-Gordon Field

**Introduction and Notation** Let $\phi(\vec{x})$ be the real Klein-Gordon (quantum) field, written as: $$\phi(\vec{x})=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}} $$ where $\omega_{p}=\sqrt{|\vec{p}|^2+m^2}$, $a_{\vec{p}}$, $a^{\dagger}_{-\vec{p}}$ the annihilation and creation operators, and let $\pi(\vec{x})$ the momentum density conjugate to $\phi(\vec{x})$, given by $$\pi(\vec{x})=\int\frac{d^3p}{(2\pi)^3}(-i)\sqrt{\frac{\omega_{p}}{2}}\left(a_{\vec{p}}-a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}} $$ **The question** The only non trivial equal-time commutator is $$[\phi(\vec{x}),\pi(\vec{y})]=i\hbar\delta^{(3)}(\vec{x}-\vec{y}) $$ As the relation between $\phi(\vec{x})$ and the $a,a^{\dagger}$ is linear, and so is between $\pi(\vec{x})$ and them, I'm going to express the commutator obeyed by $a$ and $a^{\dagger}$. I'm failing to derive the inverse Fourier transform $$ \int d^3x'\phi(\vec{x})e^{i\vec{p}'\cdot\vec{x}'}=\int\int d^3x'\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}}e^{i\vec{p}'\cdot\vec{x}'}=\\\=\int d^3x'e^{i\vec{p}'\cdot\vec{x}'} \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}}=\\\=(2\pi)^3\delta^{3}(\vec{p}')\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}}$$ **What I want is a Dirac delta $\delta^{3}(\vec{p}-\vec{p}')$** , so where is my procedure failing? I know this is a math question, but given the physical context, it may fit here better. Thanks for your time, any hint will be appreciated **EDIT 1** **EDIT 2** EDIT 1 Now as an answer


Online course on theoretical electrodynamics

I'm looking for good online course for an introduction into theoretical electrodynamics. However, it seems that the MIT opencourseware only contains partial lectures for this topic. Has anyone got a recommendation for me?


Change of basis in Dirac Notation

> `Question`: An operator $A$ is in a particular basis $|a_i\rangle$ (where > $i=1,2$), and is represented by $$A=\begin{pmatrix} 0 & -i \\\ i &0 > \end{pmatrix}$$ Now, define two new basis vectors $|b_i\rangle$ by $$\langle > a_i | b_1\rangle =\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\\ 1 \end{pmatrix}$$ > $$\langle a_i | b_2\rangle =-\frac{i}{\sqrt{2}}\begin{pmatrix} 1 \\\ -1 > \end{pmatrix}$$ > > What is $A$ in the new basis? `Attempt`: First, I defined the transformation matrix $U$: $$U=\begin{pmatrix} \langle a_1 |b_1 \rangle & \langle a_1 |b_2 \rangle \\\ \langle a_2 |b_1 \rangle & \langle a_2 |b_2 \rangle \end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \\\ 1 & i \end{pmatrix}$$ If we let $A'$ be the matrix in the new basis, we obtain $$ A'=UAU^\dagger$$ And it is a simple calculation (by hand or through mathematica) to show that $$ A'=\begin{pmatrix} 1& 0 \\\ 0 & -1 \end{pmatrix}$$ However, I wanted to check my work using Dirac notation, so I used the equation $$ A_{ij}'=\langle b_i |A|b_j\rangle=\sum_{n,m} \langle b_i |a_n\rangle A_{nm} \langle a_m | b_j \rangle$$ A sample calculation of $A_{12}'$ is shown below: $$A_{12}'=\sum_{nm}\langle b_1 | a_n \rangle A_{nm} \langle a_m | b_2 \rangle$$ $$=\langle b_1|a_1\rangle A_{12}\langle a_2 | b_2 \rangle+ \langle b_1 |a_2\rangle A_{21}\langle a_1|b_2\rangle$$ $$=\frac{1}{2}(1)(-i)(i)+\frac{1}{2}(1)(i)(-i)=1$$ Using the same method for the other components, we find $A'$ to be $$ A'=\begin{pmatrix} 0 & 1 \\\ 1 & 0\end{pmatrix}$$ My question is which method is incorrect? Do I have the formula for the matrix U wrong, or am I not using Dirac notation right? Thank you in advance.


Rank $L$ spherical harmonic tensor as a $2L+1$ dimensional Cartesian vector?

Rank two Cartesian tensors can be decomposed into $L=0,1,2$ spin like things 3x3=1+3+5. But the second equation below does not transform like a "tensor", it looks more like a vector transform in five dimensional Cartesian coordinate. $$\mathfrak{D}^{+}(R)V_{i}\mathfrak{D}(R) = \sum_{j=1,2,3}R_{ij}V_{j}\tag{1} $$ (three dimensional vector.) $$\mathfrak{D}(R)T_{i}^{(2)}\mathfrak{D}^{+}(R) = \sum_{j=-2,-1,0,1,2}\mathfrak{D}_{ij}^{(2)}T_{j}^{(2)}\tag{2}$$ (rank two spherical harmonic tensor.) $$ \mathfrak{B}^{+}(F)V_{i}\mathfrak{B}(F) = \sum_{j=1,2,3,4,5}F_{ij}V_{j}\tag{3}$$ (a vector transform in 5 dimensional Cartesian space.) $(2)$ and $(3)$ above looks the same. My question is can 3d spherical tensors be regarded as high dimensional vectors? Are there some examples?


Can there be energy with no force or energy with no power?

I think that both force (number of newtons) and power (p=ui(?)) implies that there is energy so we can't have force without energy and we can't have power without energy(?) But can there be energy that is energy and no force and no power?


Gas transformation for different molar number

When a gas at normal conditions ( **1atm, 273K, 22.4L** ) we say like the $$\frac{PV}{T} = 0.0820...$$ And by the know equation: $$PV = nRT$$ Where R is equals to $0.08205...$ and $n$ is the number of mols of the substance. When we want to compare a previous state of a gas with its next state, we use the formula: $$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$ But the $R$ is just the first state of the gas, already in a constant term ($\frac{P_1 V_1}{T_1}$ is $\frac{1 \cdot 22.4}{273}$ that is $R=0.082...$) So, instead of doing: $$\frac{1 \cdot 22.4}{273} = \frac{P_2 V_2}{T_2}$$ We do: $$R = \frac{P_2 V_2}{T_2}\tag{assuming the same n of mols}$$ But let's say we're doing this equality for a gas that has a given number $n$ mols of the substance. By what I know it is $$nR = \frac{P_2 V_2}{T_2}$$ I have $2$ questions: $(1)$ - How do I know that for a $1/2$ number of mols (for example), the quotient $\frac{PV}{T}$ is gonna be exactly $1/2$ (in other words, how do I know that they are linear) $(2)$ - Why do I have to multiply the $n$ always for the $\frac{1 \cdot 22.4}{273}$ and not for the $ \frac{P_2 V_2}{T_2}$ in the equation? (in other words, if I'm just equating two states of the gas, I guess there should be no problem with which side I multiply by $n$). I don't get what "multiplying by $n$" means. Like, the two states of the gas that I'm equating have the same amount of molecules, doesn't mean if they are $1/2$ mol or anything, they're the same quantity. Thanks!


How to directly calculate the infinitesimal generator of SU(2)

We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial U}{\partial \alpha_{i}}$$ from Lie group theory. But, where are the problems of the methods I used below? First, I parameterize the SU(2) with the $(\theta, \phi, \gamma) $ like this: $$U=\begin{bmatrix} e^{i\theta}sin\phi & e^{i\gamma}cos\phi \\\ -e^{-i\gamma}cos\phi & e^{-i\theta}sin\phi\end{bmatrix}$$ and the E is when $(\theta, \phi, \gamma)$= $(0,\frac{\pi}{2},0)$. Second, I use the definition of infinitesimal generator like this: $$ I_{1}=\frac{\partial U}{\partial \theta}|_{(0,\frac{\pi}{2},0)}=i\begin{bmatrix}1 & 0\\\ 0 & -1 \end{bmatrix}$$ $$ I_{2}=\frac{\partial U}{\partial \phi}|_{(0,\frac{\pi}{2},0)}=i\begin{bmatrix}0 & i\\\ -i & 0 \end{bmatrix}$$ $$ I_{3}=\frac{\partial U}{\partial \gamma}|_{(0,\frac{\pi}{2},0)}=i\begin{bmatrix}0 & 0\\\ 0 & 0 \end{bmatrix}$$ Here is the question... Why do I get the 0 matrix? We should expect to have the Pauli Matrix. Isn't it?   Where is the problem from?


How can primordial helium be formed before atoms?

In my physics textbook it describes the events at the beginning of the Universe. I'm confused about the order at a certain point. It says that at some point primordial helium is created, then it says that later atoms are formed. Isn't primordial helium made of atoms? Thank you :)


Does sending data down a fiber optic cable take longer if the cable is bent?

Ok, so, my simplified understanding of fiber optics is that light is sent down the cable and it rebounds off the sides to end up at its destination. Which got me thinking, if it has to bounce more times (and having a shorter travel between each bounce), does the light (data) take longer to get to the other end of the cable? Like this: http://i.imgur.com/pCHUf.jpg A part of me is saying no, because it's still the same distance to travel and bouncing doesn't take up any time, but another part of me is saying yes because the light will have further to travel the more times it bounces, and thus will take longer to get to its destination. I'm swaying towards it taking more time. Thanks!


Experimental proof of gravitational redshift of light

Has the gravitational red shift been proven for electromagnetic waves only or also for a single photon?


Reduction of a sum to the first Brillouin zone in a band structure calculation

this might be a "standard trick" for many solid state physicists, however it's one that I'm not familiar with so maybe you can help me. Here's the Problem: Suppose we're given a Hamiltonian of the form $H=\sum_{k} \epsilon_{k} c^{\dagger}_{k}c_{k} + \sum_{k} U c^{\dagger}_{k+Q} c_{k}$. Here U is some complex (!) number, k is a 2 dimensional wavevector and Q=$(\pi,\pi)$. Furthermore we impose $\epsilon_k = - \epsilon_{k+Q}$. The first Brillouin Zone is the set $\\{ (k_x,k_y) ; |k_x|+|k_y|<\pi \\} $ in 2D-k-space. Now define the 2D-vector $\Psi_k = (c_k,c_{k+Q})$. Then the Hamiltonian be written as: $H= \sum_{k}' \Psi_k^{\dagger} A_k \Psi_k $ with the k-dependent 2x2 Matrix $A_k$ defined by: $$ \left[ \begin{array}{ c c } \epsilon_k & U \\\ U^* & \epsilon_{k+Q} \end{array} \right] $$ The prime (') in the sum denotes that it has to be taken over wavevectors in the frist Brillouin zone only! Now multiplying this quadratic form out "in reverse" I obtain something like: $H=\sum_{k} \epsilon_{k} c^{\dagger}_{k}c_{k} + \sum_{k} U c^{\dagger}_{k+Q} c_{k} + \sum_{k} \epsilon_{k+Q} c^{\dagger}_{k+Q}c_{k+Q} + \sum_{k} U c^{\dagger}_{k} c_{k+Q} $ It's not clear to me why the third and fourth term are supposed to vanish in case I'm restricting my sum to the frist Brillouin zone. I hope someone can help. It should be rather technical, but still important I think. Thanks in advance.


What is the origin of the naming convention for position functions?

In physics, position as a function of time is generally called `d(t)` or `s(t)`. Using "d" is pretty intuitive, however I haven't been able to figure out why "s" is used as well. Is it possibly based on another language?


Coulomb splitting in atomic physics

In atomic physics, if a configuration with equivalent electrons in some shell (say Neodymium : $[Xe] 6s^24f^4$) gives same $L = \sum_i l_i$ and $S = \sum_i s_i$ ($i$ denoting individual spin and orbital momentum of the electrons, and in this example $L = 1$ appears twice with $S = 0$) considering the Coulomb interaction between electrons ($L$ and $S$ being the good quantum numbers in this case), how can these two $(L,S) = (0,0)$ have different energies ? I don't really see what can split these two levels if they start from the same configuration. Does someone know if they actually split ? And if they do, why ?


effects of sunlight through a glass window?

Both my father and my grandfather where drivers, and over time ended up with a wrinklier left hand compared to the right hand, due to sunlight exposure over 40+ years while holding the steering wheel with the left hand. I googled for sunlight exposure through a glass window and I get contradictory answers either stating either that most glasses filter out UV-light that affects skin tanning and ageing or that the effect of sunlight on skin is pretty much the same if it is direct or through a glass window. So, are the effects of sunlight through a glass window different from direct sunlight exposure?


Could quarks be free in higher-dimensional space than 3D?

Reading this answer, I now wonder: if quarks are confined by $r^2$ potential, could their potential allow infinite motion in higher-dimensional space? To understand why I thought this might be possible, see what we have with electrostatic potential: in 3D it is proportional to $r^{-1}$. This is just what Poisson equation tells us for point charge. If we solve Poisson equation in 2D space, we'll see potential is proportional to $\ln\frac r {r_0}$, and in 1D it's proportional to $r$. We can see that it only allows infinite motion starting form 3D. Could the same hold for quarks, but with some higher than 3D dimension? Or is their potential of completely different nature with respect to space dimensionality?


The string Poisson bracket

Where does the factor $\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from? $$ \\{X^{\mu}(\tau,\sigma),\dot{X}^{\nu}(\tau,\sigma')\\} ~=~ \frac{1}{T}\delta(\sigma-\sigma')\eta_{\mu\nu}. $$ I think I can see from remembering the definition of a Poisson bracket (for example in canonical coordinates) why in terms of momentum we have $$ \\{P^{\mu}(\tau,\sigma),X^{\nu}(\tau,\sigma')\\} ~=~ \delta(\sigma-\sigma')\eta_{\mu\nu} $$ but I don't see why this factor in the first equation has to be there. In addition to deriving it by calculation, is there an intuitive physical way how one can see why the factor of inverse tension has to be there, similar to explaining the appearance of the tension in front of the integral in the action by the fact that it costs energy to stretch the world-sheet?


What is the physical meaning of the "decay rate" in Fermis golden rule?

As far as I understood, Fermi's golden rule gives a prediction of the transition rate in a perturbed quantum system $H_0+V$ between two eigenstates of the unperturbed system $H_0$, say from $\left| i\right>$ to $\left| f\right>$ with eigenenergies $E_i,E_f$. The prediction is perturbative in the first order of $V$. The result is, that the rate equals $$\gamma_{i\to f} = 2\pi |\left<f|V|i\right>|^2 \delta(E_i-E_f)$$ This rate is defined as $$\gamma_{i\to f}=\frac{d}{dt} |\left<f|U(t)|i\right>|^2.$$ **My question:** How should one interpret this differential "rate" physically? **My thoughts so far:** Apparently it is the derivative of the _transition probability_ with respect to time, so it is not a probability itself. In literature it is often called _decay rate_ , but for an exponentially decaying probability $p(t)\sim e^{-\lambda t}$ the decay rate $-\lambda$ would not be computed as $dp/dt$, but $(dp/dt)/p$.


relationship between Fredkin gate and the swap and I gates

I need to prove this relationship: $$G_{Fredkin} = I \otimes |0\rangle\langle 0| + G_{swap} \otimes |1\rangle\langle 1|$$ with $$G_{swap} = |00\rangle\langle 00| + |01\rangle\langle 10| + |10\rangle\langle 01| + |11\rangle\langle 11|$$ So I think I need to show that both sides are a linear map on $H \otimes H \otimes H$ So what is the basis for $H \otimes H \otimes H$, and then how do I show that $H \otimes H = I$? Is this the right approach?


Why does surface charge not move?

If you have a wire with current flowing through it, and the current flowing the wrong way (not parallel to the wire) surface charge will buildup, generating a field to force the current to flow the right way. What's stopping the surface charge from redistributing? After the correct distribution has been set up for the surface charge to flow, what's stopping the surface charge from redistributing itself over the conductor to maximize distance between like charges?


Two identical particles with spin $s$. What is the spin of its corresponding "center-of-mass" and "relative" particles?

Consider a system of two identical quantum particles with spin $s$ and mass $m$. Using center-of-mass coordinates one obtains an equivalent system given by a particle of mass $2m$ and one of mass $m/2$. Can we talk about the $\textit{spin}$ of these two new fictitious particles? What can we say?


Where does this energy equation come from: $E_k=k\frac{k(q_1q_2)}{r}$

I have this equation but I have no idea where it came from. $E_k=k\frac{(q_1q_2)}{r}$ I don't understand especially since I have this other equation: $E = k\frac{Q_{source}}{r^2}$ And cannot find any relation between the two. Please any helps or suggestions would be useful. Thanks! **EDIT:** I have researched this and either am looking in the wrong place or just can't find it anywhere so please help! :)


Molecular spectroscopy for atomic spectroscopers

My experience is with atomic spectroscopy of alkali atoms, I've recently been asked by a friend to help with some advice on analyzing molecular spectrophotometry data in the context of molecular biology, To fit spectroscopic data to extract density in atomic spectroscopy you typically need to know the relative transition strengths and natural linewidths for all transitions (for hyperfine resolution spectroscopy), know the interaction path length, know the doppler broadening coefficient (and know the temperature), construct a voigt profile and fit. If I wanted to fit spectroscopic data to extract, say, ATP concentration from a spectrophotometer sweep, would I be able to do this by determining the corresponding molecular parameters? Are there caveats to single transition molecular spectroscopy that I should be aware of? It would seem that doppler broadening would be the largest contribution to the linewidth so it I may be able to fit to a simple gaussian profile instead of the more computational intense voigt. In atomic spectroscopy you usually measure a change in coherent beam intensity so the contribution from fluorescence of the atom can be made negligible in the low power unsaturated regime with polarization analyzers and other small trade-tricks. I'm not completely certain how spectrophotometers are set up with that regards, would fluorescence near the measurement frequency be something I need to take into account?


Falling chain with friction

I really need some help with a physics problem, and I guess my doubt is more conceptual, than question based. But still, let me pose it to you: > _A uniform chain of mass $M$ and length $L$ lies on a rough table, with > coefficient of friction $k$. When 1/3rd part of the chain is hanging off the > edge, it starts slipping. Find the work done by the frictional force on the > chain, at the instant the last link falls off._ Now, to find the work, I know we must use integration. The work done would be $$\int k \, p \, T \, g\, dp \tag{1}$$ Where $k$ is the coefficient of friction, $p$ is the varying length, $T$ is the mass per unit length ($= \frac{M}{l}$), and $g$ is the acceleration due to gravity. After applying appropriate limits, we get the answer. The integrand in $(1)$ simply should mean force $\times$ displacement. Now does my question come in. In the integrand, the Force quantity is defined by the variable frictional quantity, which is equal to $(k \, p \, T \, g)$, so where did the quantity of displacement of the chain go? Shouldn't there be an extra $p$ in the intgrand, such that it becomes $(k \, p^2 \, T \, g)$? Or is the notation $dp$ itself used to signify the displacement? I know the question is a bit complicated, but please help me out. It will be highly appreciated.


laser spectral width vs. linewidth

I have been racking my brains over the differences between laser spectral width and something called the linewidth. The linewidth was written about in detail by Henry in 1982. The spectral width is the width at -20dB down from peak of the wavelength spectrum of the laser. I am looking at some laser data right now that is saying that laser x has 10 kHz linewidth and 60 pm spectral width. You can convert spectral width from frequency to wavelength as in this article and I have done that calculation. By that calculation, a laser with a linewidth of 10kHz should have a wavelength width of about 1x10^-6 nm, not 0.06 nm. Linewidth is often measured with self-heterodyne technique, not a spectrum analyzer. What am I missing?


Matrix elements of momentum operator in position representation

I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} \psi(x)=-i\hbar\frac{\partial\psi(x)}{\partial x}$$ **(1)** Is it ok to conclude from this that: $$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}?$$ And what does this expression mean? **(2)** Using the equations: $$ \frac{\langle x | \hat{x}\hat{p} | x' \rangle}{x} = \frac{\langle x | \hat{p}\hat{x} | x' \rangle}{x'} = \langle x | \hat{p} | x' \rangle $$ and $$\langle x | [\hat{x},\hat{p}]|x'\rangle=i\hbar \delta(x-x')$$ one can deduce that $$\langle x | \hat{p} | x' \rangle = i \hbar \frac{\delta(x-x')}{x-x'}$$ Is this equation ok? Does it follow that $$\frac{\partial \delta(x-x')}{\partial x} = - \frac{\delta(x-x')}{x-x'}?$$


Observation of light bending spacetime

Has radiation or energy bending spacetime ever been observed? If not, is it likely that it ever will, assuming current technology? * * * **Note:** This is not a question of _space_ bending _light_ , but of _light_ bending _space_.


Limitations on the choice of axis of rotation regarding rolling wheels

Consider a situation where a wheel is rolling without friction on a level surface. Call the center of the wheel $C$, the point where the wheel contacts the ground $G$, and some arbitrary other point on the rim of the wheel $P$. The normal force has no torque when $G$ or $C$ is picked as the axis of rotation, but it does if $P$ is chosen for any point (minus the top of the wheel). If a physics problem then says to pick the force that gives the largest torque, how am I meant to choose if the axis of rotation choice changes the answer? The physics book I have states that $G$ or $C$ are points at which an axis of rotation may be placed because they both have the same angular acceleration around each other. This seems fine to me; I can visualize the concept of the $C$ rotating about $G$. The trouble is that it seems to be the same situation for any $P$. Working out the math, it does seem that every point on the rim chosen as the axis of rotation is such that the center of mass rotates about the chosen point at the same speed as the chosen point rotates about the center of mass. What is going on here? How does one 'get a feel' for these problems where it seems that the axis of rotation choice is so critical?


Thermodynamics - Sign convention

I use the sign convention:   * Heat absorbed by the system = $q+$ (positive)   * Heat evolved by the system = $q-$ (negative)   * Work done on the system = $w +$ (positive)   * Work done by the system = $w -$ (negative) Could anyone please tell me, that volume increased in system does positive or negative work?


Triangle inequality Clebsch-Gordan coeffcients

The Clebsch-Gordan coefficients can only be non-zero if the triangle inequality holds: $$\vert j_1-j_2 \vert \le j \le j_1+j_2$$ In my syllabus they give the following proof: $$-j \le m \le j$$ $$-j_1 \le m_ \le j_1$$ and $$ -j_2 \le m_2 \le j_2$$ For maximal values: $m = j$, $m_1 = j_1$ and $m_2 = j_2$ we get: 1) $-j_1 \le j-j_2 \le j_1$ which implies $j_2-j_1 \le j \le j_1+j_2$ 2) $-j_2 \le j-j_1 \le j_2$ which implies $j_1-j_2 \le j \le j_1+j_2$ Which should prove the triangle inequality. This proof looks really simple, but I don't completely understand it though. It seems that I'm missing some essential reasoning, and I can't find where. Why for instance do they take for $m_1$, $m_2$ and $m$ all maximal values? Can't I also take $m$ maximal and $m_1$ minimal? This would give bad results though. So I really don't understand it, and I hope that someone can clarify it.


Vasiliev Higher Spin Theory and Supersymmetry

Recently there is renewed interest in the ideas of Vasiliev, Fradkin and others on generalizing gravity theories on deSitter or Anti-deSitter spaces to include higher spin fields (utilizing known loopholes in the Weinberg-Witten theorem by including infinitely many higher spin fields and by working with asymptotic conditions that do not permit S-matrix to exist). There is also a conjecture for a duality for the theory as formulated in asymptotically AdS space with the O(N) vector model in the large N limit. So in this context I am curious if there are supersymmetric generalizations of the theory, and how much supersymmetry can be shown to be consistent with this set of ideas (given that the usual restriction to 32 supercharges comes from forbidding higher spin fields).


Total current of two sources in series?

Here is a question that a friend asked me. He had to an experiment in school and do some calculations afterwards. Those calculations require maximal current that the DC source can produce. He has measured the maximal currents for two separate sources but forgot to measure it for both connected in series. If maximal current of one battery is $a$ and maximal current of other battery is $b$, what will be the maximal current if one connects these batteries in series? I could imagine the equation system: $$I=\frac{\varepsilon_1 + \varepsilon_2}{r_1+r_2}$$ $$I_1=\frac{\varepsilon_1}{r_1}$$ $$I_2=\frac{\varepsilon_1}{r_1}$$ where $I$ is for current, $\varepsilon$ for electromotive force and $r$ for inner resistance. But it seems to have too many unknowns. And I'm not even sure if it's correct to describe the system in this way.


Virtual Particles and Causation

Sometimes when people debate what type of cause a universe with a beginning may have Virtual Particles has been used as an example of a thing that can arise without a cause. So my question would be what exactly is a virtual particle and do they really pop in and out of existence without a cause?


Darcy Flow in porous material - consider porosity in cross-section area?

According to Darcy's Law, the volumetric flow rate Q of a fluid occuring due to a pressure difference $\Delta p$ over a distance L and through a cross- sectional area A of a porous medium (volume V) is given by: $$Q = \frac{k\cdot A}{\mu}\cdot \frac{\Delta p}{L}\tag{1}$$ When calculating the pressure increase caused by that inflow of Q over a time $\Delta t$, it is typically considered for porous media that only a fraction of its total volume can actually be filled by the fluid, where the fraction of "free space" is characterized by the porosity $n = V_{free}/V_{total}$: $$\Delta p = K\cdot Q\cdot \frac{\Delta t}{n\cdot V_{real}}\tag{2}$$ My question now is: Why is the porosity considered in the last equation, but not in the cross-sectional area of the first equation? By the very same logic I could say, that only a fraction of the cross-sectional area is "open" for fluid to flow into, couldn't I? Especially since (1) is often used for the determination of material permeabilities, but always the total cross-sectional area is used for the calculation. This is just something that seems odd to me and maybe one of you guys has some insight on that and tell me where I'm thinking wrong.


Why is pressure gradient assumed to be constant with respect to radius in the derivation of Poiseuille's Law?

Poiseuille's Law relies on the fact that velocity is not constant throughout a cross-section of the pipe (it is zero at the boundary due to the no-slip condition and maximum in the center). By Bernoulli's Law, this means that pressure is maximum at the boundary and minimum at the center. But in the book I have it is assumed that the pressure gradient is independent of radius (distance from the center of the pipe), and the pressure gradient is thus extricated from a radius-integral. Can anyone justify this?


![enter image description here](http://i.stack.imgur.com/oar2l.jpg) How to derive the equation in my question?


Grain of sand attracting the sun?

My friend keeps telling me that according to physics... _"The sun attracts a grain of sand on the earth with the same force that the grain of sand attracts the sun"_ or _"A grain of sand on the earth attracts the sun"_ Is that true? Does in theory a grain of sand really attract the sun?


Modeling Syringes e.g. with the ideal gas law

Gentlemen I have a similar yet very practical problem that might provide further insight. I'm trying to model a moving plunger in a syringe (something like a piston in a cylinder). At time zero the plunger is at rest, the pressure (behind the plunger) is atmosphere and the volume zero. Then I open (hydrogen) gas which flows at a certain flow rate (which varies in time) and the pressure to move the plunger also varies in time. So my idea is to model using: $$\frac{\mathrm{d}}{\mathrm{d}t}(PV)=\frac{\mathrm{d}}{\mathrm{d}t}RT≡P\frac{\mathrm{d}V}{\mathrm{d}t}+V\frac{\mathrm{d}P}{\mathrm{d}t}=R\frac{\mathrm{d}T}{\mathrm{d} t}$$ If I assume constant temperature $P\frac{\mathrm{d}V}{\mathrm{d}t}+V\frac{\mathrm{d}P}{\mathrm{d} t}=RT$ Is this right? How do I go about solving this if I know the change in flow rate and pressure versus time? Note also that as the volume of gas increases so does m (the mass)....I'm puzzled...has anyone every modeled a syringe in detail?


Why not formulate Quantum Mechanics using Lagrangians?

As the title implies, why is it that the most common formalisms we use in quantum mechanics prefer to describe systems in the terms of a Hamiltionian instead of a Lagrangian? Is there some convenience to defining our systems one way over the other? Are there cases I'm not aware of where Lagrangian formalism is preferred?


Calculating the $J$ value for atomic terms, having a lot of trouble with this. Already attempted

I am trying to understand this, and want to be very very clear. This is a homework question but I **already attempted to answer it** , so please don't put this question on hold. The question > What atomic terms are possible for the electronic configuration ns1nd1? > Which term is likely to lie in the lowest energy? Here is my attempt at answering this question. Some of the things I understand like calculating Multiplicity, you do that by the following relationship: > $$S = s_1 + s_2, s_1 + s_2 - 1, \ldots , |s_1 - s_2|.$$ Then apply it to the multiplicity rule: > $$ 2S + 1$$ I got the values 1, and 3 for the multiplicity and understand that the maximum multiplicity is the one that lies in the lowest energy. I also understand that the maximum $L$ value corresponds to the lowest energy, and since $L = 2$ therefore the term Symbol will be D, and the supercript for S will be either 1, or 3. What I don't get is how to calculate the $J$ values, the $J$ values the solutions manual has are 3, 2, 1 but it didn't explain how they derived those values. Do you use the Clebsch-Gordan relationship for the $J$ values as well, as such? > $$J = L + S, L + S -1, \ldots |L - S|$$ Really having trouble. Can someone please explain this to me? **Here is my attempt:** ![enter image description here](http://i.stack.imgur.com/WYEsS.png) I don't understand how they obtained the $J$ values


Why p-wave superconductors are rare in nature?

I have the basic question that why so many superconducting materials are s-wave and d-wave pairing, but the p-wave superconductors are so rare in nature? An equivalent question may be that why topological materials are so rare in nature, but the answer to the this question may be more complicated then my first question.


Andromeda/Milky Way collision: How, and how accurately, can a galaxy's lateral velocity be measured?

Some sources suggest that the Andromeda Galaxy is likely to collide with our own in approximately 3 to 5 billion years. We can estimate the distance to the Andromeda Galaxy using various techniques, including measuring the apparent brightness of Cepheid variable stars; its distance is currently estimated to be about 2.5 million light-years. We can measure its radial velocity (i.e., the rate at which it's either approaching or receding from us) using Doppler shift. One source, the same Wikipedia article I linked to above, indicates that its radial velocity with respect to the Sun is about 300 km/s in our direction; another article says the radial velocity relative to our galaxy is about 120 km/sec, also in our direction. (Presumably the difference is due to the Sun's orbital motion around the core of the Milky Way.) But that's just the radial component of the velocity. Taking the 120 km/sec figure, it could be moving directly toward the Milky Way (more precisely, its core could be moving directly toward the core of the Milky Way) at 120 km/sec, or it could be moving at a 45° angle at about 170 km/sec, or any of a number of other possibilities. Without an estimate of the lateral component of the velocity, there's no way to be sure whether the collision will occur or not. I'm reasonably sure we can't measure the lateral velocity directly; 120 km/sec over a century would cause Andromeda to move only about 0.04 light-year (if my calculations are correct). And yet this Wikipedia article says: > The best indirect estimates of the transverse velocity indicate that it is > less than 100 km/s. with a reference to "Abraham Loeb, Mark J. Reid, Andreas Brunthaler and Heino Falcke The Astrophysical Journal, 633:894–898, 10 November 2005", but the link is invalid. So how can a galaxy's lateral velocity be measured, or at least estimated? How accurate can such an estimate be with current technology? Can we expect improvements in the near future?


Baryonic density in collision experiments

Does anyone know any way of estimating the net baryon density in collision experiments, e.g. in LHC, RHIC, or the upcoming ones at GSI-FAIR? I have comes across many hand-waving arguments, sample -   1. Electron-positron colliders start with $\rho_B = 0$, and since conservation laws have to respected, matter and anti-matter must be produced in equal amounts. Hence, barring inhomogeneities, LHC is essentially in the $\rho_B \approx 0$ regime.   2. Antikaon-proton interaction is known to be attractive, through various model-dependent calculations, so if we have a ${\bar K}-p$ collision experiment, it shall involve the formation of quasi-matter having a larger density than usual, and hence, via these kind of reactions, GSI-FAIR plans to study QCD matter at high densities (especially in the Compressed Baryonic Matter experiment), while it remains inaccessible at LHC. (Please correct me if the hand waving is inappropriate.) I am wondering, if there is any way of making this a little more calculation oriented, though I understand that a fully rigorous calculation would be very difficult. (I don't want that, too.) Also, is it possible to have any model-independent answers, since model- dependent answers can be found with a bit of googling, e.g. Phys. Rev. C 72, 034613 (2005), which even has the evolution of these densities in lead-lead collisions. Though these calculations are the order of the day, model- independent calculations, if possible, teach you much more.


Why does quantum zero point energy contribute negative mass to strings?

A string which doesn't have any kind of vibrations will have mass whose square is negative due to quantum zero point energy. But why does it contribute negative rather than positive mass to strings?


How do I predict volume loss due to evaporation when boiling water?

Suppose I have a pot with diameter $D$ containing a volume of water $V$, being heated by a flame under it. If the ambient air temperature is $T$ and relative humidity is $R$, how can I calculate the expected rate of loss due to evaporation over a period of time, $t$? Assume the period of time begins once the water begins boiling. **Edit:** For simplicity's sake, also assume the flame is just hot enough to get the water boiling.


Is speed an intensive property?

I remember being taught in elementary physics that while it makes sense to add volumes, masses, or heat, it makes no sense to add temperatures. As I wanted to use that to illustate some other issue, I checked it on wikipedia, and discovered the concept of intensive and extensive properties of materials and systems, which is apparently a century old, though I do not remember it being taught to me (it was not as old then :-). The concept seems to be particularly useful in material and systems thermodynamics. But I was wondering how far it extends, and I was keeping in mind my original problem of determining when adding values in a given unit (not necessarily a physical one) made sense or did not. I noticed the fact that the ratio of two extensive variables is intensive. So I started looking, a bit ramdomly I confess, at ratios, and the first that came to mind was speed. And I wondered whether adding speeds made sense. We do that all the time, so it should. But then, my initial problem was not about adding two quantities, but a long list of them (it was a database question). While adding a few speeds (or velocities) makes sense when I move in the bus, or analyse the motion of the Moon in the solar system, I cannot imagine it would ever make sense to add a list of a hundred or a thousand speeds (though computing the average would make sense, as it would for temperature). But I makes perfect sense to sum the masses of a thousand objects. So there is apparently something special, not quite right, about adding speeds. Of course, I know that slightly older results state that adding speeds is not done with simple addition, but I feel that my problem is elsewhere (though there may possibly be a connection). I am aware that temperature in some materials is related to speed of motions inside it, so I am not too surprised. But speed in general seems to go beyond that (sorry for the vagueness). Also speed is not listed by Wikipedia as an intensive property. So, my question is : why does it seem improper to add many speeds (or velocities)? I guess this must also be true of other physical quantities, and I am wondering what is the right way to look at this, and understand it. Is there a more general notion than intensive and extensive?


Rutherford's Gold Foil Experiment

Can anybody explain how Rutherford bombarded a 0.0004 cm thick gold foil? How did he put it in a photographic sheet? Wasn't the foil too thin to be held? How did he know that the atoms were deflected at various angles? Did he calculate every alpha particle's angle of deflection?


Variance of Nested Experimental Uncertainty

I have to find the uncertainty of a quantity $Q$ doing two mean values. For example for a set of parameters I measure ten times $Q$, I obtain a mean value $Q_1$ and variance ${\rm Var}(Q_1)$. Then for a different set of parameters I measure ten times $Q$ and obtain $Q_2$ and ${\rm Var}(Q_2)$ etc. At the end I compute the mean value which is the sum of $Q_i$ but these 10 from one set of parameter to the other are correlated so I don't know how to compute the variance. Put differently, I don't know how to compute the variance if I have two averages.


How does $F = \frac{ \Delta (mv)}{ \Delta t}$ equal $( m \frac { \Delta v}{ \Delta t} ) + ( v \frac { \Delta m}{ \Delta t} )$?

That's how it's framed in my Physics school-book. The question (or rather, the explanation) is that of the thrust of rockets and how the impulse is equal (with opposite signs) on the thrust-gases and the rocket itself. $( m \frac { \Delta v}{ \Delta t} ) = -(v \frac {\Delta m}{ \Delta t} ) = F_i$ I suppose it's a problem with how I see the transfer of impulse and exactly which part of the equation relates to which part of the physical world (gases, rocket). So we can start from there. _**Title equation:_** $F = \frac{ \Delta (mv)}{ \Delta t} = \left ( m \frac { \Delta v}{ \Delta t} \right) + \left ( v \frac { \Delta m}{ \Delta t} \right)$ _Grade:_ The equivalent of G-10 in the US.


Does an electomagnetic field "modulate" an electric field?

Lets imagine an electromagnetic wave is vertically polarized. If you ignore the magnetic component, the electric field lines would point upwards from the ground or downwards in a cyclical way and of course vary in field strength. Now lets say you attach a battery to two metal plates and you produce an electric field having an horizontal polarization/direction. Does the presence of an electromagnetic wave "modulate" the horizontally orientated electric field, in terms of it's strength and direction?


Why do we require the generators of $\mathrm{SU(N)}$ gauge theories to be $N \times N$ matrices?

I have often read that the generators for $\mathrm{SU(N)}$ gauge theories must be $N \times N$ matrices; see for instance these notes at the top of page 3: http://www.staff.science.uu.nl/~wit00103/ftip/Ch12.pdf. Why is this? I don't think this is necessary from a mathematical point of view. For instance, for $\mathrm{SU(2)}$ we can consider the $2 \times 2$ generators: \begin{equation} \begin{array}{ccc} \displaystyle J_{1/2}^1=\frac{1}{2}\begin{pmatrix} 0 & 1 \\\ 1 & 0 \end{pmatrix} \; ,& \displaystyle J_{1/2}^2=\frac{1}{2}\begin{pmatrix} 0 & -i \\\ i & 0 \end{pmatrix} \; ,& \displaystyle J_{1/2}^3=\frac{1}{2}\begin{pmatrix} 1 & 0 \\\ 0 & -1 \end{pmatrix} \end{array} \end{equation} However, also the following $3 \times 3$ generators satisfy the Lie algebra: \begin{equation} \begin{array}{ccc} \displaystyle J_{1}^1=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0\\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{pmatrix} \; ,& \displaystyle J_{1}^2=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\\ i & 0 & -i \\\ 0 & i & 0 \end{pmatrix} \; ,& \displaystyle J_{1}^3=\begin{pmatrix} 1 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & -1 \end{pmatrix} \end{array} \end{equation}


Topological quantum computation : Anyon model

Could someone tell me about Frobenius-Schur indicator and the associated cups and caps notation in context of anyon model. One possible reference could be Parsa Bonderson thesis which is freely accessible here: http://thesis.library.caltech.edu/2447/2/thesis.pdf


Why will choice of coordinates impose functional relations on the metric?

I am reading Steven Weinberg's _Gravitation and Cosmology_. On page 10 he says: > In $D$ dimensions there will be $D(D+1)/2$ independent metric functions > $g_{ij}$, and our freedom to choose the $D$ coordinates at will allows us to > impose $D$ arbitrary functional relations on the $g_{ij}$... Can anyone tell me why the choice of coordinates will impose functional relations on the metric?


Do electrons in multi-electron atoms really have definite angular momenta?

Since the mutual repulsion term between electrons orbiting the same nucleus does not commute with either electron's angular momentum operator (but only with their sum), I'd assume that the electrons don't really have a well- defined angular momentum (i.e., they do not occupy a pure $\left|lm\right>$ state). I would assume that the actual wavefunction is dominated by one such state compared to others, so it is approximately pure, but is there really a points in enumerating electrons according to their momenta, like the s, p, d, f and so on sub-shells?


How does battery cell size affect voltage drop for a fix current load?

For a fixed current load, will the voltage drop be larger in a small cell or a big cell battery? Why?


1-dimensional Ring geometry - Group of Translations

I considered a Ring-like one dimensional geometry. In this, if we fix an origin (at some point on the circumference), we can think of **set of all displacements along the circumference** to form a **vector space**. Now one vector can be denoted by (for some reasons that will become clear), $$ \left( \begin{array}{ccc} x \\\ 1 \end{array} \right) $$ Further one can obtain any other vector in the space by translating the vector, say $ x_0 \rightarrow x_0+a $. We can use the linear transformation : $$ T(a) = \left( \begin{array}{cc} 0 & a \\\ 0 & 0\end{array} \right) $$ such that $$ \left( \begin{array}{ccc} x + a \\\ 1 \end{array} \right) = \left( \begin{array}{ccc} x \\\ 1 \end{array} \right)+ T(a)\left( \begin{array}{ccc} x \\\ 1 \end{array} \right) $$ Now the set of all such linear transformations will form a group. Most important part of this transformation is that, if the circumference of the ring is some $L$, then the transformation $T(nL)$ where $ n \in \mathbb Z $ should not change the vector. Mathematically, $$ T(nL) \left( \begin{array}{ccc} x_0 \\\ 1 \end{array} \right) = \left( \begin{array}{ccc} x_0 \\\ 1 \end{array} \right) $$ Now my question is, with these definitions is the **group of Translations a Compact one** ? And if it is the generator of the translations will have some properties like angular momenta (although this is a generator of translations) ? PS : I hope I am not talking about rotations. I am just talking translations along the circumference of the circle.


Half wave plate and angular momentum

Given:   1. A half wave plate freely floating in space.    2. Circularly polarized light, falling perpendicularly to it. The plate changes polarisation of the beam to the opposite one. Therefore it receives angular momentum and starts to rotate. Where does energy comes from?   What would happen with a single photon passing through the plate?


Which is the heaviest present day lifter (rocket)? And is it comparable to the Saturn V rocket?

I know of the Ariane 5 ECA, the Delta IV rocket and a few more, but which of the present day's rockets is the top heavy lifter, say, to low Earth orbit (LEO)? Although it is not a certain fact, I would imagine that a very heavy lifter to LEO is also good for placing objects into geostationary transfer orbit (GTO) and could be a good candidate for out of Earth orbit flight (for instance a trip to the Moon).


twistor-spacetime correspondence

Could someone explain the correspondence between lines in twistor space and minkowski space-time points? a basic derivation would suffice


Normalizable and non normalizable modes of gauge fields in AdS/CFT

In Lorentzian AdS space there are both normalizable and non normalizable solutions and we also know (at least for scalar fields in bulk) what do they correspond to in the boundary. But I saw the calculation only for scalar fields. Can someone please give me a reference where people have calculated these modes for a gauge fields, say for a graviton field? McGreevy's lecture note says the relation $\Delta(\Delta-D)=m^2L^2$ gets modified to $(\Delta+j)(\Delta+j-D)=m^2L^2$ for form $j$ fields. Does this mean for other fields too the normalizable and non normalizable behavior remains the same: namely $Z_0^{\Delta_+}$ and $Z_0^{\Delta_-}$, as $z_0\rightarrow 0$ ($\Delta_{\pm}$ are two solutions of course)? How can that be?


Information escape from a black hole

_Is the following a possible scenario? If not, why not?_ Assume there is a supermassive black hole $Z$ isolated in inter-galactic space. Nearby and stationary relative to $Z$ is observer $A$. A number of light years away spaceship $X$ is travelling at, say, $0.25c$ directly towards the black hole and (eventually) $X$ falls directly into $Z$. $X$'s mass as observed by $A$ will increase as $X$'s velocity approaches $c$ as $X$ approaches $Z$'s event horizon. It seems that $X$'s velocity should be so close to $c$ as it encounters the event horizon of $Z$ that $X$'s apparent mass should cause it to have its own event horizon. I am assuming that $A$ will observe that $X$ becomes a black hole just prior to its event horizon merging with $Z$'s. Now assume that just prior to the merging of the event horizons a spaceship $Y$ is travelling a few light seconds behind $Z$ on the same path, but at about $0.5c$. Just prior to the merging of event horizons $Z$ radios $Y$ (to whom $Z$ will NOT appear to have an event horizon) and $Y$ relays this message to $A$. (Note: This could also happen in reverse, so a fast conversation may be possible.) _What is wrong with this, as it appears to anyone stationary with respect to $Z$, that $A$ and $Z$ are communicating, with information being exchanged from within a black hole?_


What is the difference between air pressure and atmospheric pressure?

I remember studying that air pressure and temperature are inversely proportional. Now I saw in book that "Atmospheric pressure decreases as we go higher and higher." But in height the temperature is less, and so the air pressure is high. But it is given atmospheric pressure increases with altitude. I googled and understood that **air pressure** and **atmospheric pressure** are different. But I can't understand how each is different from the other in properties.``


Is airspeed constant or cyclical when flying model aircraft in windy conditions?

I need help to settle an argument about aeronautics. Particularly model aircraft. It has been observed by some that when a model airplane flying with the wind turns back into the wind , some aircraft tend to pitch up and gain altitude indicating an increase in airspeed. Alternately, when flying into the wind, after turning around tend to lose altitude indicating a lower airspeed. Most model pilots say that this is an “illusion” or “pilot error” and that given no change in throttle or thrust the airspeed remains constant. I disagree! It is my theory that in the first case forward momentum is carried through the turn causing a momentary increase in airspeed and when the model turns to fly with the wind it takes a while for the model, now with a slower ground speed to get up to optimum air speed. Thus airspeed is not constant but cyclical. I realize that there are many factors involved and that some of my detractors base their belief on their training in full scale aircraft. Model aircraft tend to be smaller, have lower glide ratios and are often based on high performance aircraft designs as opposed to your typical civilian aircraft. Flying style is also quite different with faster scale speeds and sharper turns for the models. Am I right? Wrong? How do Newton's laws of motion relate to this?


Using Gauss's Law to calculate electric fields between plates

I have two earthed metal plates, separated by a distance $d$ with a plane of charge density $\sigma$ placed a distance $a$ from the lower plate. I want to derive expressions for the strength of the electric field in the regions between the plane and the top plate and the plane and the bottom plate. I'm not sure how to apply Gauss's Law to the given situation. I think the best Gaussian surface to use would be a cylinder(?), and then somehow integrate over the two regions. But this would suggest that the electric field between the charged plane and the bottom surface was independent of the total separation, $d$ of the two earthed plates, which I don't think is correct. I don't yet fully understand Gauss's Law, and need more practice with it. Please can someone point me in the correct direction, and give me a hint as to what I should be looking to integrate? With very many thanks, Froskoy.


What is the general formula for a trebuchet?

What I'm really looking for is a formula for a trebuchet that I can input the desired initial velocity after launch and mass of the object, and from that figure out how long the long arm, short arm, sling and pivot need to be, and how much mass I need to have for a counterweight. Does this exist?


Did Aristarchus take the radius of the Earth into account in calculating the distance to the Moon?

My text says that Aristarchus (310 BC – ~230 BC) measured the "angle subtended by the Earth-Moon distance at the Sun" ($\theta$ in the figure below) to establish the relative Earth-Moon and Earth-Sun distances. ![enter image description here](http://i.stack.imgur.com/YAlXy.png) I understand that he must, in fact have used the Moon-Earth-Sun angle, and then subtracted that from 90° to arrive at $\theta$; but how did he establish the Moon-Earth-Sun angle? The reference points for all three objects is their centers, yet what Aristarchus must have in fact measured was the angle between the Moon and the Sun at the _surface_ of the Earth. Did Aristarchus take this discrepancy into account in his calculations? If so, how?


Finding the charge density

I can't figure out how to find the charge density for the following problem: > A conductor is placed in an external electrostatic field. The external field > is uniform before the conductor is placed within it. The conductor is > completely isolated from any source of current or charge. > > Assume that at some point just outside the surface of the conductor, the > electric field has magnitude E and is directed toward the surface of the > conductor. What is the charge density n on the surface of the conductor at > that point? I'm supposed to express my answer in terms of $E$ and $\epsilon_0$. I know that Electric Flux = Normal Angle x Electric Field x cos(theta), but I don't know how to relate this to charge density? What am I missing to solve this problem? This is very basic, I know, but I'm stumped. The hints on Mastering Physics also aren't really helping.


Maxwell equations and Fourier decomposition

I'm currently working on maxwell equations and in order to lower the fields dimension, we perform a Fourier decomposition (according to $\theta$) due to the system symmetry. For any vector field $\mathbf{U}(r, \theta, z)$, we have, $$ \mathbf{U}(r,\theta,z) = \sum_\alpha \tilde{\mathbf{U}}^\alpha(t,z)e^{i\alpha \theta}.$$ Together with the usual Maxwell equations (in cylindrical coordinate system) we decide to work with metallic boundary conditions. Let $\Omega$ be our domain and $\Gamma$ its boundary, then, for the electric field, $$ \mathbf{E} \times \mathbf{n} = 0 \quad \text{on $\Gamma$}$$ Where $\mathbf{n}$ is the normal vector oriented outside $\Gamma$. Then, my question is how to get the boundary conditions verified by $\mathbf{\tilde{E}}$ instead of $\mathbf{E}$ ? Does it lead to $\mathbf{\tilde{E}} \times \mathbf{n} = 0$ ? An additional question would be : what about a non homegenous conditions such as $\mathbf{E} \times \mathbf{n} = f$ ? I know that's pretty dumb but I can't convice myself. Thanks in advance.


Why do we remember the past but not the future?

The question is sometimes referred to as the "psychological arrow of time" (Hawking, 1985). Here the past is understood as a moment or time when the entropy of the universe was lower, and contrarily for the future. So it is generally thought that PAOT is a consequence of the thermodynamic arrow of time of our universe. If so (maybe not?), how do the two relate? Some explanations in the literature:   1. Practical memory systems work in a way that the formation of new memories entails an overall increase of total entropy of the system and the environment. For example, to create a memory, i.e. to cause our neurons to orient in a particular fashion, requires energy which results in our body heating up a little bit, increasing the total entropy (Hawking, 1985 and 1994); The initialization of memory to make it reusable is an irreversible process that increases total entropy (Landauer, 1961. Wolpert, 1992).   2. More recently, People have argued that even reversible and non-dissipative memory systems are subject to PAOT (Mlodinow and Brun, 2014). The conclusion is arrived by imposing some constraints on what a memory system should be like. Specifically, they argue that a memory should be somehow robust to small microscopic changes in states of the system it records (what they call "generality" requirement). But the smallest changes in the future state destroy the thermodynamic arrow of time between now and the future. So any memory of the future of the system "could remember only one possible configuration of that system". This fine-tuning disqualifies it as a bona fide memory. My problem with explanation (1) is that even if it's correct, it doesn't seem to be a complete answer in itself. Yes, increase of (new) memory happens only as total entropy of the universe increases. So what? It doesn't have anything to say on the nature of that memory. Why couldn't it occasionally be a memory of the future for that matter? Explanation (2) leaves no such ambiguity. But the generality requirement seems artificial: surely a memory that records the only future configuration of the system remembers the future in a deterministic world, there being no "what ifs" regarding that state? Of course, my understanding of the problem is only preliminary. I'd like to know whether there is not some generally accepted explanation, or any other thoughts you have on it.


Etale bundles and sheaves

> Before answering, please see our policy on resource recommendation > questions. Please try to give substantial answers that detail the style, > content, and prerequisites of the book or paper (or other resource). Explain > what the resource is like as much as you can; that way the reader can decide > which one is most suited for them rather than relying on the suggestions of > others. Answers which just suggest a book or paper may be deleted. > > Also note that all answers to this question are automatically **community- > owned** , so they are often subject to major editing, often to make them > comply with the book policy. I am now going through Isham's book _Modern differential geometry for physicists_ and got stuck with the notions of _etale bundle_, _presheaf_ and _sheaf_. Could someone please suggest some other, more intuitive and more accessible references on etale bundles and sheaves, preferably the ones giving more motivation and sufficiently many explicit (and worked-out) examples and, preferably, accessible to theoretical physicists (i.e., not just mathematicians)? P.S. To make things clear, a few math texts I have managed to find so far like Godement's and Bredon's _Sheaf Theory_ (two books with the same title) seem way too tough for me. The part on sheaves in Arapura's _Algebraic Geometry over Complex Numbers_ is somewhat better but still a bit too fast going and with too few examples and not too much motivation. Pretty much the same applies to the part on sheaves (which is too brief anyway) in the Clay Institute volume _Mirror Symmetry_. If there are no suitable books, are there perhaps some good lecture notes on the subject accessible to physicists rather than just mathematicians, from which one get a reasonable intuition on sheaves and stuff?


An oscillating skateboard

A half pipe of a skateboard park consists of a concrete trough with a semicircular section of radius 5m, I hold a frictionless skateboard on the side of the trough, pointing down toward the bottom and release it; how long will it take to come to back to the point of release? Note: I am supposed to use Newton's laws and polar coordinates. The solution is already given in the book, I am just confused as to why $F_r$, the force in the radial direction, is equal to $mg\cos\theta$ minus the normal force (where $\theta$ is the angle from the origin of the half circle). Also, I am used to using the vector form of Newton's laws in polar coordinates and not using conservation of energy or free fall or anything else.


Collision of two photons

Could someone explain me how will be look like collision of two photons? Will they behave like:   1. Electromagnetic waves, they will interpher with each other and keep they wave nature   2. Particles and they will bounce like classical balls I assume that energy of that system is too small to make creation of pairs possible.


Redshift 1+z - CMB Temperature lower?

I know that $\frac{\lambda_2}{\lambda_1} = 1 + z$ Suppose a galaxy had redshfit $z=3$. Does this mean that the wavelength becomes $4\lambda$? Then by wien's law where $\lambda \propto \frac{1}{T}$, does this mean that the temperature now observed is $\frac{1}{4} \times 2.73 K$?


What is the apparent viscosity in shear thinning turbulent flow through a pipe?

The explanation of shear rate in laminar flow is straightforward: We imagine small layers of fluid that glide on each other. Now, in turbulent flow, this does not work as there are no layers. If I want to know the apparent viscosity of a shear thinning (or other non-Newtonian) liquid, I need to know the shear rate. I've asked about this before here, and received an answer. However, I can't solve the Navier Stokes equation, so someone has to walk me through it or present me with an answer. Fluid may be assumed to be a power-law fluid.


Sum of intensity of reflected and transmitted waves

**The given state** :   Let $\psi$ be a wave that passes from medium $a$ to medium $b$.   Let $A$ be the amplitude of $\psi$.   Let $R$ be the amplitude ratio of the reflected wave $\psi_r$ and the original one, and $T$ the amplitude ratio of the transmitted wave $\psi_t$ and the original one.      **The question** : What is the intensity of the transmitted wave?      **An attempt** :   The condition at the boundary of the two media demands that $1+R=T$.   The intensity of a wave is $I_\psi=|\psi|^2$.      On one hand we have $I_t\propto T^2=1+2R+R^2$.   But on the other hand, assuming that the intensity is preserved and that intensity is additive, we also have $I_t=I_\psi - I_r \propto T^2=1-R^2$.      Assuming that $R\neq 0$ there is a contradiction between the 2 answers.   Which one is the right answer?      Thanks.


Electron-Positron Scattering using the Feynman Rules - Integration Q

![enter image description here](http://i.stack.imgur.com/neYV0.jpg)   I'm doing independent studies on electron-positron scattering, specifically the annihilation diagram contribution to the M matrix in Bhabha scattering, and this is the equation I recovered with the following initial conditions: An electron, associated with external momentum p_1, and a positron, associated with p_2, annihilate to produce a virtual photon, associated with internal momentum q, that produces a positron-electron pair (p_4 and p_3, respectively). I'm just going to assume you have a running understanding of the Feynman Rules. These are the associated spinors for the particles involved in the interaction. $$u\to e^- enters\\\\\overline u\to e^- exits \\\ v \to e^+ exits\\\ \overline v \to e^+ enter$$ This is the term added when a vertex is reached, it is a 4 x 4 matrix for fermions. $$-ig_e\gamma^\mu$$ The recovered equation: $$ (2\pi)^4\int{[\overline u ^{(S3)} (p_3)(-ig_e\gamma^\mu)v^{(S4)}(p_4)}](\frac{-ig_{\mu \nu}}{q^2})[u ^{(S1)} (p_1)(-ig_e\gamma^\nu) \overline v^{(S2)}(p_2)]\ \delta^4(q-p_3 -p_4) \\\ \times \delta^4(p_1+p_2 -q)\ d^4q $$ where $g_e$ is the quantum-electrodynamic coupling constant. How do I reduce this down (I see that I can contract the index on $\gamma^\nu$) and integrate the derivatives of the Dirac delta function with arguments $q, p_1,p_2,p_3,$ and $p_4$? I've never come across having to integrate derivatives of the delta function. Is it possible to use Eq. (17) on http://mathworld.wolfram.com/DeltaFunction.html ? Any help is appreciated.


What physics does occur at short distances in QED?

Let us take the standard QED ($e^-, e^+, \gamma$) as a model of QFT and ask what is its "short-distance" physics? They say the UV infinities appear because we do not know the real physics of short distances and initially we introduce it wrong. OK, but after renormalizations, what physics does remain? Do we replace the unknown/wrong physics with certain/right one? Can anybody describe it without appealing to unphysical bare particles? Have we an idea about the real electron from QED? If so, why we cannot use it as the input to construct a reasonable theory from the very beginning? P.S. Moderators, please do not close my questions before they are answered, let people answer.


Can an object be infinitely small?

I read somewhere that the earth has to be smaller than 1 cm to become a black hole, according to Schwarzschild. Since big bang came from a singularity, I am wondering, is there any minimum volume for anything?


Formulas for compressibility of solids

I am taking a course in mechanics this semester, as well as a course in reservoir physics. Both courses have sections devoted to pressure/compressibility of solids, but the formulas look slightly different, so I wondered if they really mean the same or not. In my mechanics class I am told that: $$\Delta P = B \left(\frac{- \Delta V}{V_0}\right)$$ Where $B$ is a constant known as the **bulk modulus** of a given material. In my reservoir physics class I am presented with the formula: $$c = -\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$ Where $c$ is referred to as **isothermal compressibility**. So my question is - are these formulas basically the same, where $c = \frac{1}{B}$? And if they are not the same, can someone please explain the difference to me? I would really appreciate if someone could help me with this!


Underground explosions due to plate tectonics and natural gas pockets

I am not sure if anyone has ever researched this but I am curious about underground reservoirs of natural gas and plate tectonics. Specifically, as the Earth's crust gets pulled down to the mantle do pockets of natural gas get ignited and explode? Or do these pockets of gas get pushed to the surface or further back along the crust as the pressure increases? Given that we have such large underground reservoirs of natural gas that we have discovered I would think that some exist in or near subduction zones. I guess the same question could be asked of reservoirs of oil. But I would think that the natural gas deposits would be a bit more volatile.


Mechanics of Materials (pressure and temperature)

A solid right cylinder of rock core is surrounded by four rods made of mild steel (all-thread rods). The rods are placed equidistantly around the core in a square formation. The tops and bottoms of the four rods are fixed to the top and bottom rigid plates, but the rock is just fitted snugly between the top and bottom plates. The rock is wrapped in an impermeable jacket. The whole system is submerged into a pressure vessel holding water and the water pressure is increased. When the water in the pressure vessel is increased to a certain amount, the fluid in the pore space of the rock is increased (pore pressure) accordingly to give the rock pressure support resulting in an effective stress on the rock. After reaching final pressures the temperature of the pressure vessel and all its contents is then increased. Determine the following quantities: (a) the resulting load $P_s$ in the rock cylinder and $P_r$ in the steel rods; (b) the corresponding axial stresses $\sigma_s$, $\sigma_r$; and (c) the axial deformation $\delta$ of the assembly and (d) the principle stresses on the core. ![fixture](http://i.stack.imgur.com/ELknN.jpg) Below is my initial attempt at this problem. I assumed that lateral strain is insignificant but I don't know if that is true. If it is not true then I suppose this problem would have to be solved with iterations? I am hoping someone can tell me if the methodology is correct or how to correctly solve this problem. I've included some values to solve with, but I would ultimately like to have the equations/methodology to solve for any values of the system. Thanks. When trying to solve this problem I assumed the deformation of the rods are uniform throughout their volume, the rods are prismatic, the compressive load acts at the center of the rods, rods are homogeneous, isotropic, and behave linearly elastically. I made the same assumptions about the rock aswell. The bottom rigid plate is a circle with a diameter $d_p=15$ in., each rod has a diameter $d_r=0.4$ in., and the core has diameter $d_s=7$ in. The rods have a Young's modulus $E_r=30,000$ ksi and the rock has a modulus of elasticity of $E_s=1,450$ ksi. The starting length of both the rods and the core is $L= 20$ in. Before temperature is introduced to the system the entire fixture is subjected to the overburden water pressure $\sigma_{ob}=25,000$ psi and internal pore pressure $\sigma_{Pp}=20,000$ psi. Therefore the resulting load on the core and rods imposed by the plate is $$P = \sigma_{ob} \cdot A_{plate} + \sigma_{Pp} \cdot A_s=\sigma_{ob} \cdot \left(A_{plate}^{bottom}-A_{plate}^{top}\right) + \sigma_{Pp} \cdot A_s $$ $$=\sigma_{ob} \cdot \left(A_s+4A_r\right) + \sigma_{Pp} \cdot A_s $$ $$ = (-25\, \text{ksi}) \left(\frac{\pi}{4} \cdot (7 \, \text{in.})^2 + \pi \cdot (0.4 \, \text{in.})^2 \right) + (20\, \text{ksi}) \left(\frac{\pi}{4} \cdot (7 \, \text{in.})^2 \right) \approx -205 \, \text{k}$$ The plate is subjected to the force $P$ and the the unkown compressive forces $P_s$ and $P_r$; thus the equation of equilibrium is: $$\Sigma F_{vert} = 0$$ and $$4P_r + P_s - P = 0$$ This equation, which is the only nontrivial equilibrium equation available, contains two unkowns. Therefore, we conclude that the structure is statically indeterminate. Because the end plates are rigid, the rock cylinder and steel rods must shorten by the same amount. Denoting the shortenings of the rock and steel parts by $\delta_s$ and $\delta_r$, respectively, we obtain the following equation of compatibility: $$\delta_s = \delta_r$$ The changes in lengths of the cylinder and rods can be obtained from the general equation $\delta = PL/EA$. Therefore, in this problem the force- displacement relations are: $$\delta_s = \frac{P_sL}{E_sA_s}$$ and $$\delta_r= \frac{P_rL}{E_rA_r}$$ We can now solve simultaneously the three stes of equations to obtain the axial forces in the rock cylinder and steel rods: $$\frac{P_sL}{E_sA_s}=\frac{P_rL}{E_rA_r} \rightarrow P_r=P_s \frac{E_rA_r}{E_sA_s} \rightarrow 4P_s \frac{E_rA_r}{E_sA_s}+P_s-P=0$$ Solving for $P_s$ we have $$P_s=P \left(\frac{E_sA_s}{4E_rA_r+E_sA_s}\right)$$ Likewise, solving for $P_r$ we have $$P_r=P \left(\frac{E_rA_r}{4E_rA_r+E_sA_s}\right)$$ Substituting our values we can solve for the resulting loads: $$P_r=(-205 \, \text{k}) \left(\frac{(30000 \, \text{ksi})(\frac{\pi}{4}(0.4 \, \text{in.})^2)}{4(30000 \, \text{ksi})(\frac{\pi}{4}(0.4 \, \text{in.})^2)+(1450 \, \text{ksi})(\frac{\pi}{4}(7 \, \text{in.})^2)}\right) \approx -11 \, \text{k}$$ Similarily, plugging values in for $P_s$ we get: $$P_s \approx -161 \, \text{k}$$ The change in length of the core and rods due to the pressure increase will be: $$\delta_r=\delta_s=\frac{PL}{4E_rA_r+E_sA_s}=\frac{(-205 \, \text{k})(20 \, \text{in.})}{4(30000 \, \text{ksi})(\frac{\pi}{4}(0.4 \, \text{in.})^2)+(1450 \, \text{ksi})(\frac{\pi}{4}(7 \, \text{in.})^2)} \approx -0.057 \, \text{in.}$$ At this point the temperature is increased to $400\,^{\circ}\mathrm{F}$. The thermal strain relation $\epsilon_T$ induced on a body is $$\epsilon_T = \alpha (\Delta T)$$ where $\alpha$ is the coefficient of thermal expansion and $\Delta T$ is the change in temperature. We assumed $\alpha_r = 6.5 \times 10^{-6} \,^{\circ}\mathrm{F}^{-1}$ and $\alpha_s = 3 \times 10^{-6} \,^{\circ}\mathrm{F}^{-1}$. The equivalent stress induced due to thermal expansion is: $$\sigma_{eq} = E\alpha(\Delta T)$$ And the equivalent load can be calculated as: $$P_{T} = \sigma_{eq} \cdot A$$ Substituting values we can solve for the equivalent loads due to temperature increase in the rods and rock: $$P_T^s = (1450 \, \text{ksi})(3 \times 10^{-6} \,^{\circ}\mathrm{F}^{-1})(400\,^{\circ}\mathrm{F})(\frac{\pi}{4}(7 \, \text{in.})^2) \approx 67 \, \text{k}$$ $$P_T^r = (30000 \, \text{ksi})(6.5 \times 10^{-6} \,^{\circ}\mathrm{F}^{-1})(400\,^{\circ}\mathrm{F})(\frac{\pi}{4}(0.4 \, \text{in.})^2) \approx 9.8 \, \text{k}$$ We then add these loads to the loads due to compression to get the final loads: $$P_r + P_T^r = -11 \, \text{k} + 9.8 \, \text{k} \approx -1.1 \, \text{k (compression)}$$ $$P_s + P_T^s = -161 \, \text{k} + 67 \, \text{k} \approx -94.4 \, \text{k (compression)}$$ With these values we can calculate the axial stresses: $$\sigma_s = \frac{P_s}{A_s} = \frac{-94.4 \, \text{k}}{\frac{\pi}{4}(7 \, \text{in.})^2} \approx -2.45 \, \text{ksi}$$ $$\sigma_r = \frac{P_r}{A_r} = \frac{-1.1 \, \text{k}}{\frac{\pi}{4}(0.4 \, \text{in.})^2} \approx -8.76 \, \text{ksi}$$ We now solve for the resulting length change in the rods and core do to the temperature increase: $$\delta_r=\delta_{r}^i+\alpha_r(\Delta T)L'= -0.057 \, \text{in.} + (6.5 \times 10^{-6} \,^{\circ}\mathrm{F}^{-1})(400\,^{\circ}\mathrm{F})(20 \, \text{in.} - 0.057 \, \text{in.}) \approx -0.006 \, \text{in.}$$ $$\delta_s=\delta_{s}^i+\alpha_s(\Delta T)L'= -0.057 \, \text{in.} + (3 \times 10^{-6} \,^{\circ}\mathrm{F}^{-1})(400\,^{\circ}\mathrm{F})(20 \, \text{in.} - 0.057 \, \text{in.}) \approx -0.034 \, \text{in.}$$ From this result we can conclude that the expansion in length of the rods due to the temperature increase causes a seperation between the bottom core face and the bottom fixture plate. This result may have came about due to incorrect assumptions on the values for the young's modulus and/or coefficient of thermal expansion. This separation will cause a fluid connection between overburden pressure and pore pressure which is not acceptable. If the resulting axial stress on the rock is to be a 1000 psi less than the resulting rock lateral stress, what would be the required material properties for the rock and rods?


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