title: Linearizing Gravity to ${\cal O}(h^3)$

text: I've seen the action of linearized gravity in many places. We basically have $${\cal L} ~\sim~ \frac{1}{G_N}\left( - \frac{1}{2}h^{\alpha\beta} \Box h_{\alpha\beta} + \frac{1}{4} h \Box h + {\cal O}(h^3)\right)$$ in the gauge where the trace-reversed field is divergenceless. I'm doing some field theory, on linearized gravity backgrounds by treating $h_{\mu\nu}$ as a massless spin-2 field. I can't seem to find the ${\cal O}(h^3)$ terms in the Lagrangian anywhere. I know how to evaluate it, but it looks nasty. Are there any known references that just lists the next to leading order terms in the above Lagrangian?


title: Diffraction-Long versus short waves

text: Why is it that long waves diffract more than short waves? I know that refraction has a quantum mechanical explanation, does diffraction have a similarly complex explanation? My teacher in high school said that using ocean waves would help me understand-but it didn't. Is there something in the math, or some related physics concept that can help me better understand diffraction? Related Question: Bass and Treble-Car Steroes


title: Does a hydrogen atom today have same mass as a hydrogen atom in the future?

text: Does an atom of hydrogen today have the same rest mass energy as an atom of hydrogen a billion years in the future? Standard cosmology seems to tacitly make this assumption. But surely one can only make this assumption if the FRW metric has a time-like Killing vector? From V.Moretti's answer to my previous post the FRW metric does not have a time-like Killing vector so that we cannot assert that the energy of an atom of hydrogen is constant over cosmological time. Is this reasoning correct? I'm sorry to keep bringing this question up but I think it's an important point that is overlooked in current cosmology. **Answer to responses** I guess I am following the spirit of Sean Carroll's Energy is not conserved Blog post where he says: > When you thought energy was conserved, there was a reason why you thought > that, namely time-translation invariance. A fancy way of saying “the > background on which particles and forces evolve, as well as the dynamical > rules governing their motions, are fixed, not changing with time.” But in > general relativity that’s simply no longer true. Einstein tells us that > space and time are dynamical, and in particular that they can evolve with > time. When the space through which particles move is changing, the total > energy of those particles is not conserved. > > Energy and momentum evolve in a precisely specified way in response to the > behavior of spacetime around them. If that spacetime is standing completely > still, the total energy is constant; if it’s evolving, the energy changes in > a completely unambiguous way. According to the FRW metric spacetime is evolving so surely the energy of particles cannot be assumed to be constant? A hydrogen atom today does not necessarily have the same energy as it did a billion years ago or will have a billion years in the future. Taking the spatially flat FRW metric in Cartesian co-ordinates for simplicity we have: $$ds^2 = -dt^2 + a^2(t)(dx^2+dy^2+dz^2)$$ Now from this representation of the FRW metric one might conclude that it is only the spatial co-ordinates that are evolving. In fact we can transform to conformal co-ordinates: $$ds^2 = a^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ where $d\tau = dt/a(t)$. Now we can clearly see that spacetime is evolving so we cannot assume that the energy of massive particles is constant. However according to V.Moretti massless particles like photons that are described by conformally invariant fields do not have their energy changed by a conformally changing metric as described above. Thus we can rigorously argue that photon energy is constant if we are in fact moving forward in conformal time $\tau$. So an increasing length scale is associated with a constant energy. Thus a massive particle with a _fixed_ length scale is associated with an energy that increases with the scaling factor $a(\tau)$. In seems that the energies of all massive particles must scale with $a(\tau)$ including the Planck mass. In fact I suppose as all particle masses are ultimately linked to the Planck mass then the the scaling particle masses is ultimately caused by the Planck mass increasing.


title: volume of phase space of composite microcanonical ensemble

text: Let $N_1$ be number of particles in volume $V_1$ with momenta and coordinates $(p_1, q_1)$ and $N_2$ be particles with momenta and coordinates $(p_2, q_2)$ in $V_2$, if $E_1, E_2$ be the energies of subsystems satisfying $E< E_1 + E_2 < E + 2 \Delta$ with $\Delta << E$. My book says that > Obviously the volume of the region of $\Gamma$ space that corresponds to > conditions with total energy lying between $E_1 + E_2$ and $E_1 + E_2 + 2 > \Delta$ is $\Gamma(E_1) \times \Gamma(E_2)$ Why is the volume of phase space multiplied? Shouldn't it be added?


title: Can a fundamental particle black hole with conserved charge emit Hawking radiation?

text: Let's says there is a fundamental particle:   * That is so massive that it is a black hole by itself (Compton wavelength < Schwarzschild radius)    * That carries a conserved quantum number (e.g. charge of an exotic interaction) which no lighter particle carries Would it be able to emit Hawking radiation? If not, does it contradict with the classical arguments (entropy analogy, pair creation at the horizon etc.) regarding the origin of Hawking radiation?


title: Are galaxies "disk" shaped?

text: When you look a sphere from a fixed observation point, you can easily mistake it for a circle, so I was wondering: are galaxies really "disk" shaped or we just don't have the means to detect the "$z$-axis" of them?


title: Equation for null geodesic around schwarzschild metric?

text: I'm trying to find the path of a photon around the Schwarzschild black hole, given its initial conditions. After much tribulation, I've basically given up on solving the equations by myself. I just need equations which I can integrate for: $$ r(\lambda) = \text{function(initial direction, initial position)} $$ $$ \theta(\lambda) = \text{function(initial direction, initial position)} $$ $$ \phi(\lambda) = \text{function(initial direction, initial position)} $$ $$ t(\lambda) = \text{function(initial direction, initial position)} $$ given the initial conditions of the light ray. Surely this has been solved before?


title: Is darkness really light?

text: According to this wikipedia article > "Consequently, most objects that absorb visible light reemit it as heat. So, > although an object may appear dark, it is likely bright at a frequency that > humans cannot perceive." Can someone explain this to me in lay man's terms? Does this mean that darkness is really light?


title: Does decrease in temperature affect mass $E=mc^2$?

text: My understanding of Quantum physics and String Theory is very basic and I don't yet have a grasp on the maths, but in my research I have come up with a question. Does a decrease in temperature also signify/create a decrease in mass? I am quite willing to believe the concepts I think I am beginning to understand are totally misconceptions on my part. But, as I see it mass is a product/mark of disruption to the Higgs field as a particle moves. If these particles are ultimately composed of vibrating Strings, their mass is given by the rate at which they vibrate and how it interacts with the Higgs field. I don't have any idea if, as the temperature (as we know it) approaches 0k, the string/quark/sparticle would change its state in any way or if the changing of the state of the String would change what type of particle it represents. It seems I'm more full of questions than answers, but that's why I love these topics! I know what I'm learning is new if the best and brightest are still working on the 'basics' to just make their systems work :).


title: Could false vacuums potentially describe, in part, the Big Bang?

text: I've just read about something I had never of before, false vacuums. After reading a couple descriptions of what a drop (or rather, a _vacuum metastability event_ ) would imply, I thought of the Big Bang. I realise this might be a very ignorant idea, but is there any chance the Big Bang could have been one such change of state? If not, what do we know about one or the other that would make this a definite no?


title: Origin of Charge Neutrality

text: What is the origin of the "charge neutrality" requirement in solids? Why do we require the bulk to be charge neutral, yet the surface can have a net charge?


How much of the mass of a supermassive black hole comes from dark matter? Assuming cold dark matter exists and that it doesn't bank up and get pushed away from black holes like some baryonic matter. Assuming the supermassive BH is one of the first in the universe and has been sucking in DM since then.


title: Question about Majorana fermions

text: I have a few questions about Majorana fermions.   1. What is Majorana mass? Does it have a different value compared to the mass in the Dirac equation for an arbitrary fermion? How exactly do they differ?   2. Can the Majorana equation be rewritten in form two-component spinor equation? Or it is two-component already?


title: Spin tensor from Noether theorem and spin tensor from Pauli-Lubanski vector

text: **Spin 3-vector directly from Noether theorem** Let's have one of applications of Noether theorem: the invariance of action under Lorentz group transformations leads to conservation of tensor $$ \tag 1 J_{\mu , \alpha \beta} = x_{\alpha}T_{\mu \beta} - x_{\beta}T_{\mu \alpha} + \frac{\partial L}{\partial (\partial^{\mu}\Psi_{k})}Y_{k, \alpha \beta} = L_{\mu , \alpha \beta} + S_{\mu , \alpha \beta}. $$ Here the second summand is called spin tensor. The conservation law $\partial^{\mu}J_{\mu , \alpha \beta} = 0$ leads to conservation in time the following tensor: $$ \tag 2 J_{\alpha \beta} = \int d^{3}\mathbf r J_{0, \alpha \beta}. $$ The second summand of $(1)$ after integration $(2)$ gives spin vector. For example, in Dirac theory we have $\hat{\mathbf S}_{i} = \frac{1}{2}\varepsilon_{ijk}S^{jk} = \frac{1}{2}\int d^{3}\mathbf r\Psi^{\dagger} \Sigma \Psi $. The value $S_{\alpha \beta} = \int S_{0 , \alpha \beta}d^{3}\mathbf r$ isn't conserved in general. **Spin 4-vector (Pauli-Lubanski vector)** It can be shown that quantity $$ W_{\mu} = \frac{1}{2}\varepsilon_{\mu \nu \alpha \beta}J^{\nu \alpha} P^{\beta} $$ refers to eigen angular momentum, and also it is translational invariant. It is conserved in time if $J_{\mu \nu}, P_{\alpha}$ are also conserved (while $S_{i}$ isn't). **The question** Whichever characterizes the spin truly?


title: Single photon double slit experiment

text: In the question Double Slit experiment with just one photon or electron, one of the answers says > There have been experiments recently where one can detect which the slit the > particle went through and still the interference pattern appears. This had 6 votes. Is it correct? If you put a detector to find out which way the particle went through won't that destroy the interference?


title: Lorentz symmetry and Noether's theorem

text: I'm trying to overcome some misunderstanding that I have in Noether's theorem. There is formula in David Gross's Lectures on QFT for Noether's theorem: $$J^\mu_\alpha=\mathcal{L}X^\mu_\alpha+\Pi^\mu_i\left[\Psi_{i\alpha}-\partial_\nu\phi_i X^i_\alpha\right]$$ $J^\mu_\alpha$ - conserved vector current, $\Pi^\mu_i$-canonical momentum, $X^\mu_\alpha$ and $\Psi_{i\alpha}$ are generators of symmetry which are defined by $$x'^\mu=x^\mu+X^\mu_\alpha \omega^\alpha,$$ $$\phi'(x'^\mu)=\phi(x^\mu)+\Psi_{i\alpha} \omega^\alpha,$$ Particularly I'm working out Lorentz symmetry: $$x'^\mu=\Lambda^\mu_\nu x^\nu$$ Infinitesimal version of this transformation is: $\Lambda^\mu_\nu=\delta^\mu_\nu+\omega^\mu_\nu$, where $\omega^\mu_\nu=-\omega^\nu_\mu$ is antisymmetric. **So, my problem that I dont understand why** (they write this expression in many sources, not only in Gross) $$X^{\mu,\alpha}_\beta=\delta^\mu_\alpha x^\beta-\delta^\mu_\beta x^\alpha$$. If I'm calculating $X^{\mu,\alpha}_\beta\omega^\beta_\alpha$, I get $2\omega^\mu_\nu x^\nu$, not the $\omega^\mu_\nu x^\nu$. Also I've done this calculation: $$f(x'^\mu)=f(x^\mu+\omega^\mu_\alpha x^\alpha)\approx f(x^\mu)+\omega_{\alpha \mu}x^\alpha \partial^\mu f = f(x^\mu) + \frac 1 2 \omega_{\alpha \mu} \left(x^\mu \partial^\alpha-x^\alpha \partial^\mu \right)f $$ If $f=x^\mu$ (coordinate function), then $$X^{\mu,\alpha}_\beta=\frac 12\left(\delta^\mu_\alpha x^\beta-\delta^\mu_\beta x^\alpha\right).$$ _What exactly am I doing wrong?_


title: From Paris to ... London

text: (Excuse the pun in the title, couldn't resist) Paris and London are connected by a straight underground tunnel, as shown in the diagram below. A train travels between the two cities powered only by the gravitational force of the Earth. Find the maximum speed of the train and the time taken to travel from Paris to London. The distance between the two cities is $d$ and the radius of the Earth is $R$. Neglect friction. $x$ is the horizontal distance from Paris to the location of the train. We are inside the Earth, so the gravitational force is not constant. The distance of the train from the centre is $r$ and the gravitational force is $-mg r/R$. The component of this that accelerates the train is $-mg r/R \sin \theta$ if $\theta$ is the angle between the line segment $h$ and the position vector of the train. This gives the differential equation $$\frac{dv}{dt} = v \frac{dv}{dr} = -g \frac{r}{R} \sin \theta$$ Now, $r^2 = h^2 + r^2 \sin^2 \theta$ from the diagram, which means $\sin \theta = \sqrt{1-h^2/r^2}$ . Plugging this into the integrand and simplifying yields $$v \frac{dv}{dr} = -\frac{g}{R} \sqrt{r^2 - h^2}$$ so that $$\int_0^{ v} \tilde v d \tilde v = -\frac{g}{R} \int_R^{\sqrt{(x-d/2)^2 + h^2}} \sqrt{r^2 - h^2} dr$$ which may be solved by a hyperbolic substitution. ![enter image description here](http://i.stack.imgur.com/bC2bB.png) Do I have the set up right? When I try to put in the limits, the computation starts to get very messy. Thanks for any pointers.


title: Speed=mass=time?

text: 1. I was wondering why objects gain mass with speed?    2. How can we measure speed if there is no truly stationary point to measure from?    3. And if an object comes to a stand still will it loose its mass?    4. Also how would time be affected both at (near) light speed and at a complete stand still.    5. One more thought. Could we reach the speed of light as viewed from Earth if we set the center of the universe as our destination?    6. As I understand it we are moving away from this point quit fast and how do we know this point is not moving it self?


title: Is a compact universe consistent with the results of (for example) the Michelson-Morley experiment?

text: If the universe is compact then there is a twin paradox that is resolvable only by selecting a preferred inertial reference frame (arXiv). I was under the impression that the lack of a preferred inertial reference frame was an empirical result (e.g. from early experiments designed to measure Earth's movement against the aether). But if I've understood these first points correctly then we should not still be entertaining the notion of a compact universe at all, as in the top voted answer to this question. So: Is a compact universe theory still viable? If so, what might account for the lack of evidence supporting a preferred inertial reference frame (or is there actually now evidence of a preferred frame)?


title: Some objects seem to have the same color whether the light that we perceive is emitted or reflected

text: Is color only a property of perception, considering these two examples:   1. The glass used in a green traffic light looks green no matter how it is illuminated, either by a white light bulb behind it or a white light bulb in front and   2. A green LED (to stay with the same color) emits a green color and receptors in our eyes that are excited by green light are stimulated and we eventually perceive this emitted light as green. Does not the correspondence between the color of the objects(LEDs) and our perception indicate that some property of color is a part of the "real" world?


title: Calculate minimum energy of incident neutrino using Mandelstam variables

text: I am studying the following nuclear reaction: $v + \tilde{v}\rightarrow Z^0$ where the antineutrino is motionless and has a given mass. The $Z^0$ boson has also a known mass. I'm trying to calculate the minimum enegy of the incident neutrino for the reaction to take place. I am using the Mandelstam $s$ variable as following: $s=(p_v+p_{\tilde{v}})^2=p_v^2+p_{\tilde{v}}^2+2p_v p_\tilde{v}$ At this point I need to make some assumptions that I'm not so sure about: $p_v^2 = E_v^2-m_{0,v}^2 \simeq E_v^2$ Given than the energy of the incident neutrino is much bigger than its resting mass. What about the antineutrino? Is it ok to write $p_\tilde{v}^2 \simeq E_\tilde{v}^2$ ? And what about the $s$ variable for the output of the reaction? Based on the definition it's: $s=p_{Z^0}^2=\left(\sum{E_i}\right)^2 - \left(\sum \vec{p}_i\right)^2$ In the coordinate system of center of mass it's $\left(\sum \vec{p}_i\right)^2=0$. Therefore: $s=\left(E_{Z^0}^{cm}\right)^2$ Since the variable $s$ is invariant: $E_v^2 + E_{\tilde{v}}^2 + 2E_vE_\tilde{v}=\left(E_{Z^0}^{cm}\right)^2$ I'm looking for the least energy $E_v$.   1. Are my assumptions for the approximations correct ?   2. How do I proceed from this point on?


title: The Ward identity for EM amplitude with massive vector boson

text: Let's have theory of massive vector boson interacting with EM field: $$ L = |D_{\mu}W_{\nu} - D_{\nu}W_{\mu}|^{2} + m^{2}|W|^{2}, \quad D_{\mu} = \partial_{\mu} - ieA_{\mu}. $$ The question: how to show that this lagrangian involves the consequence of the Ward identity $q_{\mu}M^{\mu ...} = 0$ on the Feynman diagramms level (so here is possible to emit the longitudinal photons in $\zeta$-gauge)? Particularly (for a given process in a given order of perturbation theory) I may only get the expression for the vertex and then act on it by photon momentums. But I don't know how to get the non-perturbative result.


title: speed of gas molecules when they escape to space

text: I wonder if the escape velocity of gas molecules is the same as the one of rockets, which is 7.8km/s. And do oxygen escape at all or not? Are they too heavy?


title: Water bottle moment of inertia

text: I've noticed that I can make a full water bottle spin about its short axis easier than I can make it spin when it is 1/4 or 1/2 full. Also, when it is spun and is not full, the geometric center of the water bottle moves in a bizarre way (kind of circular). In general the moment of inertia is directly proportional to the mass. So why is this the case? I can see that the water in the bottle has more freedom to move when it is not full than when it is full, and this affects the spin. My idea is that it maybe has something to do with the fact that the center of mass of the system is not fixed to a certain point relative to the bottle when this bottle is not completely full. But I'd like to see a mathematical explanation of this phenomenon.


title: What is the relationship of the curl of a vector with its conservativeness?

text: My Physics teacher told me that a conservative field has zero curl.But I am not getting the logic behind it...I am even not clear why the curl represents how much the vector field swirls around the point in consideration...Please explain using mathematics and physical meanings...


title: Derivation of Van der Waal's Equation from Helmholtz Free Energy

text: From the following equation $$ f(T,v) = f_{id}(T,v) - RT \ln\bigg(1-\frac{b}{v}\bigg)-\frac{a}{v}$$ how do I derive Van der Waal's equation. In the equation above, $f$ and $f_{id}$ are the Helmholtz Free Energy of Van der Waal's gas and ideal gas, respectively. Moreover, $ a $ and $ b $ are constants, $ v $ is the volume, $ T $ is the temperature.


title: (A,B)-Representation of Lorentz Group: Coefficient functions of fields

text: I have a question regarding the construction of general causal fields in Weinberg's book on quantum field theory. In his conventions a field that transforms according to the irreducible (A,B) representation of the Lorentz group is given by (eq.5.7.1) \begin{equation} \psi_{ab}=(2\pi)^{-3/2}\sum_{\sigma}\int d^3p\left[\kappa a(\boldsymbol{p},\sigma)e^{ipx}u_{ab}(\boldsymbol{p},\sigma)+\lambda a^{c\dagger}(\boldsymbol{p},\sigma)e^{-ipx}v_{ab}(\boldsymbol{p},\sigma)\right]\, . \end{equation} Here, $a$ and $a^{\dagger}$ are the usual creation and annihilation operators, $u_{ab}$ and $v_{ab}$ are coefficients carrying an irreducible representation of the Lorentz group, and $\kappa$ and $\lambda$ are coefficients. The zero-momentum coefficients $u_{ab}(0,\sigma)$ have to fulfill the conditions \begin{equation} \sum_{\bar{\sigma}}u_{\bar{a}\bar{b}}(0,\bar{\sigma})\boldsymbol{J}^{(j)}_{\bar{\sigma}\sigma}=\sum_{ab}\mathcal{J}_{\bar{a}\bar{b},ab}u_{ab}(0,\sigma) \end{equation} \begin{equation} -\sum_{\bar{\sigma}}v_{\bar{a}\bar{b}}(0,\bar{\sigma})\boldsymbol{J}^{(j)*}_{\bar{\sigma}\sigma}=\sum_{ab}\mathcal{J}_{\bar{a}\bar{b},ab}v_{ab}(0,\sigma), \end{equation} where $\boldsymbol{J}^{(j)}_{\bar{\sigma}\sigma}$ are the angular momentum matrices in the$j$- representations of the rotation group, and $\mathcal{J}_{\bar{a}\bar{b},ab}v_{ab}(0,\sigma)$ are the angular momentum matrices in the $(A,B)$ representation of the Lorentz-group. Weinberg shows that $u_{ab}(0,\sigma)$ is given by \begin{equation} u_{ab}(0,\sigma)=(2m)^{-1/2}C_{AB}(j\sigma;ab)\, , \end{equation} where $C_{AB}(j\sigma;ab)$ is the Clebsch-Gordan coefficient and the normalization was chosen for convenience. However, when I try to calculate the coefficient $u_{ab}$ in the $(1/2,1/2)$ representation and want to relate them to the $u^{\mu}$ obtained when working directly in the vector representation of the Lorentz group I cannot reproduce them. , where \begin{equation} u^{\mu}(0,\sigma=0)=(2m)^{-1/2}\begin{pmatrix}0\\\0\\\0\\\1\end{pmatrix}\qquad u^{\mu}(0,\sigma=1)=-\frac{1}{\sqrt{2}}(2m)^{-1/2}\begin{pmatrix}0\\\1\\\\+i\\\0\end{pmatrix} \end{equation} \begin{equation} u^{\mu}(0,\sigma=-1)=\frac{1}{\sqrt{2}}(2m)^{-1/2}\begin{pmatrix}0\\\1\\\\-i\\\0\end{pmatrix}\, . \end{equation} What is the procedure of translating from (A,B) to a mixture of Lorentz indices and spinor indices in more general cases, such as in the Rarita-Schwinger field?


title: How does a vertical counter-weight system work?

text: How does a vertical counter-weight system work when you have variable weight on one side? From what I understand from this page elevators use a counter weight to counter-balance the cart and some impressive break systems to stop the elevator in emergencies. I'm assuming the motor then handles the additional weight of the passengers. However, some standing desks offer the ability to adjust the desk height, and in such a case no motor or lock-system is used. The one in the link boasts with "quick, single-motion adjustment—no knobs, levers or motors" and "solid platform remains stable during heavy data entry." Do they rely on friction alone or is there some mechanic I'm missing? I'd like to understand the physics behind this and will appreciate it if someone can explain it to me.


title: Do Magnets\ Magnetic field become weaker when exposed to para magnetic objects?

text: I was wondering whether magnets (permanent) become weaker as they are exposed to para-magnetic object? I was thinking this after seeing this: http://www.adamsmagnetic.com/blogs/2012/factors-which-cause-permanent-magnets- to-lose-strength-or-to-demagnetize/ I am buying an magnet and wish to know whether when magnet (permanent) looses its Magnetic domains structure or alignment when exposed to para magnetic materials? I don't think so however since I have no knowledge about this I am left asking for an answer. Thanks,


title: Does $E = mc^2$ apply to photons?

text: Photons are massless, but if $m = 0$ and $E = mc^2$ then $E = 0c^2 = 0$. This would say that photons have no energy, which is not true. However, given the formula $E = ℎf$, a photon does have energy as ℎ is a constant not zero and photons with a zero frequency do not exist. What am I missing here, or does the famous formula not apply to photons?


title: Photons, no energy?

text: I am only 14, so I don't know much about physics, and I would appreciate help on this topic. A photon, is said to have 0 mass. However, they carry energy, as electromagnetic radiation. Albert Einstein proved that $E = mc^2$. But if you plug the data in, $E = 0\times (299791)^2$, you try to multiply the speed of light squared by $0$. Therefore, the energy of a photon would be $0$. How is that possible? Since all the photon does is carry energy?


title: What is the correct way to take out equivalent capacitance here?

text: ![enter image description here](http://i.stack.imgur.com/XH3AP.jpg) So here I have a capacitor with two plates , one positively charged and one negatively, as shown in the figure. Now the material between them is $k_1$ for thickness d/2 and $k_2$ for the other d/2. There is no plate or anything between the separation. We have to find the equivalent capacitance. What my school teacher did was that she took them in series and drew them as two capacitors with one which has medium $k_1$ in between and the other with $k_2$ in between. But I think this is wrong. I took it as a capacitor which is in a medium $k_1$ and has a slab in between of medium $k_2$. I know that if there is a capacitor with distance between the plates as d and has air between it and has a slab in between the plates such that it's thickness is t and it's dielectric constant is k then it's equivalent capacitance is $\frac{\epsilon A}{d-t(1-\frac1k)}$. So if I imagine it as the scenario above with dielectric constant $k_1$ in the back and a slab of dielectric $k_2$, then we just put $t=d/2$,$k=k_2$ and multiply the above by $k_1$, which according to me should be the answer but if I go by my teachers method and do the math, we get $\frac{k_1 k_2 \epsilon A}{(k_1 + k_2)d}$. Who is wrong here? **EDIT:** Here A is the area of the plate.


title: why sometimes touching old flickering tube lights starts them properly

text: In my old house there are two old tube lights. Some times they don't start properly, (specially at evening time, may be it is because of low voltage), they starts flickering i.e. on and off continuously. And when my elder brother touches them, mostly at center they just goes on. Even i have tried many times, the touching works, and even after removing the hand, once it is on(working properly), doesn't stop them, and they remains on. So why and how touching them works. Sorry for my bad English. Thanks in advance.


title: Compute closed line integral of electric field in circuit

text: I have a circuit where resistor is parallel to capacitor, which is charged with voltage U. How to compute line integral around closed loop to get the result of Kirchhof second law - $U_{capacitor}+I\cdot R=0$? For beginning I splited it in two parts, starting from the capacitor positive plate through resistor:   $$\oint \vec{E} \cdot d \vec{l} = \int_{+R-}\vec{E_{R}} \cdot d \vec{l} + \int_{-C+} \vec{E_C} \cdot d\vec{l}$$ Since intensity of resistor is with the same direction as current density $\vec{j}$: $$\int_{+R-}\vec{E_{R}} \cdot d \vec{l} = \int_{+R-} \rho \cdot \vec{j} \cdot d \vec{l} = I \cdot R$$ When I do the same for capacitor I gain negative line integral, because the $E_{c}$ is opposite to $d \vec{l}$ when positive charge is moved from negative plate to the positive. So in which part I have made a mistake?


title: Approaching speed of light: why do objects appear further away in front of me?

text: A Slower Speed of Light is a video game created by the MIT Game Lab which allows users to experience what it would be like if the speed of light was closer to normal walking/running speeds and thus experience relativistic effects in an otherwise familiar environment. From the site: > A Slower Speed of Light is a first-person game prototype in which players > navigate a 3D space while picking up orbs that reduce the speed of light ... > [allowing] the speed of light in the game to approach the player’s own > maximum walking speed. Visual effects of special relativity gradually become > apparent to the player [as more orbs are collected]... These effects, > rendered in realtime to vertex accuracy, include the Doppler effect (red- > and blue-shifting of visible light, and the shifting of infrared and > ultraviolet light into the visible spectrum); the searchlight effect > (increased brightness in the direction of travel); time dilation > (differences in the perceived passage of time from the player and the > outside world); Lorentz transformation (warping of space at near-light > speeds); and the runtime effect (the ability to see objects as they were in > the past, due to the travel time of light). When I play the game and collect enough orbs for relativistic effects to really start becoming significant something happens that I didn't expect: as I accelerated towards objects they appeared to move further away from me (despite length contraction definitely being taken into effect, since I was able to get to those objects faster over a given distance than had I not collected any orbs). Luboš Motl in this question says that this is due to: > the shrinking of transverse directions if you're moving forward (or their > expansion if you move backwards) which makes object look "further" > (optically smaller) if you're moving forward. Because of this shrinking, you > may effectively see "behind your head". You also see things how they looked > like some time ago. Based on my understanding of the Lorentz transformations the directions transverse to one's motion aren't effected ($y'=y$ and $z'=z$ if all motion is in the $x$ direction). So, what exactly causes this effect that I see in the game and Luboš Motl explains? Am I misunderstanding the Lorentz transformations?


title: Unequally charged hanging pith balls

text: I have done a problem which asks us to find the charge on two equally 'massed and charged' pith balls which are left hanging on a string with a certain length that repels each other and attains an equilibrium point making an angle (the string) with the vertical. The problem was quite clear, what left me wondering was: Won't the angle be the same if the balls are unequally charged (but equally massed) ? Or does it deviate from the initial case?


title: Does the entropy of a system vary in the equilibrium?

text: I am not physicist and this question may seem trivial. But I understand that in the equilibrium the magnitudes such as temperature or volumen do not vary. Is the same for entropy? My logical says that it should not, and here is why: Entropy is proportional to the number of microstates of a macrostate In the equilibrium, a system is one of the macrostates with more microstates -> But not necessarily in the one with more of them. The system may visit macrostates with a slightly different number of microstates If this is the case, entropy would be varying (although maybe not too much, but it would be) Am I wrong here or the reasoning is good?


title: Divergence of non conservative electric field

text: I'm looking for the proof that the 1st Maxwell equation is valid also on non conservative electric field. When we are talking about a electrostatic field, the equation is ok. We can apply the Gauss (or Flux) theorem and get Gauss' law: $$\mathbf{\nabla} \cdot \mathbf{E} ~=~ \frac{1}{\epsilon _0} \rho (x,y,z).$$ The question is, why when there is a time dependent magnetic field, and then a time dependent (non conservative) induced electric field, the 1st Maxwell equation is the same? How we can prove that?


title: Symmetry for even even nuclei

text: For even even axially symmetric nuclei K=0 because even even spherical nuclei don't show rotaional spectra and therefore do not rotate about the axis of symmetry; Thus the angular momentum about a symmetry axis vanishes . > My question is : Why even even spherical nuclei don't show rational spectra > and therefore do not rotate about the axis of symmetry? Related post : > Hamiltonian in Nuclear collective model


title: Radioactivity features

text: What happens if someone would make a wire coil around nuclear reactor core? What are the possibility to capture radioactivity directly?


title: How does one derive the Lamb shift for the Hydrogen atom?

text: I've been perusing my copies of Srednicki and Peskin & Schroeder, and I can't seem to find an explanation of how one derives the Lamb shift that I can follow. How does one derive the Lamb shift? What order in perturbation theory do you have to go up to? Does one use a path integral or canonical quantization approach?


title: What are the applications of Gauss's law in technology?

text: Freshmen physics textbooks use Gauss's law plus symmetry to calculate the electric field. I was wondering if this method of finding the electric field using a symmetry is used in real applications in life, science, research, or technology. For example do researchers use symmetry to find the electric field due to a charged virus that is spherically symmetric, or a charged DNA that has some kind of symmetry or charged symmetrical objects in nanotechnology. I am looking for such specific applications.


title: What is the meaning of uncertainty in Heisenberg's uncertainty principle?

text: The Heisenberg's uncertainty principle states the following: $$\Delta p \cdot \Delta x \ge \frac{h}{4\pi}.$$ While studying for my high school physics exams, I fooled myself into believing that I understood the uncertainty principle (at least the implications). But suddenly the question that's nagging me is the following. If the uncertainty $\Delta x$ of an electron is 1.2 nm, does it imply that the probability that the x-coordinate lies within a 1.2 nm range, equal to 100%? Or does it mean that the probability is 95%? Or does it mean something totally different? I wonder why no author made it clear in the high school/junior college level textbooks. I am uncertain about what uncertainty means.


title: Why is laser light a cone?

text: What about the production of laser light disallows it to be perfectly straight as opposed to a cone? I feel like it should be a plane wave, not a very tight cone.


title: Average velocity = Arithmetic Mean

text: I am having trouble grasping how the average velocity of a particle between an initial position and final position equals to the arithmetic mean of the initial velocity and the final velocity when the acceleration is constant. My textbook offers me the following explanation: Because the velocity at constant acceleration varies linearly in time according to: $$v_{xf}=v_{xi}+a_{x}t,$$ we can express the average velocity in any time interval as the arithmetic mean of the initial and final velocity. $$v_{x, average}=\frac{v_{xf}+v_{xi}}{2}$$ (for constant $a_{x}$) I tried graphing a theoretical position-time, velocity-time graph to understand this but I still cannot see why it is the arithmetic mean. Could someone please explain this to me?


title: Using Wien's Law to show spectral distruibution function of one temperature represents all temperatures

text: This is a exercise question from Quantum Mechanic textbook by Bransden: > Using Wien's Law to show that if the spectral distribution function of black > body radiation, $\rho(\lambda,T)$ is known at one temperature then it can be > obtained at any temperature (so that a single curve can be used to represent > black body radiation at all temperatures. Wien's law being: $\rho(\lambda, T)=\lambda^{-5}f(\lambda T)$ I understand this question as that temperature $T$ only acts as a scalar to the $\lambda$ variable, so that the differentiation of Wien's Law with respect to $T$ will give a constant multiple to the whole function, but in fact it is not clear that this is true by examining only on the Wien's Law. Any help would be appreciated.


title: How are propositions concerning spacetime curvature constructed explicitly in terms of coincidences?

text: Is Einstein's insight [1] that > All our well-substantiated space-time propositions amount to the > determination of space-time coincidences [such as] encounters between two or > more [...] material points. and [2] his (so recognized) > [...] view, according to which the physically real consists exclusively in > that which can be constructed on the basis of spacetime coincidences, > spacetime points, for example, being regarded as intersections of world > lines applicable to propositions concerning spacetime **curvature** ? How, for instance, is the proposition " _spacetime containing the worldline of material point **A** is curved_" constructed and expressed **explicitly** on the basis of spacetime coincidences (in which the " _material point_ " identified as **_A_** or suitable other " _material points_ " took part) ? * * * **Edit** in response to the answers and comments presently provided (Sept. 12th, 2013):   * Trying to put my question more formally,  given that there is the set $S$ of any and all distinct " _spacetimes_ " imaginable,   and that there is the function (or proposition)   $\kappa : S \rightarrow \\{ true, false, undetermined \\}$   which for any spacetime under consideration represents whether it is " _curved_ ", or " _not curved_ ", or not an eigenstate of "possessing any _curvature_ " at all, and further given a set of sufficiently many distinct names $W := \\{ A, B, M, M', ... \\}$,   and that there is the function   $coinc : S \rightarrow \text{ powerset}[ \text{ powerset}[ \, W \, ] \, ]$   which for any spacetime under consideration represents the set of (distinguishable) coincidences of (different, and distinctly named) " _worldlines_ ", I'd like to know the explicit expression of the function (or proposition) $f : \text{ powerset}[ \text{ powerset}[ \, W \, ] \, ] \rightarrow \\{ true, false, undetermined \\}$,   for which   $\forall s \in S: f( coinc( s ) ) = \kappa( s )$. Looking at the above quotes it may be expected that $f$ has been worked out and written down long ago already; therefore, please write it down in an answer here, or point me to the corresponding reference. However, here are   * Considerations which answers would be acceptable otherwise: either a proof that such a requested function $f$ doesn't exist at all; presumably by exhibiting two distinct spacetimes $s_a$ and $s_b$ for which $coinc( s_a ) = coinc( s_b )$ but $\kappa( s_a ) \ne \kappa( s_b )$. (But recall the hole argument discussion relating to the difficulty of distinguishing spacetimes at all.) Finally, if such a requested function $f$ can neither be explicitly stated, nor refuted, then   define the notion " _geodesic_ " (which has already been used/presumed in answers below) or at least the notion "null _geodesic_ " explicitly in terms of $coinc( s )$.


title: Is it incorrect to explain the direction of a coded vector quantity?

text: For example, let's say that in a linear physics problem, all the data are given to a certain direction, and coded positively for direction to the right. So +5m/s would be a velocity of 5m/s to the right. And lets say after some calculation you get a negative answer, implying the direction was therefore to the left. (-2m/s for example), would it be therefore incorrect to state -2m/s to the left? Would this not be like saying the opposite of 2m/s to the left, and therefore to the right?


title: Unstable energy levels

text: Well, reading about "Raman Effect" I saw that when the electron absorb some energy, with frequency $ \omega_{abs} $, that is different from $ \omega_{n} - \omega_{n-1} \neq \omega_{abs1} $, it go to an "virtual energy level" that is unstable, but nevertheless we could measure its lifetime. And more, the electron could stay in this unstable level enough time to absorb another electron with frequency $ \omega_{abs2} $, such that $ \omega_{n} - \omega_{n-1} = \omega_{abs1} + \omega_{abs2} $, and than finally go to an stable energy level, that is foreseen by the Quantum Mechanics. My question is: Is this right? An electron could stay a little time in a unstable energy level?


title: A question about Dirac operator

text: The Dirac operator at 2 dimension can be written as $$ D=\sum_{k=1,2}\sigma^{k}D_{k}=\left( \begin{array}{cc} 0 & \partial_{x}-i\partial_{y}-i(A_x-iA_y)\\\ \partial_{x}+i\partial_{y}-i(A_x+iA_y) & 0 \end{array} \right) , $$ where $A_i$ is a gauge field. If we add some other terms to the operator above, such as $\sigma^{3}\Delta$, where $\Delta$ is a constant, is it still a Dirac operator? Can the last term be understood as some kind of gauge field? **Edit:** Clear the background of the question: I am reading Kane and Mele's paper on $Z_2$ topological order (pdf). For my understanding the general question is how the inner structure (in this case is time reversal symmetry) of a Hamiltonian can be used to classify the band structure. They said in this case it is related with twisted Real K theory. Is there anyone know how to apply K theory in the graphene case?


title: How does QM allow imaging of individual electron orbitals?

text: Question: Why does the uncertainty principle allow probing of characteristics specific to the electron orbital distribution? If you measure an electron's position/momentum, then after you measure it, it is obviously no longer in that exact same orbital. What's going on? Recently, a new AFM technique has allowed researchers to distinguish between different types of chemical bonds (paper link). The published results clearly distinguishes information about bonding electrons (specific to benzene structures). ![Hexabenzocoronene image](http://i.stack.imgur.com/hKxfi.gif) What does quantum mechanics state about variability of measurement of electrons in coherent states? Must the orbitals be excited to get information about it? Do they then give off an emission in order to return to the ground state? If the electrons aren't actively disturbed in these processes, wouldn't that be a blatant violation of the uncertainty principle? What are the statistical limitations of getting information about an electron orbital? Theoretically, could you experimentally map the exact 3D wave function of an orbital? I want this question to be agnostic to the method like AFM, although AFM seems particularly confusing to me. As I understand the idea, it moves a tip along a surface and detects slight movement as it moves over humps. You can imagine a single electron orbital as a hump and the probe moving over it continuously being pushed up or down due to the electrostatic attraction or repulsion. But this is clearly wrong! Such a process couldn't be smooth.


title: Why is Light invisible?

text: Why can't we see light? The thing which makes everything visible is itself invisible. Why is it so?


title: Do field and potential energy always come together?

text: 1. Is energy directly due to a field always potential energy?   2. Is potential energy always due to a field? From the two Wikipedia links: > a field is a physical quantity that has a value for each point in spacetime. > > potential energy is the energy of an object or a system due to the position > of the body or the arrangement of the particles of the system. Thanks!


title: Projectile Motion

text: I know the angle at which a projectile is launched, how far it needs to go, and also the maximum height. How can I find the initial velocity needed (disregarding air resistance)? Currently, I am using: $Vf^2=Vy^2-2g\Delta y$ From that, I can find the velocity in the y direction needed for the projectile to reach the known maximum height. Next, I am finding the time it takes to reach that maximum height by using the found y velocity.


title: Why not using cosmic rays to study HEP, since they are way more energetic than LHC?

text: Cosmic rays energies can exceed $10^{8}$ TeV, way higher than the energy scale achieved in the LHC or that can be achieved in the near future. cannot we just use them to study fundamental interactions at such extreme energy scales, instead of LHC? What are the difficulties in guiding and channeling such super high energetic particles and make them collide to probe fundamental physics at such super energy regime?


title: Why does an ice rink use a primary and secondary coolant?

text: A normal refrigeration cycle uses one coolant, so why does vapor compression cycle of an ice rink you a primary coolant and a secondary coolant?


title: Fourier Expansions for Closed Strings and Parity

text: I'm revising some string basics, and have come across the following problem. For closed strings one introduces the worldsheet parity operator $$\Pi : \sigma \mapsto \ell-\sigma$$ where $\ell$ is the string length. In order to relate the left-movers and right-movers by parity in the Fourier expansion one assumes that $X$ and $P$ are parity even. But why is this reasonable? I don't understand what forces $X$ and $P$ to have a specific behaviour under parity. Indeed surely a generic closed string will not have $X(\sigma)=X(\ell-\sigma)$! Is there something I'm missing here? Thanks in advance for any helpful thoughts. **Edit** I wonder whether the notes I'm reading are only dealing with **unoriented** string theory. Perhaps that would be the reason for assuming the evenness of $X$ and $P$. Could someone possibly confirm that I've got the correct meaning of unoriented?


title: Is there a true parallel between Gibbs' phase rule and Euler's polyhedral formula?

text: Gibbs' phase rule states: $$F=C-P+2$$ where $F$ is number of degrees of freedom, $C$ is number of components, $P$ is number of phases. Euler's polyhedral formula states: $$V+F-E=2$$ where $V$ is number of vertices, $F$ is number of faces, $E$ is number of edges. It is easy to see that these formulas are similar. Is there a true parallel between them? Otherwise, what is the mathematical meaning of Gibbs' phase rule?


title: How does a lightning striking stall a moving car?

text: I am currently an undergraduate student leading a discussion section for an introductory electricity and magnetism course. I was asked this question that I was unable to answer and I am hoping for some help in understanding what is going on. I was told by this student that he and his girlfriend were driving along a highway when lightning struck the roof of their car. Although they were unharmed by the lightning (Faraday cage), my student claimed that his car stalled and was unable to turn it back on right away. However, after about 5 to 10 minutes, the car started up immediately. So my question is **how does lightning striking the roof of a car stall a moving car?**


title: Thermographic camera vs infrared camera

text: AFAIK, all objects above 0K emit infrared radiation. It is an electromagnetic radiation, like visible light. I guess thermographic and infrared cameras work on the same principles, but their images are very different, the former is false color and very different from the visible image, the latter is black and white and usually correlates to the actual visible image, at least that's what I've seen so far. So physically, how exactly are they different?


title: What is the difference between heat and work

text: If 700 J of work is applied onto a gas, and therefore the gas contracted 0.005 m^3 (from 0.01m^3), and the temperature stayed at a constant rate of 300 kelvin, then how come when pressure is added to the system (Not by adding more molecules, but instead by increasing the kinetic energy of the molecules by adding heat) until the system becomes adiabatic (The system has as much total energy as it did from the start), we don't say that the heat applied equals to 700J? It would make perfect sense if 700J were added to a system, then in order for that system to return to the same state, 700J would be taken away.


title: A single valued function from a multi-valued function

text: In Schrieffer's book "Theory of Superconductivity", there is said when he deals with multiple connected superconductors (and discuss London equations), that if one takes the line integral of the potential vector on the boundary of a hole where there is a magnetic flux inside we have : $$\int{\vec dl\cdot \vec A} = \Phi$$ Then, if we go inside the superconductor ($\vec B=0$ thus $\vec A = \nabla \chi$ where $\chi$ is a scalar function) and again take the line integral of $\vec A$, we deduce that $\chi$ must be multi-valued : $$\int{\vec dl\cdot \vec A} = \Delta \chi = \Phi$$ But it is stated that now, if the magnetic flux is known in each hole, $\vec A$ is determined and the physics is saved. But for me, it is not clear that if we know how much $\chi$ raises with every $2 \pi$ rotation around a hole, we don't know how it varies... Could someone help me on this ?


title: How does the properties of a soil affect gamma-radiation

text: I'm doing data-analysis on airborne gamma-ray data and I have the following statement given which I don't fully understand and I would need some help interpreting it: > "Gamma radiation attenuation is affected by density, porosity, particle size > and water content of the soil. Increase of fine aggregate in the soil causes > the water content to increase, and hence the density increases as well, > causing an attenuation of gamma radiation." Is my interpretation correct: "The more dense and wet the soil is the less gamma-radiation is reflected back from the soil" Hope my questions is appropriate here. I'm not a physicist so could you please explain this to me in simple terms if possible? :) Thank you for any help


title: Sound Produced due to motion of air inside a tube

text: I'm trying to understand the sound produced by Air motion inside the respiratory system. If you have a long tube with a suction apparatus at one end , You turn on the apparatus and the air is sucked in. Now Sound waves are produced due to vibration of the molecules in the wall so that each molecule behaves like a harmonic oscillator and the whole is a system of coupled oscillators . I have some questions. If The air inside the tube is highly collaminated in its motion , That is it has zero component of velocity perpendicular to the axis , The there will not be any sound waves that are produced . If there is a very small inhomogeinity at the beginning then inside the tube will these inhomogenities be amplified ? How does The Frequency and intensity depends on the velocity of air , The diameter of the tube and the degree of disturbance in the motion of air ?(I expect that sound will be a linear combination of waves with different frequencies , the more the turbulence , the more terms will be in the series expansion , The less diameter the more velocity and subsequently the bigger values of frequencies that appear in the series) Will sound waves emitted have the same frequency along the tube, that is the velocity of and kinetic energy of air increases as they move along the axis and subsequently the energy of sound will increase or may be the higher velocity the more turbulence that occurs and subsequently sound waves will be a linear combinations of many waves with different frequencies ? Why is the sound produced from a cavity amplified and has low frequency?(I expect this is due to turbulence , but in this case why does turbulence amplify sound?, I think that the more turbulence , the more frequencies produced and when you add them together , the amplitudes will tend to vanish for some frequencies and increase for other frequencies , right? ) Lastly , What exactly is the pitch of the sound and how is it related to the frequency?


title: Enhance a home build spectrometer

text: I would like to build a home made spectrometer like it's describing here: http://coolcosmos.ipac.caltech.edu/cosmic_games/spectra/makeGrating.htm How can I improve this spectrometer? A prism from an old binocular? How would I enhance this spectrometer to improve accuracy and clarity?


title: Will acceleration rate of expansion of space become faster than speed of light?

text: From watching cosmology lectures, it seems that the space between galaxies is expanding at an accelerating rate, my question is since it is the space that is (acceleratingly expanding), the special relativity does not apply in this case? in other words since it is not anything that is accelerating with relative distance, would at some point galaxies that are far enough from each other separate faster than speed of light? Please note, I am not asking if the galaxies would move faster than speed of light, but whether the rate of expansion of space be faster than speed of light so in effect each galaxy would be as if it is behind an event horizon where even light can not escape? If yes, is there a name for such an event horizon that is cause by expanding space outside rather than the conventional black hole event horizon caused by stretching of space within the event horizon? How would one be able to differentiate between a black hole even horizon and a galactic even horizon caused by accelerating expansion of space outside? (not sure if the concepts of inside and outside are still meaningful in such cases)


title: Coriolis Acceleration and Latitude?

text: We had a lecture on the Coriolis effect today and I am very very confused, this is the first time I've heard of it as such. Anyway, I know that Coriolis acceleration is calculated by $\vec{a} = -2 \vec{\omega} \times \vec{v}$ where $\vert \vec{\omega} \times \vec{v}\vert = \omega v \sin\theta$. This gives $a = -2\omega v \sin \theta$. If I am given a latitude of say, 40°, how do I calculate the Coriolis acceleration of a moving object such as a bus in different directions like due south or due east? Also, can I use $\omega = \frac{2\pi}{T}$ to calculate $\omega$? Would $T$ be the seconds in a day?


title: weak bosons and feynman-stueckelberg interpretation

text: from Wiki _"The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle."_ Q:If each is the other's antiparticle then which is retrograde in time according to Feynman-Stueckelberg interpretation (FSI)? I mean if by the FSI antiparticles move backwards in time, and if $W^+$ and $W^-$ are each others antiparticle - then which one moves backwards in time? EDIT: after noting David's and Ron's points I've added a diagram to illustrate the problem. Suppose out of the vacuum $W^+$ and $W^-$ are created which then decay to electrons and neutrinos ![W boson creation](http://i.stack.imgur.com/H8zdl.png) It only makes sense if one of the W bosons is retrograde. I have arbitrarily placed the tilt of the W bosons, so they can be interchanged without loss of generality.


title: Pulley and Two Masses: Why is the acceleration the same?

text: Say you have a massless string going over a pulley pulley attached to a cart on a frictionless surface one end, and a hanging mass on the other end. What guarantees that the cart and the hanging mass accelerate at the same speed? The net force is the sum of the masses*accelerations, so why couldn't the cart accelerate much faster than the hanging mass. If they were connected by something rigid, I think I could understand, but in this case, I absolutely don't...


title: How does the entropy of a system change when the amount of substance changes?

text: I am thinking of a closed system consisting only of an ideal gas. Suddenly, the amount of substance of the gas changes. No other property of the system shall change. How does this affect the entropy of the system? How can I calculate the change in entropy?


title: Combining chemical elements

text: Prof Brian Cox mentioned on Wonders of the Universe when discussing chemical elements that Carbon 12 consists of 6 protons and 6 neutrons, he also mentioned that Helium consists of 2 protons and 2 neutrons. He then went on to say that by combining certain elements we can create others - can anyone explain what he means by this with a practical example? Can I get myself some Carbon and some Helium and somehow combine them? :)


title: Why is the Earth so fat?

text: I made a naive calculation of the height of Earth's equatorial bulge and found that it should be about 10km. The true height is about 20km. My question is: why is there this discrepancy? The calculation I did was to imagine placing a ball down on the spinning Earth. Wherever I place it, it shouldn't move. The gravitational potential per unit mass of the ball is $hg$, with $h$ the height above the pole-to-center distance of the Earth (call that $R$) and $g$ gravitational acceleration. Gravity wants to pull the ball towards the poles, away from the bulge. It is balanced by the centrifugal force, which has a potential $-\omega^2R^2\sin^2\theta/2$ per unit mass, with $\omega$ Earth's angular velocity and $\theta$ the angle from the north pole. This comes taking what in an inertial frame would be the ball's kinetic energy and making it a potential in the accelerating frame. If the ball doesn't move, this potential must be constant, so $$U = hg - \frac{(\omega R \sin\theta)^2}{2} = \textrm{const}$$ we might as well let the constant be zero and write $$h = \frac{(\omega R \sin\theta)^2}{2g}$$ For the Earth, $$R = 6.4*10^6m$$ $$\omega = \frac{2\pi}{24\ \textrm{hours}}$$ $$g = 9.8\ m/s^2$$ This gives 10.8 km when $\theta = \pi/2$, so the equatorial bulge should be roughly that large. According to Wikipedia, the Earth is 42.72 km wider in diameter at the equator than pole-to-pole, meaning the bulge is about twice as large as I expected. (Wikipedia cites diameter; I estimated radius.) Where is the extra bulge coming from? My simple calculation uses $g$ and $R$ as constants, but neither varies more than a percent or so. It's true the Earth does not have uniform density, but it's not clear to me how this should affect the calculation, so long as the density distribution is still spherically-symmetric (or nearly so). (Wikipedia also includes an expression, without derivation, that agrees with mine.)


title: How to reproduce highway vibration?

text: I would like to test my hardware under vibration that can appear on a highway gantry. If someone has a model of such vibration. i.e period and amplitude. In my lab I have a motor that can be regulated according to its RPM.


title: How does ATP transfer energy to a reaction?

text: This is a question for which I've found it surprisingly hard to find a good answer. Biology texts talk mystically about the ATP->ADP reaction providing energy to power other reactions. I'd like to know some more details. Is this following roughly right ?   1. Each reaction in a cell has a specific enzyme.   2. Each enzyme has binding sites for, say, two molecular species AND for an ATP molecule.   3. When a reaction takes places, the two species bind to the enzyme, and a little later, an ATP molecule binds.   4. For some reason (why ?), the ATP->ADP reaction is now energetically favourable, so the high-energy bond breaks.   5. This releases electromagnetic energy at some characteristic frequency.   6. Certain bonds in the enzyme have a resonant frequency that allow them to absorb this electromagnetic energy (the EM energy disturbs molecular dipoles ?).   7. The 3D structure of the enzyme is disturbed (i.e. it bends) in such a way that the 2 molecular species are mechanically forced together, providing sufficient activation energy for the reaction in question.   8. The newly formed species no longer binds nicely to the enzyme (why ?) so it detaches, as does the ADP, which also doesn't bind as nicely as ATP.   9. The End. So, is that an accurate summary ? Anyone care to add some more physical details ? (like all the thermodynamics and quantum chemistry I have no idea about)


title: Why do the drift and diffusion components cancel for each type of carrier if EHP generation plays such big role in p-n-junctions?

text: I have always argued to myself that drift and diffusion components of the current though a p-n-junction cancel for each type of carrier because any electron diffusing from n into p will sooner or later require the same electron to drift back. (And analogously for holes). However, it now seems to me that the generation of EHPs (electron-hole-pairs) profoundly contributes to the drift current: > "The supply of minority carriers on each side of the junction required to > participate in the drift component of current is generated by thermal > excitation of electron-hole pairs." (In "Solid-State-Electronic Devices" , > Streetman and Banerjee) In thermal equilibrium, that implies an equal amount of recombination events. Anyway, one can imagine an EHP being generated on the p-side, with the electron drifting and the hole diffusing to the n-side, where thy recombine. In that scenario, there is no hole drift and no electron diffusion, but hole diffusion and electron drift. So, how in earth does one dare to argue that "the drift and diffusion components must cancel for each type of carrier"? I feel like I am missing the real argument.


title: Noether Current when the Lagrangian depends on second derivative of the fields

text: Let a Lagrangian density for a field theory of $N$ fields $\left\\{\phi_i\right\\}_{i=1}^N$ be given. Assume that the Lagrangian density depends on the fields, their spacetime derivatives, _and_ their second spacetime derivatives: $\mathcal{L}(\phi_i,\partial_\mu\phi_i,\partial_\nu\partial_\mu\phi_i)$. Then a short derivation shows that the Euler-Lagrange equations are given by: $$ \frac{\delta\mathcal{L}}{\delta\phi_{i}}-\partial_{\mu}\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\phi_{i}}+\partial_{\nu}\partial_{\mu}\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\partial_{\nu}\phi_{i}} \,\,\, \forall i\in \\{1,\dots,N\\} $$ Using a similar derivation to the proof of the No-ether theorem, I was able to show that the conserved Noether current is: $$ j^\mu = \sum_{i}\left[\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\phi_{i}}\Delta\phi_{i}+\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\partial_{\nu}\phi_{i}}\partial_{\nu}\Delta\phi_{i}-\left(\partial_{\nu}\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\partial_{\nu}\phi_{i}}\right)\Delta\phi_{i}\right] $$ My question is: is this correct? It feels fishy to me because there is a term $\partial_{\nu}\Delta\phi_{i}$ and I would somehow expect all terms to be proportional to $\Delta\phi_{i}$ alone. I'm doing this to find the conserved Noether current (see this related question and this one which unfortunately had no answers yet) of the BRST transformation.


title: A doubt about fuchsian functions in physics?

text: I'm not sure if this is the right place (or math.stackexchange?) to ask the next   1. What is the difference between fuchsian, theta-fuchsian, and kleinian functions?   2. Please, suggest me an introductory reference to the Physics of previously asked for.


title: Neutron Diffraction Device

text: I was just wondering if we have a beam of neutrons, lets say of various energies meeting or colliding with a powder of monoatomic crystallites, what would happen to the neutron beam and the energies of the neutron after the collision with the powder sheet? If I was to say that my neutron beam has a wavelength of $\lambda$ and collides with the powder at an angle of $\theta$, then I do know what would happen to my neutrons after the collision. They would scatter according to **Bragg scattering** given by $$n\lambda =2b\sin\theta $$ I want to find a way to artificially enhance my signal by simply eliminating higher energy neutrons hence eliminating higher order scattering.


title: Units of a discrete Fourier transform

text: Normally a Fourier transform (FT) of a function of one variable is defined as $$f_k=\int^\infty_{-\infty}f(x)\exp\left(-2\pi i k x\right) dx.$$ This means that $f_k$ gets the units of $f$ times the units of $x$: $[f_k]=[f]\times[x]$. For an array of inputs $\\{f_n\equiv f(x_n)\\}$ of length $N$ the discrete Fourier transform (DFT) is normally defined as $$f_k=\sum_{n=0}^{N-1}f_n \exp\left(-2\pi i k n/N\right). $$ This means that $f_k$ has the same units as $f$: $[f_k]=[f]$. What is missing here? For the DFT to be an approximation of the FT, surely the units must be the same. In other words, if I am to compare the theoretical value of some Fourier transformed quantity to one obtained numerically from a discrete Fourier transform, how do I consistently do this? The obvious norm would be $\frac{x_0-x_{N-1}}{N}$, but is it really so?


I'm making an experiment and it is written in my older questions. Now, my question is -   Does water have the same surface tension at same temperature but different volume?


title: Thermionic Emission: Kinetic Energy Distribution of Emitted Electrons

text: I am having a conceptual problem understanding the kinetic energy of thermionically emitted electrons. I know that in order to escape the surface the electrons must have energies of at least the Fermi Level plus the work function ($\varepsilon \ge \varepsilon_F + \phi$). However, is all that kinetic energy converted to potential energy once it leaves the material so that once it leaves the surface an electron has kinetic energy $T = \varepsilon - \varepsilon_F + \phi$?


title: Can magnets destroy notification light on phone?

text: I'm posting this question here because it's about magnets and electronics. Please tell me if it's better suited to another site! I've been messing around with some reasonably strong magnets around my Nexus 4 phone (yes, I know that's stupid). Now my LED notification light isn't working. Could the cause of this possibly be interference from the magnets? The magnets are strongly attracted to each other - but not to my phone - and are like large capsules, about 5cm long and 1cm wide. I don't know exactly which metal they're made of but I'll hazard a guess at iron. If you want any more details please ask me!


title: Numb3rs TV series - Antenna

text: Most of you might have watched the TV series 'Numb3rs'. > http://en.wikipedia.org/wiki/Numb3rs In one of those episodes they were discussing about how to capture the contents on the screen of a computer using just an antenna. Now I don't remember in which episode it was. But I am curious to know whether such scenarios possible practically? > http://www.math.cornell.edu/~numb3rs/ Please add that episode name If anyone remembers it.


title: Are physical probabilities also quantized?

text: In physics there is quanta and energy occurs per this unit. Is it it then reasonable that probability also is quantized since energy is?


title: Gauge invariance in Laughlin's argument

text: In Laughlin's gedanken experiment which aims to explain quantization of Hall conductance, one takes the adiabatic derivative of the Hamiltonian with respect to vector potential. Now it seems that it is claimed that flux change by a flux quantum would leave the Hamiltonian unchanged upto a gauge transformation and so the quantization follows. My question is that, is there a general principle which guarantees flux(enclosed) changes by flux quantum has no effect on physical quantities? Let me also state that it does seem to be the case because   * Aharonov-Bohm oscillations are invariant under flux quantum change.   * All properties of an electron in a ring enclosing a flux are insensitive to a change by flux quantum. etc. But is this true when we have impurities, arbitrary potential etc? It seems to me that this is related to Byers-Yang work on superconductor flux quantization. Thoughts?


title: Clarifying what metric counts as flat space

text: In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers greater or less than zero, would this still count as flat space even though it is no longer Euclidean? I tbink the answer is yes but just want to make sure. Also, when a physicist says that the local tangent space on a manifold is flat, do they imply that the metric is locally Euclidean, or can the diagonal elements be any non-zero real number in that local tangent space?


title: Boundaries where AdS/CFT complementarity applies

text: Usually when I read about AdS/CFT complementarity as a particular case of the Holographic principle, it suggests that physics evolution on a boundary has a map to physics evolution on the bulk. But what of sort of boundaries does this apply? I've usually seen it applied where the event horizon is the boundary and the outside space is the bulk. But does it hold for imaginary boundaries as well, like the typical Gaussian surfaces one consider to study Maxwell equations as well? If not, what special properties needs to satisfy the boundary so that the holographic principle holds?


title: Field Strength Renormalisation in Peskin and Schroeder

text: In chapter 7 of Peskin and Schroeder they define the field strength renormalisation $Z$ for a quantum field to be the residue of the Fourier transform of the correlation function $$\langle \Omega | \phi(x)\phi(0) | \Omega\rangle$$ They go on to refer to $Z$ variously as $Z_1$, $Z_2$ and $Z_3$ at different points in the text. Is the subscript meant to denote the **order to which we've computed $Z$**? In other words do we have $$Z = Z_1 + O(e^2) = Z_2 + O(e^3) = Z_3 + O(e^4)$$ and so on? This would seem to make no sense because then naturally $Z_1 = Z_2$ due to the Feynman diagram stucture. Alternatively are they just meant to be $Z$s for different systems (Lagrangians)? Or are they just genuinely different quantities depending on context? I'm getting confused because P&S generally use subscripts to denote the order to which a quantity is valid, but it doesn't seem to be the case here! If anyone has the book and wants page references, see p215, 221 and 243.


title: Statistical Weighting Factor on thermal neutron importance

text: The problem is concerning the use of a thermal fluxed squared weighting factor in a thermal reactor. I have seen in sources the thermal flux in a reactor is squared as a statistical weighting factor, for example looking at temperature feedback. Say a number of thermocouples are measured in a reactor, the weighting of the importance of the thermocouples are to flux squared. Why is this when the reaction rate is equal to the flux? Is it simple statistics I'm not understanding.


title: RGEs of the MSSM - problems with Mathematica

text: I'm having some troubles with the trilinear soft couplings of the MSSM RGEs. I've used the ones written in Martin's supersymmetry primer and I run them using mathematica, if I do so without taking into account the soft trilinear terms (by putting them to zero at all scales and not considering that they get generated even for a[M_SUSYbreaking]=0) everything works well. I get unification of gauge coupling constants at some high scale and the running of other terms seems to be OK. When I consider the running of trilinear terms too things don't work at all. Not only the running of soft terms change in a strange way, also ِYukawa terms and gauge couplings change and I know this can't be true as there is no soft trilinear dependence in these adimensional couplings. This obviously points towards a bug in the code or something related but I've checked it many times and I don't know where the mistake or the typo could be. I've also tried to run the adimensional parameters first and then using the numerical solution I get solve the soft parameters, by doing this I think that, at least, I can force the adimensional parameters to have a good behaviour and then see what the result for the other things is. The problem is that I get a Mathematica error that I don't know how to solve, "Computed derivatives do not have dimensionality consistent with the initial conditions." I've seen that these can come when trying to solve a system of differential equations using solutions from the numerical resolution of another system of diff equations (solution of NSolve into another NSolve). I don't know how could I do this. Any thoughts? I copy the code here in case there is something really obvious that I'm missing. I don't know, maybe is just a typo that I haven't seen yet but I really don't see where it could be. As I guess the problem is with Mathematica maybe someone has dealt with a similar situation in the past even if is not writing RGEs.               Qmax = 35.     tanbeta = 10.     htzero = 173/174 Sqrt[1 + tanbeta^2]/tanbeta     hbzero = 4/174 Sqrt[1 + tanbeta^2]     gaugino = 1.5     sfermions = 1.5     higgsmassparam = 1.5     Exp[Qmax] .174     Xu[t_] := 2 ht[t]^2 ( mhu2[t] + mq2[t] + mu2[t]) + 2 at[t]^2     Xd[t_] := 2 hb[t]^2 (mhd2[t] + mq2[t] + md2[t]) + 2 ab[t]^2     system = NDSolve[{ht'[t] == 1/(16 \[Pi]^2) ht[t] (6 ht[t]^2 + hb[t]^2 - 16/3 g3[t]^2 - 3 g2[t]^2 - 13/15 g1[t]^2), ht[0] == htzero, hb'[t] == 1/(16 \[Pi]^2) hb[t] (6 hb[t]^2 + ht[t]^2 - 16/3 g3[t]^2 - 3 g2[t]^2 - 7/15 g1[t]^2), hb[0] == hbzero, g3'[t] == -3 /(16 \[Pi]^2) g3[t]^3, g3[0] == Sqrt[4 \[Pi] 0.118], g2'[t] == 1 /(16 \[Pi]^2) g2[t]^3, g2[0] == Sqrt[4 \[Pi]/30], g1'[t] == 33/5 /(16 \[Pi]^2) g1[t]^3, g1[0] == Sqrt[4 \[Pi]/60], M3'[t] == -6 /(16 \[Pi]^2) g3[t]^2 M3[t], M3[Qmax] == gaugino, M2'[t] == 2 /(16 \[Pi]^2) g2[t]^2 M2[t], M2[Qmax] == gaugino, M1'[t] == 66/5 /(16 \[Pi]^2) g1[t]^2 M1[t], M1[Qmax] == gaugino, mhu2'[t] == 3/16/\[Pi]^2  Xu[t] - 1/16/\[Pi]^2 ( 6 g2bis[t]^2 M2bis[t]^2 + 6/5 g1bis[t]^2 M1bis[t]^2), mhu2[Qmax] == higgsmassparam, mhd2'[t] == 3/16/\[Pi]^2  Xd[t] - 1/16/\[Pi]^2 (6 g2bis[t]^2 M2bis[t]^2 + 6/5 g1bis[t]^2 M1bis[t]^2), mhd2[Qmax] == higgsmassparam, mq2'[t] == 1/16/\[Pi]^2 (Xu[t] + Xd[t]) - 1/16/\[Pi]^2 (32/3 g3bis[t]^2 M3bis[t]^2 + 6 g2bis[t]^2 M2bis[t]^2 + 2/15 g1bis[t]^2. M1bis[t]^2), mq2[Qmax] == sfermions, mu2'[t] == 2/16/\[Pi]^2 Xu[t] - 1/16/\[Pi]^2 (32/3) g3bis[t]^2 M3bis[t]^2 - 1/16/\[Pi]^2 (32/15) g1bis[t]^2 M1bis[t]^2, mu2[Qmax] == sfermions, md2'[t] == 2/16/\[Pi]^2 Xd[t] - 1/16/\[Pi]^2 (32/3) g3bis[t]^2 M3bis[t]^2 - 1/16/\[Pi]^2 (8/15) g1bis[t]^2 M1bis[t]^2,  md2[Qmax] == sfermions, at'[t]==(1/16Pi^2)((at[t](18htbis[t]^2+hbbis[t]^2-(16/3)g3bis[t]^2-3g2bis[t]^2-(13/15)g1bis[t]^2)+2ab[t]hbbis[t]htbis[t]+ htbis[t]((32/3)g3bis[t]^2M3[t]+6g2bis[t]^2M2[t]+(26/15)g1bis[t]^2M1[t]))),at[Qmax]==0, ab'[t]==(1/16Pi^2)((ab[t](18hbbis[t]^2+htbis[t]^2-(16/3)g3bis[t]^2-3g2bis[t]^2-(13/15)g1bis[t]^2)+2at[t]htbis[t]hbbis[t]+\hbbis[t]((32/3)g3bis[t]^2M3[t]+6g2bis[t]^2M2[t]+(26/15)g1bis[t]^2M1[t]))),ab[Qmax]==0}, {g3[t], g2[t], g1[t], ht[t], hb[t], M1[t], M2[t], M3[t],mhu2[t], mhd2[t], mq2[t], mu2[t], md2[t],at[t],ab[t]}, {t, 0, Qmax}]     g3bis[t_] = g3[t] /. system          g2bis[t_] = g2[t] /. system          g1bis[t_] = g1[t] /. system          htbis[t_] = ht[t] /. system          hbbis[t_] = hb[t] /. system          M3bis[t_] = M3[t] /. system          M2bis[t_] = M2[t] /. system          M1bis[t_] = M1[t] /. system          mhu2bis[t_] = mhu2[t] /. systema          mhd2bis[t_] = mhd2[t] /. systema          atbis[t_]=at[t]/.system          abbis[t_]=ab[t]/.system          Plot[{g1bis[z], g2bis[z], g3bis[z]}, {z, 0, Qmax}, Frame -> True]          Plot[{M1bis[z], M2bis[z], M3bis[z]}, {z, 0, Qmax}, Frame -> True]          Plot[{mhu2bis[z], mhd2bis[z]}, {z, 0, Qmax}, Frame -> True]      As you can see I've taken the third family approximation and neglected lepton terms too. These code works without putting at[t] and ab[t] terms and dependencies, when not doing so, nothing works.


title: Is the convective heat coefficient always dependent on heat flux and delta T?

text: I'm having a difficult discussion with a colleague because I'm running heat flow simulations for a small heater (5 mm^2 area), and I insist that if we're going to run more electrical power through the heater then we should modify the convective coefficient. However this person claims that even though we are running different powers through the heater and hence obtaining different delta T since we reach a higher temperature, we should keep the same h coefficient, which to me doesn't make sense because every single equation for h that I've seen so far depends on heat flux and temperature in some way or another. My simulation results match experimental data of the heater if we change the convective coefficient. This person claims that there should be a single h value for all the simulations.


title: Kinematics and Motion

text: If a body loses half of its velocity on penetrating 3cm in a wooden block, then how much will it penetrate more before coming to rest? How does answer come out to be 1cm?


title: Decoherence and collapse

text: It is said that the decoherence does not solve the problem of measurement and/or the emergence of classicality, can somebody explain it with simple analogies or in a manner accessible to a non-professional. Concepts like coarse graining, diagonalization of density matrix, discontinuous jumps and so on are a way beyond and by same way comments that there is no collapse models.


title: A question about Lorentz invariance of the Polyakov action

text: I have a super basic and stupid question about the Lorentz invariance of the Polyakov action (cannot skip the disclaimer..) $$S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \gamma^{ab} \partial_a X^{\mu} \partial_b X_{\mu} $$ I may write the action as (using metric tensor $\gamma^{ab}$) $$S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \partial^a X^{\mu} \partial_a X_{\mu} $$ The Lagrangian is obviously invariant under proper Lorentz transformation. $\partial^a X^{\mu} \partial_a X_{\mu}$ is a Lorentz scalar. $d\tau d\sigma$ transforms with a determinant equals 1 for proper Lorentz transformation. $(-\gamma)^{1/2} $ also transforms with a determinant equals 1. But for the range of integration $[0,l]$, there is length contraction under a boost. Why the action is still invariant under Lorentz transformation? (or I completely mistaken something....)


title: Why did they omit the action-reaction pair force in this problem?

text: **Problem** ![](http://img577.imageshack.us/img577/2205/problemse.jpg)! **Solution** ![](http://img543.imageshack.us/img543/858/solutionq.jpg)! **My solution** ![](http://img853.imageshack.us/img853/4305/solution2.jpg)! Let me explain $n_{bg}$ means the force exerted by the block on the ground. This is because the normal force of the slab exerts (or touches) a force on the block and by Newton's Third Law, it repels with a force n. So $n_{ground}$ is actually a force exerted by the ground and the force exerted by the ground on the block Also I kinda intuitively knew to put a friction force on the slab on the right. Is this a consequence of Newton's third Law? As the block slides to the right, the friction force drags it from the left. On the slab, (imagining now) wouldn't the heat/friction be sliding on top from the left?


title: Is there a way to tell the rate of temperature change due to provided energy?

text: Let's say we have an isolated system containing two bodies with different temperatures. The bodies are touching each other. Is there a way to tell how long will it take them to reach thermodynamic equilibrium? A simpler example: we provide an amount of energy to a body with a known heat capacity. Disregarding other effects such as thermal radiation, we can derive the change in body's temperature from the formula $$C = \frac{\Delta Q}{\Delta T}$$ but is there a way to say **how fast** will it happen? The lack of confident explanations for the Mpemba effect suggest that there is yet no way to tell. Is it just a thing scientist haven't figured out yet, or maybe the rate of temperature change is dependant on too many variables to specify it in a single formula?


title: Superfluidity and superconductivity?

text: Superfluidity and superconductivity? Is there any relation between the two aside from the fact that they are somewhat analogies , I see a lot of people on the internet who claim the helium (in it's superfluid state) is a also a super conductor but searching this site and Wikipedia I couldn't find anything speaking directly of this although I found statements like this "liquid helium is used to cool the superconducting material ...." which suggest the opposite . -update: "Josephson effect has also been observed in SHeQUIDs, the superfluid Helium analog of a dc-SQUID" in case SHeQUIDs isn't a superconductor how could this be?


title: traditional transformer design improvement

text: The traditional tranformers suffer a lot of losses in various forms, some of them are as follows : 1\. Flux leakage 2\. Eddy currents 3\. Hysteresis Figure : ![Figure](http://lh3.ggpht.com/-8_dBB49n_3Q/UmPS6S2qMUI/AAAAAAAAAd4/FQy4MSOMjG8/s1600/2013-10-18%25252020.04.41.png) But as we are taught induction we come across the attached figure of two inductors placed coaxially various times. My point is that if we would actually place inductors in this way, we would have the following benefits : Note : taking inner coil as primary   1. NO NEED OF CORE : As the flux of inner coil passes from outer on its own, no core is needed for transferring flux from primary to secondary   2. NO FLUX LEAKAGE : As entire flux of inner inductor passes from outer, no flux is leaked.   3. NO EDDY CURRENTS: As no core is used no eddy currents are generated.    4. NO HYSTERESIS : Similarly due to absense of core, no loss due to hysteresis is encountered. So what really is the problem in creating transformers in such a way to reduce losses much more than the current status ? All relevant opinions and comments appreciated My original post : http://thinklo.blogspot.in/2013/10/improving- transformers.html


title: Reason for the convention about polarization states

text: I'd like to know if there is a special reason for limiting convention of _polarization state_ to waves that can be split in just two components of _equal frequency_.


title: Reference frame involved in the Schrödinger's equation

text: I have a doubt as to in which frame of reference the Schrödinger equation is written? I think it is inertial but can't reason it out.


title: Why do physicists believe protons and electrons are present in equal numbers?

text: I tended to consider that negative and positive charges are present in equal numbers in the universe to be a known, obvious fact. But is it so? How can we rule out the possibility that there is some kind of asymmetry in the numbers of protons and electrons? Of course, matter is neutral and even tiny deviations would give rise to enormous forces... But not all protons and electrons are in atoms in the universe. More importantly, is there some conceptual reason why we believe that this equality is exactly perfect or do modern theories allow tiny deviations?


title: What frequency photons are involved in mediating physical force?

text: If the force felt when pushing an object is mediated by the electromagnetic interaction and hence photons, what is their frequency?


title: calculating thermal energy [exam prep]

text: I'm preparing for a grade 11 physics exam but having trouble with a question: a 19.9kg mass slides in a straight 28.8m line experiencing friction with a coefficient of 0.105, how much thermal energy is produced?


title: 2$^\text{nd}$ law of thermodynamics: equivalence of statements

text: Consider the following statements: * * * Clausius statement (according to Wikipedia): > No process is possible whose sole result is the transfer of heat from a body > of lower temperature to a body of higher temperature. * * * Clausius equality: $$\oint \frac{\delta Q}{\theta} = 0$$ (which implies existence of entropy $S$: $dS = \frac{\delta Q}{\theta}$) * * * They both are the statements of the second law of thermodynamics. How can one prove their equivalence? This proof seems to be frequently omitted from books (I don't know why). Frankly speaking, the Clausius statement may be easily derived from the Clausius equality. The question is then the derivation of the equality from the statement.


title: Regularisation of infinite-dimensional determinants

text: Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM? Edit: I failed to make myself clear. In finite dimensions, there is a function of $\lambda$ whose roots are the eigenvalues (or inverses of the eigenvalues) of a given operator $U$, namely, the characteristic polynomial $\det (I-\lambda U)$. Is there some way of regularising this determinant to do the same thing in infinite dimensions? In general? Or at least for unitary operators which describe the time evolution of a quantum mechanical system? link to a related question What does a unitary transformation mean in the context of an evolution equation? EDIT: Perhaps the question still is not clear. The question was, and still is, ¿if you regularise $\det(I-\lambda U)$ as a complex valued function of $\lambda$, for $U$ a unitary operator, will its zeroes be the values of $\lambda$ such that $I-\lambda U$ fails to be invertible? ¿Has a non-zero kernel? ¿Or does regularising the determinant lose touch with that property of the finite dimensional determinant?


title: Might the LHC see nothing new at all?

text: There's no guarantee that supersymmetry (or more exotic new physics) will be seen at the LHC. Meanwhile, it's standard lore that a Higgsless standard model becomes nonunitary somewhere in the vicinity of 1 TeV. Are we at least guaranteed to get _some_ positive information about new physics, in the form of deviations from the predictions of this "higgsless standard model"?


title: Why are some solitons formed from bosonic fields fermionic?

text: Some topological solitons formed from bosonic fields have fermionic statistics. Why?


title: Phonon dispersion calculation based on velocity autocorrelation function in reciprocal space?

text: I would like to ask about the method of calculating phonon dispersion relation. Up to my knowledge, there are 2 methods to calculate the phonon dispersion:   1. By diagonalizing the dynamics matrix. There are an open source code that implement this method in many molecular dynamics packages such as fix phonon in LAMMPS.   2. By Fourier transforming the velocity correlation function such that the velocity is in the reciprocal space. I want to apply this method. I found some papers that list the equations of calculation, however, the method is not clear. **I wounder if some of you have experience about this topic** . One of the papers is: "Non-local modeling of epoxy using an atomistically- informed kernel" They use the following equation to convert the velocity to the reciprocal space: ![Velocity in reciprocal space](http://i.stack.imgur.com/o13VZ.png) My question is about the position $r$ value, we know that the position is vector and has three components $(x, y, z)$, in the equation it is written but not clear if it is converted to a scalar value, and how? or they took the position as the same component of velocity. Another question is: as we will have three three correlation functions, one for each component $(x, y, z)$, Do they take their average or summation? So we will have one Fourier transform. If not we will have three three Fourier transforms and I don't know how to take the peaks from three Fouriers. Please if any body have an idea about this topic, it is so helpful for me. Even if you know any code written for this purpose it will be helpful so I can compare it with my code to find the answers of my questions.


title: Moment of Inertia for a hollow cylinder

text: This question may be to rudimentary. I have found 2 formulas for calculating the inertia of a hollow cylinder. Which is correct or are they used for different circumstances? They seem to be used interchangably which simply imputing the same variables proves to be untrue. ![enter image description here](http://i.stack.imgur.com/jrUj3.jpg)


title: photon momentum

text: assume 2 space ships "at rest" in the vacuum, at about 300.000KM apart one from another, one having a laser source, the second having a receptor. Also assume both ships have synchronous clocks (they were together, synchronized their clocks and started to split apart slowly, both having the same acceleration, etc. so their clocks remained synchronous because they both were subject to similar acceleration forces, etc.). The first ship sends light pulses continuously towards the second and they are detected by the receptor on the second ship. Now both ships start to accelerate in the same direction (the laser ray is perpendicular on the moving direction.) keeping the distance between them constant at all time; we can assume there is a stretched thread linking them, the thread will remain fully stretched at all time at the same extent. Soon after they start accelerating, the laser will start missing the receptor on the second ship. Now they both stop accelerating at the same time(so they are now again "at rest" basically as observed by their astronauts, no acceleration felt, no changes in the distance between ships, etc.). I'm thinking now the receptor will start again receiving light pulses and the question I have is "Why is that ?". I mean, if instead of light it would have been a gun then it is just normal that the receptor to start receiving projectiles after acceleration stops but that is because the projectiles have a mass and the kinetic energy (so they have momentum as induced by the acceleration, etc.). Photons on the other hand don't have mass (they did not even exist before the moment they were created by the laser ray generator, etc.). So how do these "lately emitted photons" get a momentum from the emitting ship as they were regular (mass) projectiles ?


title: Energy loss of Earth

text: How does the Earth lose the energy that it gains from the sun's radiation if it is surrounded by - and in contact with - nothing?


title: Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?

text: For an observable $A$ and a Hamiltonian $H$, Wikipedia gives the time evolution equation for $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ in the Heisenberg picture as $$\frac{d}{dt} A(t) = \frac{i}{\hbar} [H, A] + \frac{\partial A}{\partial t}.$$ From their derivation it sure looks like $\frac{\partial A}{\partial t}$ is supposed to be the derivative of the original operator $A$ with respect to $t$ and $\frac{d}{dt} A(t)$ is the derivative of the transformed operator. However, the Wikipedia derivation then goes on to say that $\frac{\partial A}{\partial t}$ is the derivative with respect to time of the _transformed_ operator. But if that's true, then what does $\frac{d}{dt} A(t)$ mean? Or is that just a mistake? (I need to know which term to get rid of if $A$ is time-independent in the Schrodinger picture. I think it's $\frac{\partial A}{\partial t}$ but you can never be too sure of these things.)


title: We need to find unit vector along the reflected ray

text: A ray of light on a plane mirror comes along a vector $i+j-k$ The normal on incidence point is along $i+j$ we need to find unit vector along the reflected ray. I am not able to solve and draw the picture. could any one explain me what is going on? answer given is $-{1\over\sqrt{3}}(i+j+k)$


title: Explain the Föppl–von Kármán equations

text: I am a newbe to elasticity.   Could someone please explain to me briefly how the Föppl–von Kármán equations work?   What are we trying to solve for?   Is there some kind of intuition to the way they look?   Are the two terms being subtracted have any significance?   $$ \frac{Eh^3}{12(1 - \nu^2)} (\nabla^2)^2 \zeta - h \frac{\partial}{\partial x_\beta} \left(\sigma _{\alpha \beta}\frac{\partial \zeta}{\partial x_\alpha}\right)=P ^2 \\\ \frac{\partial \sigma_{\alpha\beta}}{\partial x_\beta} =0 $$ Each letter is explained on Wikipedia.


title: Why does chemical potential smaller than zero mean nondegeneracy and vice versa

text: In Mudelung's book, Introduction to Solid-State Theory, I have a confusion about the statement. ![enter image description here](http://i.stack.imgur.com/cxgwE.jpg) Here, $x$ should be $\frac{\mu}{k_B T}$. I am cofused about his statement. Why does $x<0$ mean nondegeneracy and $x>0$ mean strong degeneracy? I don't understand his statement, can you help me?


title: Electrochemical cell potential - Prigogine book

text: I'm trying to understand equation 10.2.14 from Prigogine book "Moder thermodynamics": $V=\frac{\tilde{A}}{nF}=\frac{1}{nF}\left(\mu_X^R+\mu_Y^L-\mu_{Xred}^R-\mu_{Yox}^L\right) - \left( \phi^R-\phi^L \right)=V_{cell}-\left( \phi^R-\phi^L \right)$. In this equation the leftmost term is a voltaje V whereas the rightmost term is precisely the electrostatic potential difference between both electrodes. Then, what is V? In the book derivation it is traced back to the electrostatic potential drop of a conductor wire connecting the electrodes, but then this should be equal to $\left( \phi^R-\phi^L \right)$ and then $V_{cell}$ would be 0. So I think it is an error of Prigogine book and in fact I haven't seen that equation with that "V" in any other book. I have figured out that the mistake is in the derivation: assuming that the entropy production of the whole system is the same that the entropy production of the connecting wire is what leads in the text to identifying that conducting voltage drop $V$ with the quantity $\frac{\tilde{A}}{nF}$. Indeed, the books says "in equilibrium, $V=0$", but having a conducting wire interconnecting both electrodes in equilibrium would be the trivial equilibrium situation when there is no EMF after the complete discharge of the cell, and then of course V=0. Could you throw a bit of light on this? Thanks in advance


title: How do determine the equation of motion of slinky?

text: ![](http://www4.picturepush.com/photo/a/13074552/640/13074552.png) I'm trying to analyze the forces exerted to slinky. First, I've divided both ends. The red vector is the force exerted by slinky itself, and the green vector is gravitational force. Force exerted in upper side $\vec{F_u}=kx\cos{\theta_u}\hat{i}+(kx\sin{\theta_u}-mg)\hat{j}$ Force exerted in down side $\vec{F_d}=(kx\cos{\theta_d}-\mu_sN)\hat{i}+(kx\sin{\theta_d}-mg+N)\hat{j}$ The functions $\theta_u$, $\theta_d$, $kx$ are respect to time variable. But I can't figure it out how to determine those functions. Or am I just going wrong way?


title: Why distinguish between row and column vectors?

text: Mathematically, a vector is an element of a vector space. Sometimes, it's just an n-tuple $(a,b,c)$. In physics, one often demands that the tuple has certain transformation properties to be called a vector (as opposed to e.g. a pseudovector). Sometimes its not a tuple, but e.g. an element of a hilbert space. In all cases, its possible to define a ordinary vector (or column vector, or contravariant vector): $$\vec v = \left(\begin{align}v_1\\\v_2\\\v_3\end{align}\right) \qquad x^\mu \qquad \left|\Psi\right\rangle $$ and a dual (row, or covariant) vector: $$\vec v^T = \left(v_1, v_2, v_3\right) \qquad x_\mu \qquad \left\langle\Psi\right| $$ You can define a scalar product that gives you a number (an element of the field $K$ the vector space is over). If you're feeling fancy, you can think about the row vector as an element of a dual vector space. This consists of functions (1-forms) mapping vectors to numbers. $$\vec v \in V_K$$ $$\begin{align} \vec u^T : V_K &\rightarrow K \\\ \vec v &\mapsto x\end{align}$$ Applying this function $\vec u^T$ to the vector $\vec v$ is the same as to take the scalar product $\vec u \cdot \vec v$. This is more manifest in the co/contravariant notation, or in the bra-ket notation. **My question is why do we distinguish between both kinds of vectors (in concept, not in notation)?** Is there any physical meaningful distinction, or couldn't we just teach that they are all "vectors" and skip the dual part? Could we say that the raised/lowered indicies are just a helpful notation to remember what is multiplied with what, or a shortcut for $x_\mu = g_{\mu\nu}x^\nu$, but that there is actually just _the vector_ $\mathbf{x} = (x^0, x^1,\ldots)$ I mean, do we loose anything by thinking $\left|\Psi\right\rangle \equiv \left\langle\Psi\right|$ and the different forms are just shortcuts for different products: $$\left\langle\Phi\middle|\Psi\right\rangle = \left|\Phi\right\rangle \cdot \left|\Psi\right\rangle $$ $$\left|\Phi\right\rangle\left\langle\Psi\right| = \left|\Phi\right\rangle \times \left|\Psi\right\rangle $$ Is there for example any physical situation where there is a meaningful variable $A$ that can be either $A = \vec v$ or $A = \vec v^T$, and by giving up the distinction (conceptually) we can't tell the two situations apart?


title: What are units actually?

text: This question is about the concept of units in physics. Firstly - do units have a formal mathematical definition? How are they different from pure numbers? Are pure numbers defined to be ratios of unit sizes? Were the seemingly implicit laws of units always there or did someone articulated them in the past? Why can't you use the $\log$ or $\exp$ functions on numbers with units? (is it undefined? why?) Why is it not ok to do math with different units, i.e. compare mass and distance, but it is ok to compare and do math with ratios of masses and ratios of distances? Are there any laws that allow conversion of different units to one another or are there conservation laws for each unit?


title: Can a diver swim a short distance in great depths without being physically crushed by the pressure?

text: I recently saw "The Abyss". Does it make sense that they do dives in these depths (700m) with soft suits? Also - what is all the depressurization talk about? Why do divers need to depressurize long periods after they resurface? Does that mean crew members can't escape drowning submarines?


title: Can we use intensities in the superposition principle?

text: In using the superposition principle to calculate intensities in interference patterns, can we add the intensities of the waves instead of their amplitudes? I think that amplitude account for the intensities so that both are the same thing and so it doesn't matter.


title: Trotter splitting and entanglement entropy

text: I have heard that a numerical solution to the Schrodinger equation using the Trotter splitting formula for a many-body Hamiltonian can cause an artificial increase in the entanglement entropy. I was not able to get a reference from the guy at the bar I was talking to before we left. Could anybody explain why this is the case?