Vacuum stability in quantum field theory

What exactly do people mean when they talk about the scale dependence of the effective potential ($V$)? I explain the motivation for my question (and hence my confusion) below. Please correct me as appropriate. * * * If one defines the effective potential as the non-derivative part ($p^2 \rightarrow 0$) of the 1-PI effective action, then something like the Callan- Symanzik equation will imply that $$\frac{d}{d \log \mu} V = 0 \implies \left[ \frac{\partial}{\partial \log \mu} + \beta_i \frac{\partial}{\partial \lambda_i} + \gamma \frac{\partial}{\partial \log \phi} \right] V = 0$$ If the value of the effective potential at any field value (and zero momentum) is scale independent, then shouldn't the vacuum be stable at all scales, if we know it's stable at some scale? * * * All that I've said above holds so long as perturbation theory has been used correctly. I've also seen some people "RG-improve" the effective potential by resumming the leading logs across all loop order. But then, the moment you do a partial resummation across loop orders, is there any reason for the thing you calculate to be scale independent? It's also not clear what the physical interpretation of such a scale-dependent quantity should be -- so why should one take the appearance of another vacuum seriously -- after all, you don't see any such thing when you do your calculations at some "low" scale -- and physical observables (at zero momentum) better not be RG-scale dependent?


Optimal placement of support joist under shelf

Assume an ideal board, supported by two joists. Where should those two joists be optimally placed? Instinctively, I'd say at somewhat less than 25% and somewhat more than 75% of the extend of the board. Presumably this also depends on whether the board can move freely with respect to the joists, or whether the board is attached to the joists. I'm unclear even where to start analyzing this problem ...


Do quantum states contain exponentially more information than classical states? It might seem so at first sight, but what about in light of this talk?


Uncertainty principle and commutation relations

What connection exists between the uncertainty principle and commutation relations amongst the operators representing observables in Quantum Mechanics?


Heisenberg Uncertainty Principle scientific proof

Heisenberg's uncertainty principle states that: _if the x-component of the momentum of a particle is measured with an uncertainty_ $$\Delta \vec p_x$$ _then its x-position cannot, at same time, be measured more accurately than_ $$\Delta\vec x=\frac {\hbar}{2\Delta\vec p_x},$$ $$\Delta\vec x\Delta\vec p_x \ge \frac {\hbar}{2}.$$ What is the scientific proof of this principle? Operators Uncertainty


Can T-duality resolve spacelike singularities?

Schwarzschild singularities are described by the Kantowski-Sachs metric with a contracting S2. Of course, T-duality doesn't apply to S2. But what about a Kasner-type singularity with two contracting spatial dimensions compactified over a torus T2, and an expanding spatial dimension? The T-dual of the torus gives rise to a geometry which is expanding in all spatial directions.


Grip of the train wheels

How do the wheels of a train have sufficient grip on a metal track? I mean both of the surfaces are smooth (and not flexible) and it is okay if there is no inclination, but how about on an inclined track?


Why can you see virtual images?

In optics it is widely mentioned real images are projectable onto screens whereas virtual ones can only be seen by a person. Isn't that contradictory? I mean in order to see the virtual image it has to be _projected_ onto the retina (ultimately acting as a screen). So, why can you see virtual images in the first place?


conceptual meaning of "virtual image"

I am trying to learn about optics and I am having a hard time understanding the meaning of "real" vs "virtual" image. My understanding is that for a concave mirror, the image focuses on the same side as the object so it is a real image. For a convex mirror, the image focuses on the opposite side of the object from the mirror so it is a virtual image. However, we can both see either real or virtual images, so how are we seeing something that focuses on the other side of the mirror in the convex case ? I am very confused.


What are virtual objects, Reflection of light?

While studying reflection through a plane mirror, I have been told that when the object is real the image will be virtual and **the image will be real while the object is virtual**. What are virtual objects? Do they really exist in nature?


What is the difference between virtual object and virtual image?


Help me understand virtual images versus real images

Simple question; Is a virtual image that is produced by a lens always - in front of or in back of the lens?


Virtual vs Real image

I'm doing magnification and lens in class currently, and I really don't get why virtual and real images are called what they are. A virtual image occurs the object is less than the focal length of the lens from the lens, and a real image occurs when an object is further than focal length. By why virtual and real? What's the difference? You can't touch an image no matter what it's called, because it's just light.


Why do I get readings from my radiation meter with magnetized pieces?

What causes those radiation readings in my radiation meter (See the linked video)? Metal pieces were magnetized. Used radiation detector is RADEX RD1503. "Radiation" is not generated if the tube has been wrapped inside an aluminium foil. Should induction occur through thin aluminium foil? At least magnet holds through that foil and plastic tube.


If non-zero cosmological constant interpreted as a repulsive field, what would be the properties of this field's quanta?

If non-zero cosmological constant interpreted as a repulsive field, what would be the properties of the excitation of such field, i.e. the particle which serves as the field's quantum? What would be its spin, mass, possible interactions and other properties?


Protection of the electron mass by chiral symmetry

In many textbooks it is said that mass renormalization of the electron mass is only logarithmic $\delta m \sim m\, log(\Lambda/m)$ because it is protected by the chiral symmetry. I understand that in case of massless fermions to keep them massless in the renormalization procedure it must be like this. Or differently said, the renormalization procedure respects the axial current conservation. But is there a compulsory reason for the renormalization procedure to respect the axial current conversation ? Does every renormalization procedure respect that ? Apparently Pauli-Villars and dimensional renormalization do it, but what for other procedures ? I also know that in triangular Feynman diagrams anomalies occur which do break the axial current conversation. So why can't it happen for something simpler like the electron mass respectively self-energy ?


When we get an electrical shock, the ciruit is incomplete but still the current flows, why?

If we touch the live wire and ground at the same time, we will get a shock. But the current goes from live to ground and not to neutral i.e, circuit is open. Then how can we get a shock? ![Image](http://i.stack.imgur.com/fLKbj.png) This is the same circuit: will the current flow?   ![Image](http://i.stack.imgur.com/nryo4.png) I think current should not flow. So, what is the reason that we still get shocks when the circuit is open?


thrust-bursting towards a destination in space

<\--burst of thrust------|ME in a ship>----->>--<<\--laser radar--<<\---->>---||destination In the above, I imagine myself in a ship, marking my speed to a far off destination with laser radar. In addition, someone at the destination is clocking me as well with a separate laser. So, moving toward my destination at speed S1, I'm at rest. Then I fire a burst of thrust and experience acceleration, a, for time, t, and arrive at a new speed S2, right? This is all just simple physics right? My question is, where does special relativity play into this and how would it be perceived by A) me in the ship and B) the person at the destination clocking the ship? Would it be reasonable to want to graph velocity and distance to destination over time, given regular bursts of thrust, for both me in the ship and the observer at the destination? If so, any tips/equations to use? PS: Although this might sound like a homework problem (or maybe I'm giving myself too much credit), it's actually just a lunchtime conversation :)


Tension on an object

According to a website, $T_3 = T_2 + T_1$. Also, the rope and pulley device are mass less. But I don't really understand this. Let me try and explain how I see the problem. $M_2$ receives tension in the up direction. This tension is the result of the resultant force of $M_1$ on the rope. $M_1$ receives tension in the up direction. This tension is the result of the reactant force of tension in $T_2$ The forces on the rope connected to the masses are: $T_{1m}$, $T_{2m}$ and these are both resultant forces from said masses. But since they are technically in different directions, they cancel each other out. So the net force on the rope is zero. But why is $T_3 = T_2 + T_1$? I do not understand this at all. It just doesn't make sense to me. ![enter image description here](http://i.stack.imgur.com/lRZc2.gif)


Explanation of binding energy in decays

Everyone knows that the mass of a system is less than the mass of its components, with the equation: $M = \sum_i m_i - BE(M) $ Now, if we consider a general decay, lets say $A \rightarrow \sum_i B_i$ then, for the conservation of the first component of the four-vector momentum we obtain a necessary condition for the decay: $ M(A) \ge \sum_i M(B_i) $ Isn't this last condition in contradiction with the first one? The only answer I get is a particle that can decay is not an eigenstate of the system. For this reason it shouldn't have a defined binding energy. So, for example, Tritium, that can decay, hasn't a BE? This appears so illogical if we think about nuclei with an half-life of years or more. Moreover I know this energy is experimentally defined and is bigger that 3He's BE.


Christoffel symbols and Dirac matrices mathematical similarities?

Maybe mine is a silly question, but are there **mathematical similarities** or **common roots** between the Christoffel symbols: $ \nabla - \partial = \Gamma $ and the Dirac matrices $ ( \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}I )$ obtained through: $ (\gamma^\nu \partial_\nu)(\gamma^\mu \partial_\mu) = \partial^\mu \partial_\mu $ **EDIT:** What I mean is that the Dirac matrices are obtained by trying to match two different derivatives, so I was wondering if that had some common ground with the Christoffel symbols that are defined as the difference between the connection and the coordinate derivative.


What is the difference between Quantum Physics, Quantum Theory, Quantum Mechanics, and Quantum Field Theory? Are they the same subject? I believe that they are not the same subject! Maybe there is not big difference between those subjects but I need to know what is main difference between those subjects and what is main intersection? Also I need to know which one is big subject relative to another?


Calibrating an electronic temperature sensor based on power consumption

I'm working with an electronic temperature logger that is being affected by heat generated internally. How does one come up with a calibration equation to calculate a more accurate reading of ambient temperature based on what the temperature sensor reads, taking into account its own power consumption? details: After a few hours and in equilibrium, the sensor reports values that are actually 1 degree Celsius higher than the ambient room temperature (22C) measured by a calibrated device. The sensor is accurate to 0.1 degree C at reporting the temperature of the device itself (which due to heat generated by the electronics has gotten warmer) The device consumes ~0.1 watts of power, weighs about 200g and has an average specific heat capacity of 1.0 j/g (weighted mix of glass, abs, fr-4, copper). Dimensions are 1"x 3"x 4". What I've got so far is this heating calculation: 200g * 1c * 1.0j/g / 0.1w / 60s = ~33 minutes to heat up 1 degree. I'm assuming what we need is to figure out Sensor value - Heat-generated + Heat-dissipated to arrive at actual temperature. Which will require measure the K in newton's law? then what? I'd really appreciate you help here.


Why does bad smell follow people (assuming they are not the source)?

When you are sitting in a room where there is a source of bad smell, such as somebody smoking or some other source of bad smell, it is often a solution to simply move to another spot where bad smell is not present. Assuming you are not actually the source of the smell, this will work for a while until you notice the smell has somehow migrated to exactly the spot where you are now sitting. Frustrating. This got me thinking about the fluid mechanics of this problem. Treat bad smell as a gas that is (perhaps continuously) emitted at a certain fixed source. One explanation could be that human breathes and perhaps creates a pressure differential that causes the smell to move around. Is there any truth to this? Please provide a reasoned argument with reference to the relevant thermodynamic and/or fluid quantities in answering the question. Theoretical explanation is desired, but extra kudos if you know of an experiment.


How to write the Fröhlich Hamiltonian in one dimension?

I am currently working on a (functional) analysis problem refining Pekar's Ansatz (or adiabatic approximation, as it is called in his beautiful 1961 manuscript "Research in Electron Theory of Crystals"). Anyways, I have two related questions, which the members of this community may find simple. The Fröhlich Hamiltonian is given as follows in three dimensions $$H=\mathbf{p^{2}}+\sum_{k}a_{k}^{\dagger}a_{k}-\biggl(\frac{4\pi\alpha}{V}\biggr)^{\frac{1}{2}}\sum_{k}\biggl[\frac{a_{k}}{|\mathbf{k}|}e^{i\mathbf{k\cdot x}}+\frac{a_{k}^{\dagger}}{|\mathbf{k}|}e^{-i\mathbf{k\cdot x}}\biggr]$$ The physical scenario here is an electron moving in a 3-dimensional crystal. Each $k$ signifies a (vibrational) mode of the crystal. If we restrict ourselves to just a 1-dimensional crystal, why is it that the Hamiltonian can be written as follows: $$H=\mathbf{p^{2}}+\sum_{k}a_{k}^{\dagger}a_{k}-\biggl(\frac{4\pi\alpha}{V}\biggr)^{\frac{1}{2}}\sum_{k}\biggl[a_{k}e^{i\mathbf{k\cdot x}}+a_{k}^{\dagger}e^{-i\mathbf{k\cdot x}}\biggr]$$ Namely, why do we drop the $|\mathbf{k}|$ factor in the third term? Furthermore, I see how the creation and annihilation operators work on the (bosonic) Fock space (referring to the crystal here), especially when we write the creation operator in the form $\sum_{k=0}^{\infty}\frac{(a^{\dagger})^{k}}{\sqrt{k!}}\left|0\right\rangle =\left|k\right\rangle$. Namely, the creation operator is jumping from one tensored state in Fock Space to the next. However, I also see the form $a_{k}=\frac{1}{\sqrt{2}}\bigl(k+\frac{d}{dk}\bigr)$. How are the two forms connected? How do you intuitively think of the latter form? For example, I thought of the former form as the creation operator jumping from one state in fock space to the next, but the latter form I am not quite sure.


Wrinkling paint - soluto/thermocapillarity - is it due to the primer or solvent

This is really a one-and-a-half part question. I know that when paint is mixed with a solvent or used with a primer, it sometimes wrinkles. As I understand, a key physical phenomena here is a non- uniform evaporation of the primer or solvent that gives rise to different evaporation rates at the paint surface. This can create a non-uniform temperature difference. This in turn can create a thermocapillarity (surface tension effects due to non-uniform temperature distribution) or solutocapillarity (surface tension effects due to non-uniform solvent/solute concentration) effects that wrinkle the paint surface. So is this a solvent problem or a primer problem? Am I understanding the physics right but the nature of primer and solvent wrong?


Constructing a maximally entangled qutrit state from $n$ Bell states

I've read that maximally entangled qubit states are a good "unit" of bipartite entanglement since it is possible to create any other entangled state from them using local operations and classical communication (LOCC) provided sufficiently many copies are available. What would the protocol be to construct a maximally entangled qutrit ($\vert \psi \rangle_{AB} = \frac{1}{\sqrt{3}}(\vert 00 \rangle + \vert 11 \rangle + \vert 22 \rangle)$) between two space separated parties from a set of $n$ Bell states ($\vert \Phi^+ \rangle^{\otimes n}_{AB} = 2^{-n/2} (\vert 00 \rangle + \vert 11 \rangle)^{\otimes n}$) initially shared by those parties using only LOCC? If you can, please include in your answer why local operations alone would not be sufficient.


Expansion of multi-particle state vector as a sum of n-entangled states

Physically, quantum entanglement is ranged from full long-range entanglement (Bose-Einstein condensate), described by a basis of states that look like this: $$ |\Psi\rangle = |\phi_{i_{0} i_{1} ... i_{N}}\rangle $$ to full-decoherence (a Maxwell-Boltzmann ideal gas) which a basis of states that look like this: $$ |\Psi\rangle = \prod_{i}{ |\phi_{i}\rangle } $$ And in the middle of this range we have 2-particle entanglement terms, 3-particle entanglement, etc. So it seems natural to arrange the wavefunction as a series looking like: $$ |\Psi\rangle = \sum{ \prod_{i}{ |\phi_{i}\rangle } } + \sum{ \lbrace \prod_{i_{0} ,i_{1} > i_{0}}{ |\phi_{i_{0} i_{1}}\rangle } \rbrace } + \sum{ \lbrace \prod_{i_{0} ,i_{1} > i_{0} , i_{2} > i_{1}}{ |\phi_{i_{0} i_{1} i_{2}}\rangle } \rbrace } + \dotsb $$ **BEGIN EDIT** I feel that i need to put in a bit more clear footing the mathematics behind this separation. Let's take an arbitrary state vector $|\Psi\rangle$, we might write it like this: $$ |\Psi\rangle = \prod_{i}{ c^{0}_{i} |\phi_{i}\rangle } + |\Psi_{Remainder}\rangle $$ That is, we write the state vector as a vector that is completely separable and a remainder that is not. _This decomposition is unique_. Proof: take another decomposition with $\widehat{c^{0}_{i}}$, take the difference between both decompositions and verify that both remainders are completely separable, which is against the definition The idea is that one should be able to further this decomposition of the remainder state vector, for instance lets take a vector with zero completely separable part (that is, we are on the equivalence class of our remainder above) and attempt to write it like: $$ |\Psi_{R_0}\rangle = \prod_{i_{0} ,i_{1} > i_{0}}{ c^{1}_{i_0 i_1} |\phi_{i_{0} i_{1}}\rangle } + |\Psi_{R_1}\rangle $$ Analogously, one can prove that the $c^{1}_{i_0 i_1}$ are unique and depend only on the $|\Psi_{R_0}\rangle$ vector. And we don't need to do any symmetrization operation to get this result, so this is an universal decomposition of the state vector (for bosons and fermions) (note: i'm aware that this leaves out a lot of products with mixed entanglement, i.e: some single-particle states multiplied by two-particle entangled states, but since they don't add anything to this particular argument i choosed to leave them aside) **END EDIT** **BEGIN 2ND EDIT** separability of states is a property that is invariant under unitary transformations, so a separable state is not equivalent to a separable one in any basis you choose. I've investigated a bit more and the problem in general of knowing if a state is separable or not is known as QSP (quantum separability problem). For a definition please look at this paper **END 2ND EDIT** Question: Where can i read more about this sort of expansion and do you know if there is a computational framework to estimate the relative magnitudes of each term (for instance, i would expect that for Bose-Einstein condensates you need to keep all the terms of the expansion, while for relatively high- temperature solids you would be able to get away with 3 or 4 terms)


Can the mass of an orbiting object and the object being orbited be determined by the distance and orbit velocity alone?

This question is the inverse of: "Could an object orbit while moving at twice the speed, but at the same distance, if it had half the mass?" I'm curious about the nature of orbits, but am not well enough versed in mathematics to understand Kepler's laws well. I have been wondering if the mass of a planet and a star it orbits could be determined based solely on the distance and speed of the orbit, or if the ability to orbit at a given speed/distance was based _relatively_ on the mass of both objects (i.e., we could determine the ratio of the mass of the two objects, but not the actual mass).


What is the relationship between mass, speed and distance of a planet orbiting the sun?

After reading this fascinating story about a new exoplanet, I was wondering about how mass, speed and distance determine a circular orbit of a planet around a star. Given the mass of the sun and star, and the distance between them, is there only one possible orbital velocity? Given any 2 of the following, is it possible to calculate the 3rd?   * Relative mass of the planet to the sun   * Orbital velocity   * Distance between the 2 masses And, given 1 of the above, are there an infinite number of possible values for the other 2?


Transformation Law for Covariant Derivative in $SU(2)$ Yang-Mills

In page 488 of Peskin and Schroeder, it is stated (emphasis mine): > It is not difficult to check using (15.27) and (15.21) that, even for > _finite_ transformations, the covariant derivative has the same > transformation law as the field on which it acts. I was trying to indeed verify that. This is what I have tried:   1. $V\left(x\right)\in SU\left(2\right)^{\mathbb{R}^4}$   2. $\psi\left(x\right)\mapsto V\left(x\right)\psi\left(x\right)$.   3. $D_\mu\left(x\right)\equiv\partial_\mu-igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}$.   4. $igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\mapsto V\left(x\right)igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\left[V\left(x\right)^\dagger\right]-V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)^\dagger\right]\right\\}$ Thus: \begin{align} D_\mu\left(x\right)\psi\left(x\right) \mapsto & \left\\{\partial_\mu -V\left(x\right)igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\left[V\left(x\right)^\dagger\right]+V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)^\dagger\right]\right\\}\right\\}V\left(x\right)\psi\left(x\right) = \\\\\ &= \left[\partial_\mu V\left(x\right)\right]\psi\left(x\right)+V\left(x\right)\partial_\mu\psi\left(x\right)-V\left(x\right)igA_{\mu,\,j}\left(x\right)\frac{\sigma^j}{2}\psi\left(x\right)+V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right)\psi\left(x\right) = & \\\\\ &= V\left(x\right)D_\mu\left(x\right)\psi\left(x\right)+\left\\{\left[\partial_\mu V\left(x\right)\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right)\right\\}\psi\left(x\right) \end{align} So as far as I understand, $\boxed{\left[\partial_\mu V\left(x\right)\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right)\stackrel{?}{=}0}$ should be zero. To prove that I have used the fact that $VV\dagger=1$: \begin{align} \partial_\mu\left[ V\left(x\right)\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right) &= \\\ \partial_\mu\left[ V\left(x\right)1\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right) &= \\\ \partial_\mu\left[ V\left(x\right)V\left(x\right)^\dagger V\left(x\right)\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right) &= \\\ \left[\partial_\mu V\left(x\right)\right]V\left(x\right)^\dagger V\left(x\right) +V\left(x\right)\left[ \partial_\mu V\left(x\right)^\dagger \right]V\left(x\right) +V\left(x\right)V\left(x\right)^\dagger\left[ \partial_\mu V\left(x\right)\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right) &= \\\2\left\\{\partial_\mu\left[ V\left(x\right)\right] + V\left(x\right)\left\\{\partial_\mu\left[V\left(x\right)\dagger\right]\right\\}V\left(x\right)\right\\}\end{align} Is this all correct?


Galilean Transform

I tried to solve a problem using two different ways and I had some trouble, the problem is: We define a symmetry transform of the expected value of $\vec{P}$ like this: $$\langle \psi|\vec{P}|\psi \rangle \rightarrow \langle \psi|\vec{P}+m\vec{V_0}|\psi \rangle$$ being $\vec{V_0}$ a constant vector. If we define the generators $\vec{K}$ like $$\vec{K}=-m \vec{T} $$ prove that we obtain the following relations. $$ [K_j,P_k]=-im\delta_{jk} $$ $$ [K_j,K_k]=0 $$ **And my two solutions are** Using $$U^-1 \vec{P} U = (1-i \vec{\epsilon} \vec{T}) \vec{P} (1+i \vec{\epsilon} \vec{T}) = \vec{P} + m\vec{V_0} = \vec{P} + \vec{\epsilon} $$ i end up with (where $\vec{\epsilon} \equiv m\vec{V_0} $) $$ [\vec{K},\vec{P}] = -im $$ My problem is if we expand those vectors like this $$ \vec{K}\vec{P}-\vec{P}\vec{K} = K_iP_i+K_jP_j+K_kP_k-K_iP_i-K_jP_j-K_kP_k = [K_i,P_i]+[K_j,P_j]+[K_k,P_k]=3[K_i,P_i] $$ and this gives $[K_i,P_i]=\frac{-im}{3} $ wich is wrong But if I start with $$(1-i \epsilon_i T_i) P_i (1+i \epsilon_i T_i) = P_i + m V_{i0} $$ we obtain the good answer $$[K_i,P_i]=-im $$ so, my question is, whats wrong with the first method and why it doesn't give me the right answer.


Has a metric formulation of electromagnetism ever been attempted?

I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such a way that EM charges impart an EM geometry onto spacetime that is only experienced by EM charges. That is, the EM geometry of spacetime would be a function of charges such that charges themselves produce EM curvature and the motion of charges produces EM torsion, and that charged objects move according to how their charge (electrical or magnetic, positive (N) or negative (S)) experiences that geometry. For example, a positive electric charge would appear as a "hill" to other pe charges, a "valley" to ne charges, and either a left or a right hand "whirpool" to either nm or sm charges (I'm not immediately sure which would match with which) if it had some velocity. This formulation would be directly analogous to how energy creates gravitational curvature in spacetime (and if one accepts Einstein-Cartan, gravitational torsion comes from intrinsic angular momentum), and then this resulting geometry is experienced by the energy in that region. If Einstein is correct about gravity, would it be too much of a stretch to suppose separate metric functions for each of the fundamental forces, considering that they were a single force moments after the big bang? I'm not currently concerned with the quantum mechanical approach to electromagnetism. I understand that to be truly fundamental, EM has to be formulated quantum mechanically, but right now I want to limit my question to macroscopic charged objects.


Magnification for a concave mirror

Here is the question as given in my textbook: > Find the distance of the object from a concave mirror of focal length 10 cm > so that the image size is 4 times the size of the object. The solution in my textbook has the following data stated: $u=-x$ as it is assumed that the object is real. $v=-4x$ as it is assumed in case 1 that a real image will be formed and $|\frac{v}{u}|=|m|=4$ So now I am not able to understand why the image distance $v$ is taken $-4x$. In the question it is given that the _object size_ is magnified 4 times, and not the distance of the of the object from the mirror. And the magnification formula is $\frac{-v}{u}=m$, so why does the solution include the modulus of the formula? Am I missing something in analysing the solution?


Does relativistic mass violate the conservation laws?

When an object's speed increases, its (relativistic) mass increases. Are new atoms created inside the object by its increased speed? or is its "gravitational charge" increased by its increased speed, without more atoms?


How to properly construct the electromagnetic tensor in curved space-time?

How do I properly construct the electromagnetic tensor in curved space-time? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is right (if there is one). First way:   1. Compute the magnetic field $B$ from the curl of the magnetic vector potential $A$: $$ \mathbf{B} = \nabla \times \mathbf{A}. $$   2. Place the resulting components directly in the contravariant electromagnetic tensor definition in cylindrical coordinates: $$ F^{\mu\nu} = \begin{pmatrix} 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & B^z \\\ 0 & 0 & 0 & -B^r \\\ 0 & -B^z & B^r & 0 \end{pmatrix}. $$ Second way:   1. Define the electromagnetic four-potential ($\phi$ is zero in my problem): $$ A^\alpha = (\phi, \mathbf{A}). $$   2. Lower the four-potential index by contracting it with my covariant metric tensor.   3. Compute the electromagnetic field components with the formula $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$ I replaced the ordinary derivatives with covariant derivatives.   4. Raise the indexes of this covariant electromagnetic field tensor to compare with the first way. The problem is that I can't seem to have the same results with both methods, which says pretty clearly that I am doing something wrong. Is there something fundamentally wrong in taking these steps?


What is the terminal velocity for a mobile phone

You may have seen the story of the iPhone which was dropped from perhaps 13,500 feet by a skydiver - it survived. This made me wonder how to work out the terminal velocity for something like that. Obviously calculating terminal velocity for a sphere can be relatively straightforward, but with a flattened oblong, what factors come into play? End on will be fast, flat will be slow, but is there a stable configuration?


Why is charge $q$ symmetrically distributed?

Simple question: Why is charge $q$ outside symmetrically distributed?   The material is a conductor.   ![enter image description here](http://i.stack.imgur.com/8Tutg.png)


Are quantum mechanics and determinism actually irreconcilable?

As a preface, I am not a physicist. I'm simply interested in abstract physics and fundamental principles of the universe and such. As such, if you can provide an answer for the layman (as non-academic and unjargonized as possible), it would be very, very appreciated so that I can actually understand it. Everything I ever learned about physics seemed to be built off of an assumption that the universe and everything in it behaved deterministically. So it should always be a (theoretical) possibility that, given perfect knowledge of every particle and force in the universe at a given moment, we can calculate with 100% accuracy what the state of the universe will be in the next moment. This of course assumes omniscience and unlimited computational capacity, which is why I said this is only a theoretical possibility. However, we can define our closed system to be much smaller -- say, a bottle full of nitrogen and helium -- and apply this principle more directly. And it seems like this assumption is absolutely necessary for scientific experiments to even take place or have any validity, since without this kind of determinism, the observations and the results inferred from them can't ever actually be trusted. I don't understand quantum mechanics very well, but it seems like this theory breaks this assumption completely. From what I understand, there is no way to predict what the state of the particle will be at the next moment. The most I can know is that, given that a particle is in state `A`, it will next be in state `B` or state `C`. There is absolutely no way to know for sure, and the only way to find out is to observe it actually change. Furthermore, observations of this kind don't yield any insight into what other particles in state `A` will do. So, in classical physics, laws used to look like this:               A -> B   [A implies B]      But with quantum physics, all of this is gone, and our laws can at best look something like this:               A -> ((B v C) v D) v E   [A implies B, or C, or D, or E, or ...]        How does this not break everything physics is built upon? The implications of this are seriously troubling to me, and I feel like it destroys everything I thought I knew. Can anyone explain how this works in slightly lower-level terms, or show how it's still possible for the theories and laws of classical physics to hold any weight?


Given mean insolation, can one place an upper bound on the daily peak insolation?

Photovoltaic panels are rated under standard conditions: eg a $100\mathrm{W}$ panel if irradiated with $1000\mathrm{W}/\mathrm{m}^2$ at $25^\circ\mathrm{C}$. When the cost effectiveness of photovoltaics is discussed one is often presented with daily irradiation maps averaged yearly or possibly monthly. This is fine for cost effectiveness calculations. But in designing the power conversion electronics or effects of grid injection one cannot work with average values - they must be able to sustain peak powers that are possibly larger than the nominal power of the panels, right? If so, by what factor are these electronics oversized? I cannot find any similar maps of the average daily peak irradiance. Given the daily solar energy incident ($\mathrm{kW hr /m^2\cdot day}$) averaged either annually or monthly, can one place an upper bound on the daily peak irradiance ($\mathrm{W / m^2}$)? I am particularly interested in Mediterranean countries.


Magnetic field due to a conducting sphere in unpolarized light

From Jackson, problem 10.3: > A solid uniform sphere of radius $R$ and conductivity $\sigma$ acts as a > scatterer of a plane-wave beam of unpolarized radiation of frequency > $\omega$, with $\omega R /c \ll 1$. The conductivity is large enough that > the skin depth $\delta$ is small compared to $R$. (a) Justify and use a > magnetostatic scalar potential to determine the magnetic field around the > sphere, assuming the conductivity is infinite. (Remember that $\omega \neq > 0$.) We'd like to show $\nabla \times {\bf B} = \nabla \cdot {\bf B}=0.$ I have two questions. First, what does it mean for a plane wave to have a definite frequency and be unpolarized? For example, are there many sources out of phase, all at a given frequency, radiating with varying amplitudes and in all directions? If this is true, then is it possible for the superposition of all these waves to vary faster than the original frequency due to interference? Assuming that the above question is resolved and that we can make the long- wavelength approximation so the magnetic field is roughly constant over the sphere, why does $$\nabla \times {\bf B} =0\neq \frac{1}{c^2}\frac{\partial {\bf E}}{\partial t}$$ (assuming that the electric field is just like an oscillating dipole and has a term free from powers of $\omega$)?


what is the kinematics of a particle with complex mass?

* particles with real-mass have time-like kinematics ($ds^2 > 0$).   * particles with zero-mass have light-like kinematics ($ds^2 = 0$).   * particles with imaginary-mass have space-like kinematics ($ds^2 < 0$) (tachyons). So the question is pretty simple: **What would be the kinematics of a particle with both non-zero real and imaginary parts?**


Allowed transformations in General Relativity

So in Special Relativity we have: $$ \Lambda \eta \Lambda^T=\eta $$ Is there an analagous formula for the metric in General Relativity?


Simultaneous Charging and Discharging Capacitor

sorry if I sound little noobish. Though I have a fairly good understanding of physics, I sometimes don't understand the electrical aspects. Say there is a capacitor. This capacitor is expected to act as a storage buffer. By extension, the capacitor will have a "charge" interface and a "discharge" interface. There may or may not be an electrical circuit between these interfaces and the capacitor. The expected behavior of the system is, electrical energy may be input to the system via the charge interface, which will charge the capacitor, and energy may be simultaneously drawn also, via the discharge interface, which will draw energy from the capacitor, and this process can happen so long as the energy stored within the capacitor is within its maximum and zero. The actual path of the energy may be from the interface, through the circuit, to the capacitor, and back through the circuit to the other interface; or part of the energy may be routed from one interface to the other by the circuit, and the net energy difference between the two interfaces be actually sent to, or drawn from the capacitor. Though I may think that this is possible, I'm not aware if any such system exists currently. The home inverter seems to be doing quite the same thing, but both the cycles don't happen at the same time though. Edit: Yeah, a diagram will help me also to explain better what I have in mind. ![Energy Storage Buffer](http://i.stack.imgur.com/cXRGC.jpg) The two interfaces are part of the circuit which shields the capacitor. This circuit may work in two possible ways, which I've mentioned as flow 1 and flow 2. In Flow 1, "all" of the energy which flows into/out of the system, does so through the capacitor. In Flow 2, the circuit redirects part of the energy flow in one interface to the other interface, and only the net difference between the energy flows is actually transmitted to/from the capacitor. Hope this makes it more clear. Now, let me restate my question. Is such a system possible, importantly, such a circuit possible. Are any systems available today, which do exactly the same thing. And your own views on this is really welcome.


Having trouble understanding how the centrifugal force works

I thought that I understood the centrifugal force earlier, but I can't seem to grasp how it interacts when considering that everything is relative? Let's imagine that you are the only one in the entire universe, and that you are spinning with high angular velocity, with the rotation axis pointing in the same direction that your eyes are directed. Surely you would feel the centrifugal force pulling your feet and head apart, wouldn't you? A problem with this, though, is the following: Since you are the only object in the universe, there's no way to tell if you're rotating. Your angular velocity isn't even defined, since you aren't rotating in relation to anything else. How, then, can one know how what the centrifugal force is? Is it defined in relation to all the other mass in the universe, in such a way that it's negligible in classical mechanics problems?


Quantum fluctuations in a classical domain?

"In the presence of chaos, even small ﬂuctuations (including quantum ﬂuctuations) can be ampliﬁed to produce large uncertainties in later behavior"(http://arxiv.org/pdf/gr-qc/9210010v2.pdf) Is there some experimental evidence for the amplification of the quantum fluctuations in a classic domain (typically m, s, kg)?


Stark Effect on the 1st excited state of Hydrogen

I know the ground state of hydrogen is unaffected by the Stark effect to first order. And I also know that the 1st excited state is split from 4 degenerate states to 2 distinct, and 1 degenerate state like this: ![Hydrogen splitting](http://i.stack.imgur.com/zO0Zt.png) But I don't quite understand why. I imagine it is something to with (anti)symmetry of the wavefunctions and selection rules. Can anyone explain?


Why are the orbits of planets in the Solar System nearly circular?

Except for Mercury, the planets in the Solar System have very small eccentricities. Is this property special to the Solar System? Wikipedia states: > Most exoplanets with orbital periods of 20 days or less have near-circular > orbits of very low eccentricity. That is believed to be due to tidal > circularization, an effect in which the gravitational interaction between > two bodies gradually reduces their orbital eccentricity. By contrast, most > known exoplanets with longer orbital periods have quite eccentric orbits. > (As of July 2010, 55% of such exoplanets have eccentricities greater than > 0.2 while 17% have eccentricities greater than 0.5.1) This is not an > observational selection effect, since a planet can be detected about equally > well regardless of the eccentricity of its orbit. The prevalence of > elliptical orbits is a major puzzle, since current theories of planetary > formation strongly suggest planets should form with circular (that is, non- > eccentric) orbits. What is special about the Solar System that orbits of planets here are nearly circular, but elsewhere they are moderately or highly eccentric?


Thermodynamic process when nebula is heated

The basic thermodynamics problem is stated as follows. > The nebula contains a very tenuous gas of a given number density (atoms per > volume) that is being heated to a given temperature. What is the gas > pressure?   1. What are the basic assumptions that should be taken in solving this problem? There is no sealed container, obviously, but if nebulæ were to allowed to expand indefinitely then it would be an isobaric process, would it not? Seen how that would not allow us to determine the pressure (?), then some sort of constraint is to be placed. If we are to assume that it is, in fact, an isochoric process, would it be a simple matter of finding molar mass from given number density and plugging it in the formula of Ideal Gas Law ($p = nRT / V$) assuming the volume of 1 metre cubed?   2. Given the numbers ($1 × 10^{8}$ atoms per m$^3$, 7500K) what should be realistic order of magnitude for the answer (in $Pa$ or $atm$) for the purposes of assessment as many certainly would not have intuitive grasp of your average nebular pressures? In general, for questions like this (advanced question from introductory chapter) is it detrimental to overthink the problem, that is: do I look for more difficulty than I should?


Is it possible to create an electrified silicon gas vortex?

Is it possible to create, and sustain, an electrified silicon gas vortex?   If it is possible: would it produce an electromagnetic field?   And how would that field affect the vortex?


Explicit solutions to simple one-dimensional fluid flow problems?

In order to test and verify numerical simulation results, I'm looking for explicit (or approximate) solutions to simple fluid flow problems to recalculate (like e.g. the pressure development during 1-dimensional flow through a planar crack etc.). However, I've found it quite hard to find solutions like that online, because the sources mostly either refer to really old (pre-pdf) publications or expensive textbooks. Does anybody know a good online source, that might be able to help me? Any links or other suggestions? Edit: I'm looking for solutions in a Eulerian frame of reference, since that would be the easiest to compare with simulation results where I would typically get the pressure as a function of time at distinct coordinates.


Rigid bodies - the wheel

As I've been taught lately in my mechanics course: > the wheel has a unique property: at every moment of motion, the touching > point between the wheel and the ground is not in movement and therefore no > work is done by the friction force. Now, many of those problems are solved by using the 2nd Newton law and its rotational analog. For instance consider having a wheel with a mass $m$ and a radius $R$ rolling on a slope that creates an angle of $\theta$ and we want to calculate its acceleration then we can start by writing: $$ma=mg\sinθ−F_f$$ and the analog equation for torque: $$F_f R=I\alpha$$. where $F_f$ is the frictional force. Now, the first equation is the 2nd Newton law applied on the centre of mass of the wheel, and as we see, one of the forces is the external frictional force. Now, though the touching point is not in movement at the moment, the center of mass is, and in the equation we assume there is a friction force on the center of mass and therefore work is done indeed. Now, after thinking about this for a while, I've come to the conclusion that this makes sense, cause if we see the wheel as point of mass located in the center, then energy is not preserved because some of it is transfered to the spin and that's why we have the second equation. > The question I'm having trouble with is whether the "work" of the friction > force on the center of mass is equal to the energy transfered to the spin of > the wheel?


How close can two extremal black holes with the same charge can get?

Here's a puzzle I have been pondering over. If we have two extremal black holes with the same charge, the electrostatic repulsion between them ought to cancel the gravitational attraction between them. Without any net attraction or repulsion between them, how close can we bring these two black holes to each other without merging? A peculiarity of the metric seems to suggest the event horizon is always infinitely far away for extremal black holes $\int_R^r dr' \frac{1}{r'-R} = \infty$. Does this give enough elbow room for both black holes to get arbitrarily close to each other without merging?


Is there any potential associated with magnetism

Can anybody please tell me if magnetism is a conservative force or if there is a field associated with it? How to reason? One thing I know is that the work done by a magnetic force is $0$.


Is magnetic force non-conservative?

If magnetic field is conservative, then why not the magnetic force? My professor thinks it is non conservative but he couldn't explain to me why?


Work done by the Magnetic Force

The magnetic part of the Lorentz force acts perpendicular to the charge's velocity, and consequently does zero work on it. Can we extrapolate this statement to say that such a nature of the force essentially makes its corresponding work independent of the choice of path, and hence that the magnetic force is conservative?


Why is it that the splash pattern for every drop is never the same?

This was just a random thought that crossed my mind, and I'm sure there is a simple physics explanation. Say we have a leaky faucet 20 feet above the ground that lets out the exact same amount of water every few seconds or so. Why is it that the splash pattern for every drop is never the same as any other drop? Now it's not like I've sat there and observed every splash, but sometimes its obvious, like a certain splash's water reached a farther distance than another splash. If we are letting the same amount of water come out everytime, and nothing else changes, why do the splashes vary? (pretend the water disappears off the ground after every splash, that way every splash has the same environment.)


Why is a vacuum cleaner not as good heater as an electric radiator?

I've read this question and answer: How efficient is an electric heater? , but still don't understand. If I have an electric radiator it heats the room with 1000 Watts of power. And I feel the room's getting warmer. In contrast, if I turn on a vacuum cleaner which consumes 1000 Watts as well as the radiator, it doesn't seem to heat the room as well. Why? Won't all kind of energy transform into heat ultimately?


Currents and magnetic fields produce forces that affect things they're not touching?

I was doing this past paper and am a little confused by question 5) part c)ii) ![Diagram from past paper](http://i.stack.imgur.com/3MBvP.png) ![force on current carrying wire from magnetic field](http://i.stack.imgur.com/3p3C8.png) ![how this force affects the balance](http://i.stack.imgur.com/PzDp1.png) I correctly calculated that the force acting on the rod due to the magnetic field is 0.016N but I can't see how this could affect the scales as they're not touching. I had a look at the mark scheme but I can't understand why Newton's 3rd law would apply when the two objects aren't touching? Here is a print screen of the mark scheme for part c): ![mark scheme](http://i.stack.imgur.com/gfeWK.png) Could anyone explain this please? Thank you :)


Contradiction of a scalar product

Can anyone resolve this contradiction: $$\vec{r}\cdot\dot{\vec{r}}=\frac{1}{2}\frac{d}{dt}\left(\vec{r}^2\right)=\frac{1}{2}\frac{d}{dt}\left(\left|\vec{r}\right|^2\right)\equiv\frac{1}{2}\frac{d}{dt}\left(r^2\right)=r\dot{r}, \qquad r=|\vec{r}|.$$ But the velocity $\vec{v}=\dot{\vec{r}}$ has not to be parallel to $\vec{r}$, so actually: $$\vec{r}\cdot\dot{\vec{r}}=r \dot{r} \cos{\angle\left(\vec{r},\dot{\vec{r}}\right)}$$ What am I doing wrong? Has anyone an idea? P.S. I have this problem from the book "Electromagnetic Theory" from Ferraro (p. 543).


Definition of the entropy

In physics, the word entropy has important physical implications as the amount of "disorder" of a system. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable $X$ is defined as $$H(X)=-\Sigma_x P(x)log_2 [P(x)]$$ bits, where $P(x)$ is the probability that $X$ is in the state $x$ , and is defined as $0$ if $P=0$. **Question:** Can anyone explain to me why "disorderness" of a system defined as this? especially, where has "$log_2$" come from in this formula?


A simple question about the stationary wave and fundamental frequency

Suppose we have 2 fixed end connected with a wire and now we insert a vibrator **in the middle** of the wire, and resonance occur. How would the fundamental frequency looks like? I know the case when the vibrator is at one ends and another ends are fixed while in this case, there are 2 fixed point and the vibrator is at the middle. ![](http://i.stack.imgur.com/WMmgR.png) Is the fundamental frequency like this? I imagine half of the original wire acts like a wire with a vibrator at one end and get the result. Would the fundamental frequency be different if the vibrator which is on the string is vibrating with a very large amplitude?


My model of Conductors in "static" condition - Please analyze

My textbook presents an idealization of a conductor as made up of infinitesimal units of charge and derives results. I was not convinced, so I started thinking of how electric fields are in real metals. Here is what I think now: i) My description of a "static" situation - There are electric fields inside a conductor, but no net electric field that does anything meaningful. Electrons are flying around left, right, up and down, but no net current is present. If you take a reasonably macroscopic chunk of metal, no net current will flow. ii) The entire metal is at a lower potential energy than let's say, air, that surrounds it. Therefore, there exists a work function to pull out electrons. It's like a well. You need to pull hard enough to yank an electron out. Precisely, you have to do more work than the force that holds the electron in. iii) The interior of a metal cannot have excess charge in a static condition - If the interior does have charge, the charges will repel until they reach a point of equilibrium, which means the metal isn't static. This is almost like a definition. iv) Excess charge lies at the surface of a conductor, in static situation - Since the metal is static, all the inside of the metal should behave like a regular neutral metal. For a sphere for example, this means that excess charges must form a ring of equal charge density. This is similar to saying that electric field is zero inside a ring. The excess charges will not contribute any field v) The excess charges exert a field on each other. For a sphere for example, every charge exerts charge on every other charge, so that net electric field points outside. The existence of the work function means that any attempts to pull out charges will be countered. This countering force (essentially due to protons in atoms) will counter outside field. An equilibrium is reached and therefore, those charges are more or less still. vi) The electric field inside is more or less zero, and the excess charge on the outside experiences zero field too, so the metal is an equipotential surface. vii) Any external electric field is countered using by an arrangement of charge that yields zero field inside, and these charges do not fly out because they are held in. I know there are many approximations in this model. But I think it beats the unrealistic idealization given by my textbook isn't really connected to real life. Are there any flaws in it? I will learn better models when I do solid state physics, right?


First and Second Moment of Mass

I recently came across the definition of the Center of Mass of a system as the point about which the **first** _moment of mass_ is zero. Further, it defined Moment of Inertia as the **second** _moment of mass_. My question is, What is this ' **moment of mass** '?


Exist some relationship between irradiance units and wavelenght of the incident sunlight?

Exist some relationship between irradiance units and wavelength of the incident sunlight? What about irradiance? I want to establish a relationship between wavelength and irradiance, because I would try to model photosynthesis on Vensim.


Lasing in a 2-Level system?

What exactly is the difference between 2-Level, 3-Level and 4-Level systems? Why can we not achieve stimulated emission in a two-level system using optical pumping?


First principle calculation of boiling point of water

How can we theoretically calculate the boiling point of water at given pressure (other subtle parameters as well, if any)? What is the most accurate (minimum discrepancy with experimental value) computation that can analytically predict the boiling point of water? Possibly we need to invoke quantum mechanics for this. I anticipate that many answer would say that EXAXCT prediction is computationally infeasible, but please give the outline in algorithmic form regardless of computational cost. My main goal is to learn how quantum mechanics can be applied to phenomena which can be observed by layman. Other example where quantum mechanics is used to predict physical properties of small molecules from first principle are also welcome.


Does two Neodymium magnets stick together twice the power?

Assume I use a coil to create B-flux and put ONE Neodymium cylinder magnet (1"diameter & 0.25" thick) close to the flux, it would create 10 lbs force. Does it mean that I put TWO cylinder magnets (1"diameter & 0.25"+0.25" = 0.5" thick) would create 20lb force?


Why is $U(\Lambda)^{-1} = U(\Lambda^{-1})$ for a unitary representation?

This is from the beginning of Srednicki's QFT textbook, where he writes (approximately): In QM we associate a unitary operator $U(\Lambda)$ to each proper orthochronous Lorentz transformation $\Lambda$. These operators must obey the composition rule $$U(\Lambda'\Lambda) = U(\Lambda')U(\Lambda).$$ So far OK. But where does he get the following from? $$U(\Lambda)^{-1} = U(\Lambda^{-1})$$


Battery and current confusion?

How exactly does a battery produce a current in the circuit connected across its ends? I dont want to know the chemical reactions in the battery core, but just the essence of it. I believe it doesn't do this by creating an excess of electrons at the -ve terminal and a deficit at the positive terminal. Moreover, how is the voltage and the EMF different in their definitions and value. Electric field being a conservative field, can the work done in motion of electrons in the conducting wire and all the components be compared to the work done in any other path across the terminals of the battery? And on a side note, how can we theoretically derive an relation between the potential difference and the electric current?


Why a person with a further near point experience a larger magnification with a magnifier

> Two people, Micah and Lyra, with different near points are equally close to > an object. Both inspect the object through the same magnifier by holding the > lens close to the eye. Micah's near point is located farther away from his > eye than Lyra's near point is located relative to her eye. Micah will > experience a larger magnification for which of the following reasons? > > The answer is: > >   * When the object is located at his near point, the angular size of the > object is smaller for Micah than for Lyra. > >   * The angular size of the image relative to the angular size of the object > at the near point is greater for Micah than for Lyra. > > * * * My question is why the 1st point? Not really able to visualize whats happening ... Also, why is "Micah can see the image clearly from a larger distance than Lyra can." not correct too? He has a larger near point so he can see the image from a larger distance?


Water entering hole at a depth, surface tension

The following is the question that very commonly appears in all HS textbooks.   A hollow sphere with a hole is taken to a depth of 40cm when the water starts entering the hole. if the surface tension of water is 70dyne per cm find the radius of the hole.   All the textbooks solve it in a similar manner as given below:- When water enters the sphere, radius of the bubble of air formed, will be equal to radius of the hole say r. Then excess pressure inside the bubble equals the external pressure at the depth of 40cm and thus on equating 2T/r to h*d*g, they find the answer. but i dont find this explanation satisfying at all. why should a bubble form only at that depth and that too of the radius of hole. cant it form at any depth and the radius depend on the depth with maximum being equal to the radius of the hole.?


Why can we see the cosmic microwave background (CMB)?

I understand that we can never see much farther than the farthest galaxies we have observed. This is because, before the first galaxies formed, the universe was opaque--it was a soup of subatomic particles that scattered all light. But before the universe was opaque, the Big Bang happened, which is where the cosmic microwave background (CMB) comes from. If the opaque early universe scattered all light, and the first few galaxies are as far back as we can see, why is the CMB observable? Where is it coming from?


All significant objects in the universe?

Taking it to the bottom of layman's terms, what would be the shortlist of significant _things_ in the universe? A list I could think of myself would put   * **Energy** (at whatever wavelength travelling through space)   * **Gas** (Atoms and molecules floating interplanetary, interstellar and intergalactic space)   * **Rocks** (Anything from dust to planets)   * **Stars** (In more general terms, anything that is or were a star... would include also black holes, magnetars and novae)   * **Galaxies**.. Things that does not fit very well in this classifications however are:   * The super massive black hole of galaxies (although I could simply put it in galaxies)   * Dark matter (but I do not worry too much until it has been properly measured :) )   * Wormholes (if they even exist..)   * Anything else What else is to be put in either list? cheers!


How to determine the positions of two points in a radial line by an intensity level dB?

The following is the question from my school. **A source emits sound uniformly in all directions. A radial line is drawn from this source. On this line, determine the positions of two points, 1.00m apart, such that the intensity level at one point is 2.00dB greater than the intensity level at the other.** I have no idea what to do because I haven't met a question about determining the positions by dB. How can I deal with this question? _**Thank you for your attention._**


How does gravity escape a black hole?

My understanding is that light can not escape from within a black hole (within the event horizon). I've also heard that information cannot propagate faster than the speed of light. It would seem to me that the gravitational attraction caused by a black hole carries information about the amount of mass within the black hole. So, how does this information escape? Looking at it from a particle point of view: do the gravitons (should they exist) travel faster than the photons?


Graviton through the horizon and force felt outside a black hole

Gravitational force is mediated by graviton exchange. If I am standing outside a black hole, I can of course feel the attracting force towards the black hole. This should correspond to gravitons mediated between the matter inside the horizon and myself; but then these gravitons should cross the horizon from the inside to the outside. My question is: how is this paradox precisely solved? I guess a starting point for an answer is that these gravitons are off-shell, much like in QED where photons exchanged between electrons that feel each other are virtual. But still, this confuses me a bit - is there some references explaining this in detail?


Is there any proof that the speed of gravity is limited?

I must warn that though I'm argumenting with black holes I'm **not asking how does gravity escape the black hole!**. I want to know if the absolute speed of gravity waves were **proven** bu an experiment. We know that anything with $v\le c$ cannot escape the event horizon of a black hole. But the gravity itself seems quite unaffected by itself. There can be the explanation that the gravity does **not** interact with itself - but no matter what, in this case it directly violates the rule that no information can leave the black hole.   This brings me to he question itself - I'm having serious doubts that the speed limit was proven for the gravity. I've found nice explanation of the problem with the black hole though: > As I understand it, to the outside observer the black hole takes an infinite > time to form, although in its own frame, the black hole forms right away. In > any case, the gravity felt by the outside observer is that of the object > becoming a black hole.


Speed of light versus pull of gravity - Is $c$ really the limit?

The understanding I have is that the speed of light is considered to be the highest attainable speed in physics. Of course there are theories of tachyons but since those haven't been proven we'll dismiss them for this statement. So, thinking mathematically, the speed of everything else is relative to the maximum value (using this logic I've always had an issue with believing there IS a max value instead of just infinity). Using this line of thinking, how is it possible for some force to be able to _overcome_ this universal constant? For example, the gravity of a black hole. Shouldn't that _at most_ be able to equal the amount of force carrying a photon (or other body traveling at the speed of light) in the opposite direction? By this logic the only place that gravity could possibly even match the speed of light is at the center of the singularity, so it wouldn't trap any photons at all that weren't generated at that exact point. I guess my point is that mathematically speaking, the largest "number" is infinity, and you can't have "infinity minus one" so why should the absolute largest possible speed be able to be influenced at all, let alone in a way that it can be completely negated and then some? Doesn't that completely invalidate $c$ as being the fastest possible speed? Perhaps it's just the fastest speed we've observed? People far smarter than me have developed these theories and built whole areas of physics off of them, so I don't expect to be right. But can someone please explain to me why that's wrong?


> **Possible Duplicate:**   >  How does gravity escape a black hole? If nothing in the universe can travel faster than light, how come light can't escape a black hole? I mean, Einstein's relativity says nothing can travel faster than light, but yet, light can't escape a black hole. Does this mean that light really _isn't_ the fastest thing? That the pull of the black hole is really _faster_ than light? That Einstein was wrong, even though it's been backed up by scientific evidence? I'm very confused. If anyone would be able to answer my question, I would appreciate it: Why can't light escape a black hole if nothing can travel faster than light?


Blackholes gravity, why it has effect outside the event horizon?

If gravity is a curvature of space time, and the event horizon is the collapse of this curvature on itself, then if the blackhole gets more matter after it is formed we should see no effect at all (i.e. increased gravity field) outside the event horizon, apart from the effect of matter collapsing before entering the event horizon itself (energy relaeased), is this correct? Does this mean that the law of conservation of matter and energy is violated? Or does this mean all the matter entering the event horizon is trasformed in energy before entering? Viceversa, if we see an effect outside the event horizon, does this mean that the interaction of gravity (gravity waves) is not following the law of "no information outside event horizon"? Same question if gravity is modelled with gravitons: how can the effect of increased mass be seen outside the event horizon if the gravitons are like any other particle and cannot move >c ? thanks in advance


If gravitational radiation (or anything) cannot escape a black hole, how can it produce redshift or curve spacetime?

There is an apparent paradox in a Black hole. Keenan Pepper wrote: > _Electromagnetic radiation cannot escape a black hole, because it travels at > the speed of light. Similarly, gravitational radiation cannot escape a black > hole either, because it too travels at the speed of light._ If this is true, then evidently a BH cannot exert a pull at the centre of a Galaxy, neither it can suck off the energy out of light trying to excape from it , etc? EDIT: _should I ask a new question about this_? @John Rennie, saying that gravity is curvature, seems like **shifting the problem** (,like saying that electrostatic pull is the electric field). Relativity says that mass curves spacetime, but that supposes interaction. How can spacetime _know_ that a BH is there, if it cannot communicate? a planet modifies surrounding spacetime in some way, and that , in its turn, affects bodies and light. The question now becomes: **how does a BH acts on surrounding spacetime** to make it curve, since nothing can excape it, except Hawkings radiation?


Nothing escapes BHs, gravitons mediate gravity, so why do BHs gravitate?

> **Possible Duplicate:**   >  How does gravity escape a black hole? Nothing escapes black holes, gravitons mediate gravity, so why do black holes gravitate? My question is, "where is the hole (no pun intended!) in the above understanding?"


Why aren't train headlights brighter?

I suspect a valid scientific, physics answer for this question, because I'd venture that train, insurance companies would've calculated and contemplated this question. Yet the train headlights at http://www.youtube.com/watch?v=0TY9Kdj5PJI don't appear to brighten/illuminate sufficiently far ahead. If an obstacle or hindrance were in front, then wouldn't the limited range of the headlights prevent a safe stop? Also, as can be seen at the 8:15 mark, the train's lights don't corner or turn together with the train. Why not? Wouldn't they befit and help such a train? Shouldn't such heavy, menacing trains be equipped with headlights of the 'lumens' resembling http://www.youtube.com/watch?v=jPKy1KAz8OM and http://www.youtube.com/watch?v=bWS30yHBuLQ _(same car and driver)_ , how gleaming so ever it is for a car?


Fiber optics with broadband, incoherent light

I wonder if someone could help clarify waveguiding with broadband, incoherent light please. If we take a telecomms fiber, which is single-moded above ~1.4 μm and couple a laser beam in, we assume this excites the only mode, which is the fundamental mode. The mode-field diameter of the transverse mode is determined by the core radius and wavelength of light. I assume laser light here is coherent. But what happens if you couple light in from a broadband incoherent source (e.g. an incandescent bulb with spectrum from 1.5-3 μm)?. Can you say that a single mode will be excited containing all wavelengths from 1.5 to 3 μm? And what about if the spectrum extends from 1 to 3 μm - will there be multiple modes excited, since 1.4 μm is the cut-off for single mode behavior? Or is this picture invalid since the light is incoherent?


Why do high current conductors heat up a lot more than high voltage conductors?

120 volts x 20 amps = 2,400 Watts However, if I increased the voltage and lowered the current, you can also use a smaller wire size (more inexpensive), also have less heat and achieve the same watt Power. 1,000 volts x 2.4 amps = 2,400 Watts   1. Why doesn't it heat up like current?   2. To me this approach seems more efficient and less costly because you don't use as much material, so why isn't this common?


Experiment to find structure of water

Who first determined the structure of water (two hydrogen atoms stuck to an oxygen atom at approx 105 degrees), and, more importantly, how was this done?


How to calculate the evaporative cooling rate needed to protect a house from forest fire

Recently in our area there has been a large forest fire and I've been looking into home defense from such things. I am not a physicist - but can do some basic math. I was wondering how I could calculate if a 'evaporative mister' type system could be used for such to reduce the ambient air temp to stop the house catching fire (dont care about cosmetic damage). The goal would probably be to reduce temperature of air/surfaces of the house by approx 1000F to keep them below combustible temperatures. The area is very low humidity between 4% to maybe 15% during wildfire season. How can I calculate how much water/mist I need to put into the air to reduce temperature below 400F. Very rough simplest equations are fine - I know its not an exact science when dealing with wildfires. I found this formula on wiki but I don't know how to adapt it to use it to calculate water need for a temp drop of               TLA = TDB – ((TDB – TWB) x E)     TLA = Leaving Air Temp     TDB = Dry Bulb Temp     TWB = Wet Bulb Temp     E = Efficiency of the evaporative media.       Anything I forgot/missing - be appreciated to know. Some restrictions/thoughts I had   1. The roof would need to stay generally wet light continuous layer of water   2. Misting/irrigation could not use more then 12 gallons per minute (max well output)   3. Be nice to second use the misting system for outdoor aircon on off time (ie 20% capacity)   4. Windows/glass would need some of IR shielding to stop ignition of furniture inside the house.


An astronaut camps on the moon for a period of one month as per the earth’s calendar. What would be the path of the earth seen by the astronaut in the lunar sky? (A) The earth remains approximately at a fixed altitude and direction. (B) The earth completes one revolution parallel to the lunar horizon in one month. (C) The earth completes one revolution from east (direction of rising sun) to west (direction of setting sun) in one month. (D) The earth completes one circle around the Pole Star in one month but never goes below the horizon.


A different proof for 6 degrees of freedom

I want a different proof of 6 degrees of freedom of a solid object made of $\ N$ particles. I am thinking along these lines: Definition of rigid body is $\ modulus[\vec{r_i}-\vec{r_j}]=constant \ \forall\ i,j$ This gives me $\ ^NC_2$ constraints. There exist in total $\ 3N$ equations. So the number of free variables should be $\ n= 3N- \ \ ^NC_2=\frac{N(5-N)}{2}$ Which is clearly not the answer as $\ n$ is $\ N$ dependent, but it should be $\ 6$. **What I want to do is show that :** $$\ number\ of\ constraints \ actually\ required= 3N-6$$ which is the correct answer since I know $\ n=6$ _I am aware of the proof given in Goldstein, Rana Joag etc. What I am asking is how to do it following this approach._


Spin-Statistics Theorem - Proof

Is this proof of spin-statistics theorem correct? http://bolvan.ph.utexas.edu/~vadim/classes/2008f.homeworks/spinstat.pdf This proof is probably a simplified version of Weinberg's proof. What is the difference? What is the physical meaning of $J^{+}$ and $J^{-}$ non-hermitian operators? I'm especially interested in the beginnig of proof of second lemma. How to get this: \begin{eqnarray} F_{AB}(-p^{\mu}) = F_{AB}(p^{\mu})\times (-1)^{2j_{A}^{+}} (-1)^{2j_{B}^{+}} \\\ \nonumber H_{AB}(-p^{\mu}) = H_{AB}(p^{\mu})\times (-1)^{2j_{A}^{+}} (-1)^{2j_{B}^{+}} \end{eqnarray} Also why under CPT field transform as \begin{eqnarray} \phi_{A}(x)\rightarrow \phi_{A}^{\dagger}(-x) \times (-1)^{2J_{A}^{-}} \\\ \nonumber \phi_{A}^{\dagger}(x) \rightarrow \phi_{A}(-x) \times (-1)^{2J_{A}^{+}} \end{eqnarray} conjugation is from charge reversal, - from space inversion and time reversal. What about $(-1)^{2J_{A}^{-}}$?


What is the coefficient $\mu_\text{air}$?

I am answering some exercises in thermodynamics, and I am still in trouble with some values and constants. In the exercise, it is given: $\mu_\text{air}=1.8\cdot10^{-5}~\text{kg/m s}$. What for is this value used? In the case, I need to find some tensions in given points, and I know the speed profile of the air. Other question: As I need the tension in some points, and it is about air, is it the same thing as pressure?


Two masses attached to a spring

I'm trying to understand the solution of the following problem. > Two masses $m_{1}$ and $m_{2}$ slide freely in a horizontal frictionless > track and are connected by a spring whose force constant is $K$. Find the > the frequency of oscilatory motion for this system. In the solutions manual, it is considered that $x_{1}$ and $x_{2}$ are the coordinates of $m_{1}$ and $m_{2}$, respectively, and the length of the spring at equilibrium is $l$. Then, it is defined that, the equations of motion for each mass are: $$ m_{1}\ddot x_{1} = -k(x_{1} - x_{2} + l), \\\ m_{2}\ddot x_{2} = -k(x_{2} - x_{1} - l).$$ And, with some algebraic manipulations, we arrive at the answer. My question is about the right hand side of these two equations. Why is the displacement in the restoring force equal to the difference between the positions plus the length of the spring in the first equation and minus on the second? What is the behavior of the system during the move? What changes in the analysis when I consider the frictional force? Could I consider that this motion has some relationship with the center of mass of the system?


Period of oscilation

Two masses $m_1$ and $m_2$ are connected by a spring of spring constant $k$ and slide freely without friction along horizontal track. What is period of oscillation? No force influence.


Is the electric field zero inside an ideal conductor carrying a current?

By an ideal conductor, I mean one with zero resistance. Inside an ideal conductor with no current, the electric field is zero, but is the electric field still zero with the ideal conductor carrying a current?


Maximize Magnetic Self-Inductance Through a Wire

You are given a long length W of copper wire. How would you arrange it to obtain the maximum self-inductance? Why? I am trying to use the equation $$L=\mu_o n^2 l A$$ I try to solve it using a fixed length wire of 10 units, width 1mm and winding it into a solenoid. I plug in values of circumference 10, 5, 2.5 and finding the inductance through number crunching. However, I am getting a larger values for multiple loops but the answer is a single loop (ie. a circle) rather then a solenoid. Here are the sample values I got: $n=1; C=10; r=1.59; L=0.079 \mu_o$ $n=2; C=5; r=0.79; L=0.156\mu_o$ $n=4; C=2.5; r=0.3978; L=0.318\mu_o$ If anyone could enlighten me on the proper way of solving this, I'd appreciate it.


Blasius boundary layer solutions

I'm trying to understand the Blasius boundary layer solution, but I'm having some difficulties. Using wikipedia, I wonder how they get the first formula: http://en.wikipedia.org/wiki/Blasius_boundary_layer. $$\frac{U^2}{L}\approx\nu \frac{U^2}{\delta^2}$$ And how they get to the fourth formula: $$\delta(x) \approx \sqrt{\frac{\nu x}{U}}$$ I feel that those formulas are correlated somehow, but I don't really see how they derive those. I hope someone can help.


container with low pressure inside of the high pressure container

How will act low pressure container, or what would happen inside the low pressure container when it is placed inside the container with high pressure? And if it is high pressure container inside the high pressure container? Will the container which is inside stay in the centre?


accelerated charged particles and interaction with magnetic field

In high school we are taught that magnetic field perpendicular to velocity of an charged particle experience perpendicular force that causes it to move in circular path by relation $$qvB=\frac{mv^2}{r}$$ but in drawbacks of Bohr's theory it was proposed that accelerated charged electron orbiting along nucleus will immediately loose energy in form of electromagnetic waves and collapse into nucleus.So My question basically is are we taught wrong about this relation that charged particle according to Lorenz's force will perform circular motion as far as required conditions in equations are provided but will it eventually loose energy in form of electromagnetic radiation and halt it's circular orbit ?


Potential energy during vertical fall

Suppose I have a weightless spring connected perpendicularly to the ground, and it has on top of it some weightless surface. Now, I release some sticky object from height $h$ above the system of light spring-surface. The object eventually hits the surface and the spring is starting to contract till all the kinetic energy of the object is transformed to elastic potential energy. And the system continues to oscillate harmonically. I want to find the maximal contraction of the spring, therefore, I did: (we assume that no energy is lost during the collision) $U_{GR}=E_{k,max}=U_{SP, max}$ $mgh=\frac{1}{2}kx_{max}^2$ where $k$ is the stiffness coefficient. Therefore: $x_{max}=\sqrt{\frac{2mgh}{k}}$ I've chosen my 0 level for gravitational potential energy to be on the same level as 0 level of potential elastic energy. Suppose I want to choose another 0 level for GPE, e.g. at maximum contraction. Then my equation would be: $mg(h+x_{max})=\frac{1}{2}kx_{max}^2$ It obviously leads us to a slightly different $x_{max}$ value. According to the numbers in the answer in my book it seems like the second equation is correct. My question is - how then correct equations should look like if I want to arbitrary choose my PE 0 level at any point? Or why the first one is wrong? Also, I'm wondering why the amplitude of the oscillations is not $x_{max}$ (according to the book it is slightly less the correct $x_{max}$ value, even though we assumed that no kinetic energy was lost. Interesting coincidence: the amplitude on the contrary, according to the book, equals to the $x_{max}$ value of my first equation).


The relation between Gauss's law and Coulomb law and why is it important that the electric field decrease proportionally to $\frac{1}{r^{2}}$?

My question relates to the third MIT's video lecture about Electricity and Magnetism, specifically from $21:18-22:00$ : http://youtu.be/XaaP1bWFjDA?t=21m18s I have watched the development of Gauss's law, but I still don't quite understand the link between Gauss's law and Coulomb law: How does Gauss's law change if Coulomb law would of been a different one. I also don't understand why is it so important for Gauss's law that the electric field decrease proportionally to $\frac{1}{r^{2}}$ ? For example, what would of happened if the electric field decrease proportionally to $\frac{1}{r}$ , or $\frac{1}{r^{3}}$ ?


How we can find the kinetic energy?

A solid disk of mass m is rolling along a surface its center has velocity v what is the kinetic energy of disk?I cannot solve the problem,


Why is there a potential difference?

![enter image description here](http://i.stack.imgur.com/R70Fu.png) The question then asks for the potential difference between $X$ and $Y$, which is claimed to be $3.6\text{ volts}%$. Why would there be a potential difference in this case? If I connect a lightbulb on $X$, and another on $Y$, it seems very obvious that the brightness would be the same. Or is the standard direction of current important? So the current "encounters" the 60 ohm "first" when flowing to $X$, and the 30 ohm "second" when flowing to $Y$? That seems a little crackpot to me. So if we change the resistor to the left of $X$ to 100 ohms, and the resistor to the right of $Y$ to be 0 ohms, the differing definitions of current flow would actually cause a difference in measurement of potential?!?! The correct answer is worded: ![enter image description here](http://i.stack.imgur.com/kpT1i.png)


Speed to run a loop

So this guy was the first to run a loop and in this (german) article (and also in the video) a certain speed (13.8km/h) is mentioned. Why must he run at this speed and not just "as fast as possible"? My intuition says it has to do with to much centrifugal force and him not being used to it $\rightarrow$ trip hazard.


phospholipid bilayer

I'm a high school student. The head is hydrophilic, the tail is fatty acid,in other words hydrophobic. Here is the thing I don't understand, all textbooks state that water is repelled by hydrophobic tail, why? The hydrophilic head is composed of polar molecule,and water is polar,so they will attract each other. But why hydrophobic tail will repel polar molecule?Isn't hydrophobic tail is just composed of nonpolar molecules?


Water pressure in free fall

The increasing water pressure as you go deeper is generally explained in terms of the weight of the water column above the observation point pressing down. The question, then, is what would happen if you had a big blob of water in free fall, say 100m in diameter-- not big enough to produce large gravitational forces on its own-- and swam to the center of it. I'm imagining some sort of contained environment, here-- a giant space station, or the planet-sized envelope of air in Karl Schroeder's Virga novels, that sort of thing-- so you don't worry about the water boiling off into vacuum or freezing solid, or whatever. Would there be any pressure differential between the surface of the blob and the center of the blob? At first glance, it would seem not, or at least not beyond whatever fairly trivial difference you would get from the self-gravity of the water. But it's conceivable that there might be some other fluid dynamics thing going on that would give you a difference. For that matter, what difference would you expect between the pressure inside the water and outside the water? We know from shots of astronauts goofing around that water in free fall tends to stay together in discrete blobs. This is presumably some sort of surface tension effect. Does that lead to a higher pressure inside the water than out? How much of a difference would that be? Or would you not particularly notice a change from sticking your head into a blob of free-falling water? (Other than, you know, being wet...) (This is just an idle question, brought on by thinking about the equivalence principle, and thus free-falling frames. It occurred to me that somebody here might know something about this kind of scenario, so why not post it?)


Why does ebonite rod gets negatively charged when rubbed with fur

What I think is : The protons are present at the centre of the atom with rotating electrons around it so when it is rubbed by fur the electrons get passed from the ebonite rod to the fur leaving the rod negatively charged. But in reality the rod is negatively charged Why ?


electrons in elecrtostatic force(please read specification given below)

when comb is rubbed with hair, and brought near small pieces of papers, small paper pieces gets a towards comb. Here the comb gets negatively charged and when brought near piece of paper where do electrons go because after sometime attraction of paper pieces towards comb stops and comb doesnt have negative charge then. Why does it happen so ? What is the reason for it ? If a glass rod is rubbed with silk, rodna and silk gets positive or negative charge respectively, an when they are seperated after charge gets nuetral. Why does it happens? What is the reason behind this ?


How does rubbing cause the transfer of electrons from one object to the other?

I have just learnt about electrostatics. Why would there be a transfer of electrons? Is it because of the difference of the materials (i.e. triboelectric series)? So in the case of two different materials **contacting** each other, will there still be transfer of electrons?


understanding the oscillating part of the Gutzwiller trace

given the density of states according to Gutzwiller's trace formula $ g(E)= g_{smooth}(E)+ g_{osc}(E) $ i know that the 'smooth' part comes from $ g_{smooth}(E)= \iint dxdp \delta(E-p^{2}-V(x)) $ for one dimensional system however how it is the oscillating part of the trace obtained ?? :D i mean the sum over lenghts of the orbit (in the phase space) also how does the condition for WKB energies appear ?? $ \oint _{C} p.dq= 2\pi \hbar (n+ \alpha) $ from the Gutzwiller trace ??


can silence happens when 2 sound waves destroy each other

Hi is there any possibility that you located between 2 sound sources and u hear nothing? as we know 2 wave in opposite direction will destroy each other...


Where does energy go in destructive interference?

I have read that when two light waves interfere destructively, the energy contained within is transferred to other parts of the wave which have interfered constructively. However, I am having some trouble grasping this. While in experiments such as Young's Double Slit experiment, there are visible bright bands of higher energy, I would imagine that it be possible to configure light waves to propagate linearly such that the waves interfere only destructively and not at all constructively. Is such an arrangement possible? And if so, to where is the energy in the wave transferred? Similarly, how does the energy transfer from one part of a wave which is interfering destructively to another part which is interfering constructively? These regions may be several meters apart for long wavelength light, and I find it strange that energy can travel between these potentially distant and non-interacting regions.


How accurately can we measure human electromagnetic fields?

How accurately can our current technological tools measure the human bio- electromagnetic field emitted by a person? Or, to put it differently, does each person have a different electromagnetic field signature, and can we measure that with enough accuracy to tell two people apart by their frequencies?


Numerical aperture of a lens

Using a laser setup, I was asked to determine the aperture of a given lens and then use some geometrical arguments and compare the theoretical value from the manufacturer and the experimental value. However, when I did this, the values _differed by orders of magnitude_. The way I got the aperture was to let the laser light pass through the lens onto a white screen and then keep the screen at a fixed distance from the lens and measure the radius of the light spot obtained. Then I used geometry to determine the angle and thus its sine. Could diffraction be the reason of such a large error?


When we calculate electric field intensity for a point charge at any point inside electric field the field intensity is $E = F/q$ where $F$ is the force acting on charge $q$. In this case, the charge $q$ should be very small. The practical value of $q$ cannot be so small as needed. In defining electric field by measuring value $F/q$ is smaller than the actual value. **My question:** How can the accurate value of electric field intensity be calculated?   I'm quite confused. Can someone point me out?


Deriving the Angular Momentum Commutator Relations by using $\epsilon_{ijk}$ Identities

I've been trying to derive the relation $$[\hat L_i,\hat L_j] = i\hbar\epsilon_{ijk} \hat L_k $$ without doing each permutation of ${x,y,z}$ individually, but I'm not really getting anywhere. Can someone help me out please? I've tried expanding the $\hat L_i = \epsilon_{nmi} \hat x_n \hat p_m$ and using some identities for the $\epsilon_{ijk} \epsilon_{nmi}$ which gives me the LHS as something like $-\hbar^2\delta_{ij}$ but I've got no further than this.


How to design a deliberately biased coin?

For demonstrating basic probability concepts, it would be nice to have a coin- like object that lands heads/tails not in 50/50% ratio, but biased in a way that can be revealed in a short experiment. What I'd like is to make an object satisfying:   * Thin disk shape, say thickness around 1/10 to 1/20 of diameter.   * Lands heads/tails with some given lopsided ratio such as 60/40.   * Feels evenly balanced to the students' hands.   * Can be made by the average machinist, hobbyist woodworker, or 3D printing designer.   * Size not important, but maybe 10-20cm diameter. It's meant to be a theatrical prop visible to a small audience, not an actual coin. Would a disk with an interior hollow zone closer to one side than the other do the job? I doubt it, since the coin will still rotate uniformly in the air, exposing both sides equally to any direction, including the floor. Making both sides "heads" is too obvious a cheat, and I don't want 100/0% probabilities anyway. Note I'm not asking for any practical how-to workshop details, just the physical principle for designing such an object.


Why/When can the gauge superfield and/or chiral superfield kinetic term in $(2,2)$ SUSY be ignored?

This is in reference to the argument given towards the end of page $61$ of this review paper. There for the path-integral argument to work the author clearly needed some argument to be able to ignore the kinetic term(Kahler potential). But I can't get the argument fully. Just above equation 5.5 in that review the author says, _"..since the Kahler potential is irrelevant we may choose it at will,and in particular we may choose it very small. To a first approximation, in fact, we may ignore it..."_ And on Page 62 in his point 1, he says, _"..Nothing in our argument assures that the resulting kinetic term in the Calabi-Yau sigma model will be sufficiently “large” to ensure that sigma model perturbation theory will be valid.."_   * It would be great if someone can help make precise as to what the above argument means (..espeically the second part about "alrge"..) and flesh out its contents in may be an equation form or something.. Conceptually I would think that trying to ignore the kinetic term of the chiral fields is _different_ than trying to ignore the kinetic term of the gauge fields. like say as written in equation 2.22 on page 11 of this paper.   * Now one can see that in the above paper (unlike in the initially linked review) the author is trying to ignore the gauge kinetic term by trying to take the gauge coupling $e \rightarrow \infty$ limit as in Page 30 of the above paper by Witten.  ..but naively one might think that large gauge coupling is precisely the region where perturbative arguments would have started to fail and the entire idea of trying to minimize the classical potential would have started to become bad..   * But unlike in the review as linked to earlier, Witten's argument crucially needs the kinetic terms of the scalar chiral superfields (..from where the $F^2$ and the Yukawa terms come..). And Witten's argument also crucially needs the gauge kinetic term (from there the $D^2$ terms come..) Then why is he trying to justify the $e \mapsto \infty$ limit? Like he comments towards the end of Page 60, _``..the effects of finite e, are believed to be “irrelevant” in the technical sense of the renormalization group.."_ It would be great if someone could explain these two arguments about ignoring the kinetic term of the gauge fields and the chiral fields.


How to make a moondial?

A question came up on Outdoors StackExchange, How to tell time at night. I wrote an answer to that question, and in my answer I said that a standard sundial wouldn't work without some fiddling, but that an astronomer would know what fiddling was required. (I invite/encourage any qualified astronomers to go there and answer the question, BTW.) So I did some searching around, and some thinking about it. Here's the question. Suppose we have an ordinary horizontal sundial, at a fixed latitude (in the Northern hemisphere), which has been calibrated to correctly show solar time at that latitude. The sundial _works_. Now, suppose there's a Full moon. What adjustments would be necessary, what additional information is needed, what would it take to transform/convert the ordinary sundial into a moondial? The definition of "success" would be that we could look at the shadow and get the "correct" time, where correct might mean up to the precision of an ordinary sundial which might neglect the Equation of Time (is a different equation needed?). One website I read said that no adjustment was necessary, but I'm not sure I agree. If that's not too bad, how would the answer change if we were some days _off_ of a full moon? (Obviously, if we are sufficiently far off then there will not be enough light to make a discernible shadow.) **EDIT:** I've been thinking about it a lot, and thinking about the E.o.T.. The EoT has two (dominant) parts: the Kepler eccentricity part, and the oblique axis part. At first I thought the Kepler part wouldn't matter, but then I learned that the Moon/Earth eccentricity (0.055) is even bigger than the Earth/Sun eccentricity (0.017). So it would seem any reliable moondial would need to be corrected for this problem. I'm still wrapping my head around the obliqueness part. I understand that the style of the gnomon must be angled from the horizontal (equal to the latitude), but that's to make sure the style points toward true north. In fact, even if a horizontal sundial is used at an incorrect latitude, we can fix it by angling the whole sundial so the style points north. Clearly the Moon is in another orbital plane (apparently by about 5 degrees, I'm guessing that's an average), but if we change the angle of the style to compensate for the Moon then it wouldn't be pointing north anymore. So, at the moment, I'm thinking to leave the sundial as-is and then figure (somehow) the sine-wave that describes the Moon's obliqueness. The further this goes the more it looks like I'm talking about figuring an Equation of Moon Time or similar.


Josh Hill, 9, Oakdale Elementary has always talked Theory of Relativity and Astrophysics etc., I can answer most but lately he has stumped me and has been begging me to ask a pro, so here it is.... "What happens when Dark Matter comes in contact with the event horizon of a large Black Hole?” Josh Hill & Mike Hill (DAD) P.S. we should probably anticipate “ What happens when Dark Energy comes in contact with a Black Hole?” as the follow up question , just so we kill 2 birds with one stone , Thanks.


periodic boundary conditions for vortex in a square lattice

I am trying to follow this paper and track the dynamics of vortex motion on a discrete (square) lattice. The idea is to simulate the time evolution of the Gross-Pitaevskii (GP) equation, which reads (in rescaled units) $$i \partial_t \psi = - \nabla^2 \psi + |\psi|^2 \psi.$$ As initial condition one takes a wavefunction of the form of Eq. (10) in the above-mentioned paper, $$\psi(z) = \prod_{z_+ \in [Z_+]} \frac{(z-z_+)}{|z-z_+|} \prod_{z_- \in [Z_-]} \frac{(z^*-z_-^*)}{|z-z_-|}$$ where $z=x+iy$ are complex coordinates. This wavefunction describes a set of vortices (V) with coordinates $z_+$ in $Z_+$ and a set of anti-vortices (AV) with coordinates $z_-$ in $Z_-$. This is equivalent to using a wavefunction of the type $e^{i \phi}$ (polar coords.) for vortices and $e^{-i \phi}$ for anti- vortices, with $\phi$ the polar angle. One can calculate the velocity as gradient of the phase of the complex wavefunction. An example of the associated velocity field, for a V at (x,y)=(0,2.5) and AV at (0,-2.5) is shown below. ![velocity-lines](http://i.stack.imgur.com/nLZqZ.png) This should correspond to Fig. 1 in the paper. However, there are several things I do not understand. First, why do the authors say that they cannot look at the dynamics of just one vortex, but must include an anti-vortex in the initial conditions to satisfy the periodic boundary conditions? Second, apart from the initial V-AV pair, they also seem to include "a few mirrow images with respect to the boundary". However, they do not specify how many nor where they are positioned. (I assume that the mirrow image of a V is and AV and vice-versa, please correct me if I'm wrong). Lastly, they mention "evolving the system in imaginary time" in order to "cool it down" (although no temperature is included in the model!) and then "turning on a superflow" and continuing the evolution in real-time. I though all was needed was the discretization of the GP equation and following its evolution given the initial wavefunction and periodic boundary conditions ($\psi(\text{left margin})=\psi(\text{right margin})$ and $\psi(\text{top margin})=\psi(\text{bottom margin})$). **Edit** Following the suggestion of @BebopButUnsteady, I repeated the 'unit' of my system, the V-AV pair, until obtaining the following tiling pattern (vortices are circles, AVs are squares) ![tiles](http://i.stack.imgur.com/OQSXa.png) Let us first look at the real (left) and imaginary (right) parts of the wavefunctions for the initial V-AV pair: ![initial-boundaries](http://i.stack.imgur.com/ECgmG.png) One can see that, especially in the imaginary part, the values on the boundaries are not equal. Now we start adding copies of the "unit cell" along the x axis, and plot a cut though the imaginary part of the wf: ![boundaries-tiled](http://i.stack.imgur.com/VVDSm.png) It appears that, as we increase the number of copies, the values at the ends get closer and closer to 0, which would be the ideal periodic case. Of course this is just numerics, I have no formal proof yet that this procedure actually converges.


Intensity of particles with zenith angle dependence

The intensity of particles come to sea level depend to the zenith angle $I=Iv (\cos(z))^n$, where $z$ is the zenith angle and $n$ depend to the particle. I know the vertical intensity depend to the energy or momentum of the particle, for exemple, to muons, $n$ is about 2 and vertical intensity is $0.0094\, \mathrm{cm}^{-1}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ to $>0.35 \mathrm{GeV/c}$. I would like to know what is the vertical intensity and the $n$ for electrons, positrons, protons and hadrons.