Particle physics conservation law checking tool

I'm just starting out with simple particle physics, and I'm doing a ton of exercises where I have to check if a certain reaction is allowed, from the point of lepton/baryon/energy conservation and such. Now in principle I am able to figure most of it out of course, as it's simply checking a few numbers and whatnot, but from time to time I'm asked to 'add' the missing particles to the equation, and I'm not always sure if my answer is sensible. Now, as I'm doing approximately 40 of these exercises posting them here and asking if they are sensible is neither time-efficient nor very useful for anyone other than myself, so I was wondering if there's something out there such as a simple tool that checks if the conservation laws are obeyed. Something where you'd say I begin with an antitau and I end up with a tau neutrino, an antimuon and a mu neutrino, and it'll tell you yes this is possible, or no this or that conservation law is not satisfied. I'm assuming something like this should be relatively simple to make (as it simply has to know a few particles and count some numbers), but I haven't been able to find one as of yet. Does anyone know if one exists?


BBC radiation: What is it?

I'm reading articles about blazars, and in a discussion about models concerning Blazar SEDs (Spectral Energy Densities) there are mentions of different processes: Sychrotron scattering, inverse Compton scattering, etc. Now concerning models for blazar emission, there are a lot of acronyms. I managed to figure out that SSC stands for "Self Synchrotron-Compton," but I have no idea what the acronyms "XC" and "BBC" stand for. And it isn't easy to google out "bbc + anything" without finding just references to... you know: http://en.wikipedia.org/wiki/BBC Does anyone know or have a good sense of what those acronyms might stand for?


Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = \frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y}.$$ I end up with a second order ODE $$\frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y} = \left( \frac{\partial ^2 f}{\partial y' \partial y'} \right) \frac{d^2 y}{dx^2} \+ \left(\frac{\partial ^2 f}{\partial y \partial y'}\right) \frac{dy}{dx} \+ \frac{\partial ^2 f}{\partial x \partial y'} \- \frac{\partial f}{\partial y} = 0$$ Now every higher order ODE can be broken up into a system of first order ODEs in $y$ and the derivative $M = \frac{dy}{dx}$, giving $$ \frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y} = \left( \frac{\partial ^2 f}{\partial y' \partial y'} \right) \frac{d M}{dx} \+ \left(\frac{\partial ^2 f}{\partial y \partial y'}\right)M \+ \frac{\partial ^2 f}{\partial x \partial y'} \- \frac{\partial f}{\partial y} = 0. $$ From this perspective, Hamilton's equations $$\left\\{ \begin{array} & \frac{dy}{dx} = \phantom{-}\frac{\partial \mathcal{H}}{\partial p}\\\ \frac{d p}{dx} = - \frac{\partial \mathcal{H}}{\partial y} \end{array}\right.$$ are merely a system of first order equations making my above system of first order ODEs look more symmetric, after a suitable change of variables. My question is, looking at $$\frac{d}{dx}\frac{\partial f}{\partial y'} \- \frac{\partial f}{\partial y} = \left( \frac{\partial ^2 f}{\partial y' \partial y'} \right) \frac{d^2 y}{dx^2} \+ \left(\frac{\partial ^2 f}{\partial y \partial y'}\right) \frac{dy}{dx} \+ \frac{\partial ^2 f}{\partial x \partial y'} \- \frac{\partial f}{\partial y} = 0$$ it should be possible to see why the Legendre transformation arises, first because its a transformation using derivatives to change variables but also because it should just make some terms go to zero in this second order ode so that everything looks nicer, but how do you see this explicitly? It'd be great if you could use my notation, ie. $J[y]$ etc... as you see I snuck in $\mathcal{H}$ in above which shouldn't really be there, I'd love to see how that comes about in my notation - thanks!


A confusion about notation in Goldstein

On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation $$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)}+\sum_{j=1}^k {\bf F}_{ji}$$ Where ${\bf F}_{i}^{(e)}$ is the external force on the $i$-th particle and ${\bf F}_{ji}$ is the force that the $j$-th particle applies on the $i$-th one. Now if we compute the work on moving the system along a curve $\gamma$ we have: $$W=\sum_{i=1}^k\int_\gamma {\bf F}_i = \sum_{i=1}^k \int_{\gamma}{\bf F}_i^{(e)}+\sum_{i,j=1}^k\int_{\gamma}{\bf F}_{ji}$$ Now he says the following (on p. 10) > In the special case that the external forces are derivable in terms of the > gradient of a potential, the first term can be written as $$\sum_{i=1}^k > \int_{\gamma} {\bf F}_i^{(e)} = -\sum_{i=1}^k \int_{\gamma} {\bf > \nabla}_iV_i=-\sum_{i=1}^k V_i\bigg|_{a}^b$$ where the $i$ subscript on the > del symbol indicates that the derivatives are with respect to the > coordinates of $\mathbf{r}_i$. Now I can't understand why this $i$ on the del symbol. Basically we are considering the configuration manifold to be $\mathbb{R}^3$ so the natural coordinate functions are $x,y,z: \mathbb{R}^3\to \mathbb{R}$ which are just projections onto the axis and with those coordinates, there's just _one_ del operator, namely: $${\bf \nabla} = \dfrac{\partial}{\partial x} e_1 + \dfrac{\partial}{\partial y}e_2+\dfrac{\partial}{\partial z}e_3$$ So what really means "differentiate with respect to the coordinates of $\mathbf{r}_i$? And why do we need it? Does it have any relationship with the idea of taking the configuration space for a system of $k$ particles as $\mathbb{R}^{3k}$?


Force Between Two Conducting Spheres

**Aakash PHYSICS JEE (Main & Advanced) Study Package - 5 & 6 (Class XII)** **Chapter -** Electric Charges and Field               Assignment (page 12)      _SECTION - A; Q.no - 1_ > The force of repulsion between two point charges is F, when they are d > distance apart. If the point charges are replaced by conducting spheres each > of radius r and the charge remains same. The separation between the center > of sphere is d, then force of repulsion between them is > > (1) Equal to F > > (2) Less than F > > (3) Greater than F > > (4) Cannot be said               Answer - (2) Less than F      What I expected the answer to be was (1). The electric field due to a conducting sphere of charge Q is equivalent to the same due to a point charge at the center of the sphere as total charge on it appears as concentrated at the center for the points outside the charged sphere. So, the force should have remained the same. Any suggestions?


Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. This is a result of mathematics (topology), but I am interested in applications. I already visited wikipedia and cited references, but I need a little more explanation in magnetic fields, meteorology and applications in computer graphics. If any other application would be excellent. I've also read some questions on this site and some articles uploaded (an article of John Milnor for example, in math.stackexchange), but none deals specifically with applications. Maybe it's a bit difficult because I seek not know much about magnetic fields or computer graphics, but I'm just interested I would like to know if this result is really effective to model these situations, it is mentioned briefly in some texts, but not how they actually work. Thanks for any help.


flavor splitting of quark masses

It seems pretty clear that flavor symmetry splits the quark masses. It seems extremely odd that a global gauge symmetry of the Standard Model fields could or would produce such extreme splitting as is calculated in the typical models and confirmed in experiment. The range seems more suited to the magnitude of the color force itself, and yet the color force is flavor agnostic. Does this bother anyone else? Perhaps it's neatly explained somewhere that I've missed?


Bose Einstein condensation and macroscopic occupation

If have been thought, that Bose Einstein condensation occurs of the ground- state is occupied macroscopically, so $n_0\in \mathcal{O}(N)$ when performing the thermodynamic limit. So naively, this seems odd as of course, if I put in more particles without increasing system size this will always be fulfilled (when keeping the temperature constant). The problem with my imagination is now, that of course when performing the thermodynamic limit, I have to change the systems volume while I put in new particles and thus "create" more low energy states. How do I see, that above the critical temperature these effects cancel, so that due to the additional states $n_0$ does not scale linearly with $N$?


Do you need energy to exert a force?

Is force dependent or independent of energy?


Electric current in quantum mechanics?

Quick question: I want to find an expression for the (electric) current density of an electron, in quantum mechanics. Either a single electron or a general charge distribution $\rho$. Classically **j** =$\rho $ **v**. What should I use here? Maybe the electric charge multiplied by the probability current? Thanks.


Feynman propagators for scalar fields

If there are few massless scalar field, are the propagators of those different massless scalar fields indistinguishable?


Crystal momentum and the vector potential

I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\frac{i}{\hbar}\int_{\partial\gamma}q A_\mu dx^\mu}$. I also recognize that electrons in a periodic potential gain a phase factor given by $e^{\frac{i}{\hbar}k_ix^i}=e^{\frac{i}{\hbar}\int k_idx^i}$. I recall that $k_i$ plays a role analogous to momentum in solid state physics. I also recall that the canonical momentum operator is $P_\mu=-i\hbar\partial_\mu-qA_\mu$. Notice that when you operate with the momentum operator on a Bloch electron, $\psi(x)=u(x)e^{\frac{i}{\hbar}k_ix^i}$, you get $e^{\frac{i}{\hbar}k_ix^i}(-i\hbar\partial_i+k_i)u(x)$. My question is whether a parallel can be drawn between the crystal momentum, $k$, and the vector potential $A$. It seems they play a similar role quantum mechanically, but I have never seen Bloch's theorem described in terms of vector potentials. I suppose one does not even need a nontrivial vector potential for Bloch's theorem to hold. Still, crystal momentum and the vector potential play very similar roles in phase factors and I wonder whether there is any deeper meaning to that.


Deriving an equation involving Killing vectors

I'm currently studying Carroll's GR book _Spacetime & Geometry_, and ran into some trouble understanding the text. When discussing Killing vectors, Carroll mentions that one can derive $$K^{\lambda}\nabla_{\lambda}R=0$$ That is, the directional derivative of the Ricci scalar along a Killing vector field vanishes (here, $K^{\lambda}$ is a Killing vector). He remarks that the only necessary ingredients to derive this equation are: $$\cdot\;\;\text{The Killing equation:}\ \nabla_{(\mu}K_{\nu )}=0$$ $$\cdot\;\;\text{The Bianchi identity:}\ \nabla_{[\mu}R_{\nu\rho ]\sigma\lambda}=0$$ $$\cdot\;\;\ \nabla_{\mu}\nabla_{\nu}K^{\rho}=R^{\rho}_{\lambda\mu\nu}K^{\lambda}\longrightarrow \nabla_{\mu}\nabla_{\nu}K^{\mu}=R^{\mu}_{\lambda\mu\nu}K^{\lambda}=R_{\lambda\nu}K^{\lambda} $$ With $R_{\mu\nu\rho\sigma}$ the Riemann curvature tensor and $R_{\mu\nu}$ the Ricci tensor. Although I was able to derive all of the equations involved, I don't see how to put them together to get the sought-after result. The only way I saw to get started is the following: $$K^{\lambda}\nabla_{\lambda}R=\nabla_{\lambda}(K^{\lambda}R)-R\nabla_{\lambda}K^{\lambda}=g^{\mu\sigma}(\nabla_{\lambda}(R_{\sigma\mu}K^{\lambda})-R_{\sigma\mu}\nabla_{\lambda}K^{\lambda}) $$ which immediately runs me into trouble because I don't see how to simplify/manipulate either of the terms to even make use of any of the three 'ingredients'


Is there a contradiction of the theory of relativity here? -- Length contraction and EMR amplitude

Suppose there is a laser beam powerful enough to burn through iron aimed at a piece of iron. You observe this event while you are in the same frame as the piece of iron and the laser-beam generator. In this frame, there is a certain part of space that you know that the light is traveling through. Now say that you get in a rocket that travels a few meters away from the laser beam in a direction perpendicular to eventually at a constant speed arbitrarily close to the speed of light. As you do so, the area that you knew the light was traveling through contracts in the direction that you are traveling in. Say that the amplitude that you think that the light has decreases to the point where the light would no longer be carrying enough energy to burn through the piece of iron. If you see the iron stop being burned by the laser, our universe is seriously weird. I don't think this would happen. Would you see the laser continue to burn through the iron even though it does not seem to you to have the energy necessary to do so? Would this mean that, instead of having the length contract, that there is such a thing as absolute distance? (If we suppose that the amplitude you perceive the laser to have remains constant regardless of what inertial frame you are in, then the laser would appear to have a constant amplitude even as objects around it continued to contract in the direction you are traveling through. This would mean that the laser would have to seem to affect more and more of space as you traveled faster and faster, so it seems it would have to seem to burn through more and more objects as you went faster, which doesn't seem right to me.) Does this contradict the theory of relativity? Is there an error in here somewhere? Thanks.


Optical element is heated by laser: is it possible to get oscillating heat distribution?

Imagine that we have thin optical element, which is irradiated by laser. Laser heats element, so there is some heat distribution in element. There is a heat sink through upper and lower element faces proportional to temperature. There is also a heat conduction in element. The most important characteristic of the model is that heat consumption depends not only on power of laser, but on heat of the element too and relation between current heat and heat change may be very complex. With given parameters after some time heat distribution in element usually sets to some stable state (with equal heat income and outcome in all element points). Stable heat distribution can vary depending on initial heat distribution. So, the question is: if laser radiation power distribution depends on element coordinates and heat consumption is related to temperature in current part of element, but all the parameters do not change in time, is it possible that heat distribution will not come to some stable state but instead will oscillate, or behave in more complex way?


Materials about S-matrix and S-matrix theory

What is the best book or paper to learn about analytical structures of S-matrix and S-matrix theory? I already know books as _The Analytic S-matrix_ by RJ Eden, PV Landshoff, DI Olive, JC P and _Quantum Field Theory_ by S. Weinberg.


Can the electroweak/strong forces, and/or quantum mechanics be thought of as geometric?

Can the electroweak and strong forces be written as geometric theories? - Why and why not? Can quantum mechanics in general? For example, the Kaluza-Klein theory explains the electromagnetic field as "twists" that include an extra dimension of space. (As written by Lubos Motl in a previous question).


Thermodynamics, bullet fired into ice

I'm beginning to study basic thermodynamics and I want to solve this exercise: It fires a lead bullet from $3g$ to $30^oC$ with a speed of $240\frac{m}{s}$ to a block of ice at $0^0C$. How much ice melts? (the bullet remains embedded in the ice). I think that the energy becomes kinetic friction: $$Q=K=\frac{1}{2}mv^2=86,4J$$ Then, I can find the mass of ice from the ecuation of latent heat: $$Q=mL_{f}$$ $$\implies m= 0,26 g$$ Is this correct? Do not influence the temperature of the bullet? Thanks for your help.


$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$

Please see the next link: http://www3.kis.uni- freiburg.de/~peter/teach/hydro/hydro02.pdf In (2.13), he used: $$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}.$$ Is this is a formula? can someone let me some details/link about this formula? Any help appreciated!


What is the "inner" force behind attracion/repulsion?

First of all, I'm sorry for any grammatical big error - English is not my native language. I have a question that maybe does not have an answer beyond those which we already have, but maybe there is someone who can help me understand deeper than high-school level: what is exactly an electrical charge? I already know something is charged (negative or positive) when it attracts or repulses another object or particle. But, stoping at that point, the electrical force coming from a charged body seems like magic. I know it's analog to gravity: given at least 2 bodies with really big masses, they will one attract each other. So to friction/rubbing (similar forces to opposite sides)... But, why do all these happen? Sintetizing: what exactly produces electrical attraction or repulsion? How can a particle move itself in one or another direction depending its "charge"? Sorry for confusion... Thanks


Doubt on the analysis of $RC$ circuit

I'm studying $RC$ circuits and I'm in doubt on how to deal with one kind of situation. Well, first when we have simply a circuit with just one resistor and one capacitor in series, or a circuit that is equivalent to one of those simplest ones I know how to deal with it: we just use Kircchoff's loop rule and then it leads to a first order linear differential equation which can be easily solved: from here I obtain many results, like the final charge when charging is $Q\varepsilon$ and so on. However, there are some cases in which it's not possible to reduce to this simple case, and so I'm unsure on how to proceed. The example I've trying to tackle for instance is the same I've asked here: ![enter image description here](http://i.stack.imgur.com/cwNW7.jpg) Now, as I've been answered on that question I made it's not possible to reduce what's on the right to an equivalent resistor. I cannot work with impedance, just resistance and capacitance really. I've tried using Kircchoff's loop rule again, however the currents are different and I don't know how to tackle this. Can someone give a little help on how do we tackle this kind of problem? Thanks very much in advance.


How far is the range of an ultrasonic sensor as far as it can transmit?

How far can ultrasonic waves propagate in the air? (if the energy is enough), how far can ultrasonic waves propagate in the ground (concrete floor)? Are there any articles or experiments about attenuation of ultrasonic waves in different mediums?


What is the difference between Radiation and Electromagnetic Radiation

Are the two equivalent or is Electromagnetic Radiation a subset of Radiation. I am further confused by the fact that electromagnetic radiation includes both ionizing and non ionizing types of radiation. The closest answer I can think of is that Electromagnetic radiation is the set of all forms of radiation except alhpa and beta particles, because they are **not** massless photons. The only site which I found which supports this is this one. Could someone please give a definitive distinction of the two. I would also appreciate an exhaustive list of everything that is, and is not Electromagnetic Radiation.


Why does the comb attract the pieces of papers if they're neutral?

When we rub our hairs with a comb, and then try to attract small pieces of paper, they're attracted by the comb. The pieces of the paper were not electrified before they were attracted. Then they might be neutral. Why does the comb attract the pieces of the paper if they're neutral, and have no influence of a specific charge on them (Pieces of paper).


Inertial Frames of Reference - Description of an Inertial Frame of Reference

An inertial frame of reference is described as being a frame of reference in which the first law of Newton (the law of inertia) holds. This means that all events as described with respect to this frame of reference must have a zero net force acting on it and therefore traces a straight line with a uniform non-translatory motion. But, I have read in some books, especially "Introduction to Special Relativity" by the well-known Robert Resnick, wherein his definition of an inertial frame of reference also refers to such a frame of reference as being an unaccelerated system. This is where I am confused.      How can we describe a frame of reference as being unaccelerated if we occupy the frame of reference itself? No mechanical experiment conducted solely confined to a single frame of reference can determine the absolute motion of the frame of reference relative to another frame of reference. All that can be understood is that there is a certain uniform relative motion between frames of reference and no more. Is Robert Resnick saying that the inertial frame of reference is unaccelerated with reference to another frame of reference?


PhD or master in physics

I am 35 years old and I've just got a bachelor in Science. I will pursue higher education in Physics but I have my doubts if either a PhD or a master is more convenient for me. My areas of interest are Plasma-Fusion Physics, Condensed Physics, and Applied Physics. If I enter a PhD program, will take me around 6 years to complete it and probably I would be kind of old to take advantage of it...in the other hand If I choose the master (which is shorter in time) probably won't help 'cause Universities and Companies in these areas mostly look for doctors. These are my doubts and concerns. Please help with your expertise.


Are communications between computers faster by electrical signals via copper cables or electromagnetic signals?

Assume that there are two computers which are connected with a copper cable, e.g. Ethernet. Also, there is a radio connection between them, e.g. AM radio, in order to exchange data. When we try to send data from one to the other, which one would be faster? Assume they are the same distance away. As far as I know, the propagation speed of electromagnetic signals in the air is better than the copper. On the other hand, frequency of the AM signal is smaller. Is there any other factors which affects the speed?


Protons (as opposed to neutrons) to mediate nuclear fission?

I am just wondering why are protons (as opposed to neutrons) not used to mediate nuclear fission? Is it because it is charged, so we will have to input more unnecessary energy to overcome the Coulomb force?


What is heat and how does it effect an atom?

What happens in the atom when it gets heated or cooled and turns into gas, liquid or solid? I just want to know how does heat affect an atom.


Are there any causeless phenomena from the mainstream physical viewpoint?

**EDIT:** The orginal version did not produce any answers about physics. I know what life is, I have studied that for decades. I wanted to hear how the border between matter and spirit looks from the physical point of view, and I would prefer references to published works and widely accepted terminology rather than some hand-waving. If we conclude by symptoms that what everyone else calls "life" is called "chaos" in the physics parlance, then the answer is simply "no", physics does not deny life, it calls it **chaos**. Probably, what I called freedom is called "lack of causality" in physics parlance and it corresponds well with my definition of life in the sense that there cannot be something half-spirit-half-matter. Either there is a **causal relationship** between two phenomena, or there isn't, no third option. (I will leave for future the related question whether there is an original cause of all causes.) * * * Life means **freedom** and freedom means laws can be broken. If the laws can never be broken (including physical laws), there is no freedom and no life. Everything is completely **determined** and the time dimension is a mere decoration. **EDIT:** I missed a big option here, that physics might not describe the entire world. Besides the physical world of causal relationships there might be other parts of existence also, which are free from causal relationships. The might not break the laws of physics, but also not be subjected to them. There might even be some interaction between the physical and non-physical worlds. In that context my question becomes very precise: **is it possible to describe all phenomena with mathematical-physical-logical laws** or (as I expect it) there will always be some phenomena that in principle cannot be described with such laws (chaos, spirit, life etc.). * * * From Michio Kaku's Physics of the Impossible I gather that the single biggest problem in physics is radioactive decay, there is only **half-life** , but no causality whatsoever for a particular particle to decay at a particular moment. Is it the loophole through which life enters physics? In related question the accepted upvoted answer starts with **"physics deals with the explanation of observable phenomena."** That subjects physics to the ill-defined concepts of _explanation_ and _observation_ but I want a strict definition (probably it is not possible to define in mathematical-physical- logical terms what physics is, that would be a perfectly sensible conclusion and might even be provable). In that post the proponent presumes that all consciousness is **limited** (and probably the entire physical existence is limited). Is it really a necessary conclusion of the mainstream physics? The life-accepting eastern philosophies unequivocally declare both the physical nature and the consciousness to be **infinite**. That brings about the Goedel's incompleteness theorem, Banach-Tarski paradox and other conclusions common to both eastern and western sciences. I am myself mathematician and consider the **Goedel's incompleteness theorem** one of the greatest achievements in mathematics. Am I correct to interpret that in context of physics it means: "not all phenomena can be described by physics"?


about the 1D singularity of black hole

I saw some responses here saying that the singularity into the black hole is one dimension object so my question is : is it possible that the singularity is simply a merger of the 4 dimensions of the spacetime ? Is it possible physically and mathematically that many dimensions can merge into one ?


Why can we use Gauss' law to compute electric field?

For simplicity I'm considering only the sphere case. In the Gauss' Law formulation we have some field $E$ introduced by charges $Q$ inside some sphere, then we compute flux and integrate, and we get result $Q/e_0$. Right. But this formulation doesn't take into account any possible outer charges, because $E$ used in this law comes only from $Q$. **I suppose that any outer field effects will cancel out, because flux coming from it must go through surfaces with opposite normals. But we didn't make any proof of that, as far as I have seen in Gauss' Law proofs.** So, when we measure electric field inside the sphere with uniformly distributed $Q$, we make some assumptions on uniformity of $E$ inside this sphere and we consider hollow Gauss' sphere with radius smaller than radius of the outer sphere. And then we use $E$ to compute total flux going through inner sphere and conclude that it must be zero. Right. But why can we use this $E$? We don't know that there aren't any effects from outer charges fields. So why can we use Gauss' Law?


What is known on violations of unitarity or locality?

Recently the amplituhedron become a hot topic. I realized that two of the central pillars that QFT is based on, unitarity and locality, are no longer playing an important part (due to gravitational interactions). I would like to get more details on why we should give up these seemingly innocent assumptions. In particular: **are black holes the only source of concern here?**


Are probabilities really tangible physical real numbers?

Probabilities are usually considered to be a real number between 0 and 1. A real number has an infinite decimal expansion. Are probabilities really real numbers? Is the infinite decimal expansion really physical? Ordinarily, when we deal with probabilities in practice, we only deal with the first few significant digits. Are the later digits physical? If one wants to measure the 100th decimal digit of a probability, one has to sample an ensemble of size slightly over $10^{200}$ to measure the right answer with probability near 1. Does the necessity for such a large ensemble mean thinking of probability as a real tangible physical quantity is incorrect? There are two cases to consider here. If the probability is "actually" around say $4.83 \times 10^{-100}$, then one would expect the 100th decimal digit to be a lot more physical compared to the case where the "actual" probability is say $.804\cdots 3\cdots $, if such a precise probability even makes any sense. But even in the former case, isn't the probability effectively zero for all practical purposes? Effectively, isn't $3\times 10^{-145}$ indistinguishable from $7\times 10^{-82}$? How can one distinguish between both cases _in practice_? If probabilities are even more tangible than that, shouldn't we be able to set up a gadget which behaves in one way if the 100th digit is even, but in a different manner if it's odd? That's just not how probabilities behave in practice. This question leads up to the nature of the complex coefficients of the wave function in quantum mechanics. Are they really physical tangible complex numbers? What about their absolute square (a real number), or relative phases? What if we set up a case where we have a nearly exact destructive interference, with the coefficients of two basis terms nearly cancelling up to $10^{-50}$? In a Bayesian sense, it is ridiculous to suppose our knowledge or ignorance of a system can be quantified so exactly. If considered as betting strategies, a typical "rational" agent might as well flip a coin to decide between bets when their expectation values match up to the 100th digit. In a frequentist sense, an ensemble of size at least $10^{200}$ is needed. Only propensitist interpretations can make the real value of a probability physical. _Theoretically_ , for _idealized_ systems, one can have actual real numbers with an infinite decimal expansion for probabilities, but do such concepts apply to the real world? If not, then what are probabilities really?


What is the easiest way to stop a star?

On long enough cosmological time scales, hydrogen and helium nucleii will become scarce in the Universe. It seems to me that any advanced civilisations that might exist in that epoch would have the motivation to try and prevent the stars from using them up, in order to burn the fuel more slowly and extract a greater proportion of the energy as usable work. One might say that, on time scales measured in trillions of years, the stars are an unsustainable use of the universe's fuel. This question is about whether such civilisations would have the means to do something about it. My questions are:   1. What would be the most energy-efficient way (using known physics) to blow apart a star or otherwise prevent or greatly slow the rate at which it performs fusion? We're assuming this civilisation has access to vast amounts of energy but doesn't want to waste it unnecessarily, since the aim is to access energy from the hydrogen the star would have burned. In order for this to be worthwhile, the energy gained from doing this would have to be substantially more than the energy the process takes.   2. What would be the astronomical signature of such an activity? If it was happening in a distant galaxy, would we be able to detect it from Earth? **Update** I'm still interested in this question. There are some great answers below, but currently I don't feel that any of them approach the most efficient way to solve the problem, _i.e._ a way likely to be attempted by a civilisation with the resources and motivation to do so. Luboš' answer makes it clear that the energy requirements are not too stringent in themselves (you just need to skim off a small fraction of a nearby gas giant and fuse the hydrogen into heavier elements; this question already presupposes the ability to extract power from fusion on a literally astronomical scale) but points out that no "bomb" made of atoms can enter a star and blow it up. However, this does not rule out other methods, such as increasing the star's angular momentum through some means (as in AlanSE's answer) or somehow (perhaps electromagnetically) removing plasma from its surface rather than blowing it up from the inside. If removing plasma from the surface is possible then ideas based on this have a certain appeal, because the resulting hydrogen could be fused, releasing energy that could be used to skim off more, and so on exponentially. (Kind of like the opposite of that bit in the film _2010_ when a self-replicating monolith turns Jupiter into a star.) Some of the resulting energy could be put into increasing the star's angular momentum as well. However, I don't know enough about plasma physics to know whether there's a way to suck the plasma up from the star's surface. (It can be done gravitationally, of course, but then the hydrogen just ends up being depleted by runaway fusion on the surface of another star, so this isn't very useful.) Given this, my current specific questions are:   * Is there a plausible way to remove material from the surface of a star at an appreciable rate, either electromagnetically or through some other means? If so, is there anything other than the (virtually unlimited) availability of energy that limits the rate at which this could be done?   * Given a sufficiently huge supply of energy, is there an effective way to rapidly increase a star's angular momentum? It seems (per AlanSE's answer) that redirecting the star's radiation would take too long, but perhaps firing massive objects or particle beams at it would accomplish the same effect in a shorter time.   * Aside from these ideas, would adding heavier elements than hydrogen to the star suppress fusion by absorbing neutrons? What quantities would be required in order to affect the star's evolution by this method? (A civilisation that eats stars will generate a huge supply of stable nucleii as a waste product, so maybe it makes sense to simply dump some of them into the next star.)


Where do theta functions and canonical Green functions appear in physics

In the beginning of Section 5 in his article, Wentworth mentions a result of Bost and proves it using the spin-1 bosonization formula. This result provides a link between theta functions, canonical Green functions, and Faltings' delta invariant on a compact Riemann surface of positive genus. Are there any other situations in physics which involve canonical Green functions on compact Riemann surfaces?


How can a wave interfere with itself?

We know that waves can interfere with themselves to form interference patterns for example photons and electrons. How is this possible? I know it does happen but is there a proper representation or a proof of this? It would be good if someone could represent it with a diagram. I just need to understand the reason for the concept and not the concept. **EDIT** I strongly mean a single photon interfering with itself. By a single wave I meant a single photon. I just do not understand the whole concept of a wave packet and a wave interfering with itself.


Tensile strength of a braided steel wire vs. a solid steel bar?

I am wondering which one is the strongest between a steel rope and a steel bar with same diameter and made of the same steel? **EDIT :** The test could be: attach rope/bar to ceiling and suspend a mass until it break **EDIT2 :** Can I have also some order of magnitude? For example, is the rope 2 times stronger than the bar?


How is dark matter meant to explain the faster than expected rotation of galaxies?

The stars on the outer edges of galaxies go around faster than they should be. How is dark matter meant to account for this? if you just add more mass wont that just give it a greater gravitational force? So we would still expect the outer stars of galaxies to be going slower than the ones near the center. Whats the flaw in my reasoning?


Can we disprove Artificial Gravity (like in the movies) with a thought experiment?

Suppose you have a Device that creates an 'artificial gravity' (AG) inside a Box, with these properties:   * the system does not use inertial forces (like centrifugal force) or a huge mass to create AG   * the AG 'feels' just like gravity/inertia over most of the volume of the box   * the Device, the Box and its content are still subject to any gravitational forces that may be present, but the AG force adds to other gravity/inertia forces, net force may be zero   * Matter, Energy and Information can leave and enter the box   * there is a smooth but strong gradient in the AG at the walls of the box -    * conservation of energy still holds: if the AG does work, this energy has to be supplied to the device in some way, vice versa for work done against the AG   * conservation of momentum still holds: for every force due to the AG on an object, there is a corresponding counteracting force on the device There's two ways this could be ruled impossible:   * Something in our current understanding of gravity rules out such a system - though our current understanding    * AG like this could be used as a perpetuum mobile, a time machine that can create paradoxes or another device that can be shown to be impossible.  Even if AG can be easily ruled out by the first way (our understanding of Gravity), I'd be interested in the kinds of shenanigans one could pull of with such a system! Edit: I'll rephrase what I want to achieve with this question. I'm trying to sketch an AG system that 'looks' like the plot device known from many SF- Movies, and I try to define it in a way that does not obviously violate basic laws of conservation. Now, I ask you, esteeemd SE-Physicists to take this device and find ways to break the laws of physics (or maybe even logic) with it. In other words, I'm looking for a thought experiment to test the idea of AG to destruction.


How to calculate incident solar energy on earth in a given day with a spectral filter

The daily solar radiation irradiation is often quoted as being between $3.2 \text{ kW hours}/\text{m}^2$ and $7 \text{kW hours}/\text{m}^2$ per day . If you filter out a range of wavelengths, how can you calculate the energy from the remaining wavelengths? To make it concrete, say we filter everything below 450nm or alternatively above 700nm.


How much force would be necessary to hold a 6-foot steel staff weighing 20 lbs from one end of it, parallel to the floor?

This isn't a homework problem. I'm asking cause I'm wondering how strong a person would have to be to pull it off. Their arms would be extended, both hands gripping the staff at one end, and the staff is facing directly away from them, parallel to the floor, going through their 'z-axis' as it were (with x being left-right and y being top-bottom).


Polar coordinates explanation needed on calculation

This is the question. ![enter image description here](http://i.stack.imgur.com/feXTM.png) ![enter image description here](http://i.stack.imgur.com/PyOt3.png) Here is the answer. But honestly I cant figure it out. Maybe my lecturer's handwriting is quite illegible too (just kidding). Sorry if I ask too simple question but seriously I want a clarification on this. Thanks :) ![enter image description here](http://i.stack.imgur.com/H907B.jpg)


Does Celestia take the lunar orbit precession into account?

Does the lunar orbit plane rotate (8 year period precession)? Does Celestia take this into account while drawing the moon orbit?


How to check Einstein-like equations on their correspondence with the real world?

Physicists studying the grounds of physics and some mathematicians often come to a theories which are like a general-relativity, but do not coincide with it. Often these theories contradict the real world. What should one do in order to check one of such theories? Which papers/books should one read in order to be able to understand about some mathematical theories, that "the mathematical theory is nice, but this is not our world"?


Where do magnets get the energy to repel?

If I separate two magnets whose opposite poles are facing, I am adding energy. If I let go of the magnets, then presumably the energy that I added is used to move the magnets together again. However, if I start with two separated magnets (with like poles facing), as I move them together, they repel each other. They must be using energy to counteract the force that I'm applying. Where does this energy come from?


Why do electrons emit radiation?

I know how you can emit light with an alternating current, running back and forth, creating an electric field in addition to the magnetic field. But why does an electron emit light when it gets accelerated, there is no positive end to create an electric field. Further more, as opposed to talking about currents, talking about acceleration seems to be a different beast entirely.


Critical radius in heterotic string theory

If I have understood the basics of heterotic strings correctly, one requires a critical radius of compactified dimension to reconcile the apparent difference in spacetime dimension of left and right movers, $$ R=\sqrt{\alpha} $$ Why would the radius of the compactified 16 dimensional torus be related to the string tension/length etc in such a way? They seem completely unrelated to me.


If there was no Earth, what would be the acceleration of gravity from here from the Sun?

Suppose there were no Earth to pull us back and we're hanging in the solar system. Assuming no other force of gravity from other bodies, what would be the $g$ from the Sun?


$L_+$, $L_-$ Old terminology and Quantum mechanics

Two Quantum mechanics Questions one simple one, and one stupid one: For angular momentum: Considering the appropriate                   i) Mathematical motivation      How do we derive $L_+$ and $L_-$   I think I am looking for an explanation that would mention something about roots and weights. May be mention something about restricted simultaneous diagonalization.                  ii) Physical motivation      Defining some nice properties with associated upper and lower bounds. Just to mention I just want to know where $L_+ $ and $L_-$ come from. I know its almost trivial but I'm looking to see something. . . I know its not defined, its derived in two ways. For some general stuff about the importance of preserving the old language. In Weinberg's text on QM(It's ok to claim I don't understand it), the questions in chapter 2 seem to employ a lot of old, and largely still used terminology. Question 2p and 1s type language, and Question 6 has something along the lines of "Lyman $\alpha$" transitions, which is fairly understood language among experts. I have heard these things before when taking QM, but I just feel so put off by it that I have to keep reminding myself what these things mean when I see them. I know "par excellence" Weinberg is correct to promote such language, but his reputation aside, is this the best possible language* to describe these things? I recall feeling very stupid until I reminded myself of what this meant. I want to know if this sort of language* is used by modern researchers working in AMO, or if perhaps the language has evolved. Should I just stop complaining and memorize these balmer, Lyman . . . . and other historically relevant words, that are still everywhere including Record exams and other places I don't want to find myself in. The reason this bothers me is I don't want memorize values of l m n ; s p d f and other such things for specific cases and make associations along these lines. Is there some modern cure to this problem I know its a really stupid question. In the event the question is too childish just leave some references


If Black Hole never forms, how important will be to study Black Hole paradoxes?

I recently came across a paper Black Hole - Never Forms, or Never Evaporates. It is claimed that under general evaporation conditions, before particles come into the Black Hole, the Black Hole itself will completely evaporate. In other words, Black Hole cannot form from gravitational collapse. If this paper was correct, (I haven't went through all the details, but did not find a technical error yet. And I would like to know your opinions about this paper, though i am not sure if it will be an opinion-based question), do we still need to study many paradoxes of Black Holes?


Can random constructive interference with sound waves cause damage?

Any two sound waves have a random chance to constructively interfere with each other at a given spot and this will cause an increase or decrease in pressure. So is it possible that a large number of local noises in a loud environment could interfere perfectly at a single point, such as on a person or building, and cause damage? I'm not worried, just curious if this is a daily occurrence or if the probability of it is extremely low or if it can even happen.


Which is the most accurate way to model the fundamental atmospheric parameters of the Earth?

I need to calculate the fundamental atmospheric parameters, I mean the actual ones, this is not an exercise but a model for a real world application so I need to take account of the humidity of air. I need to calculate the density of the air, and I've solved this by implementing this calculator in MATLAB. But now I need to find a way to calculate the input parameters which are:   * Air Temperature   * Altimeter Settings (Which is the barometric pressure right?)   * Relative Humidity given a certain altitude in meters above the sea level. If it's not possible to calculate all the three parameters, given an altitude, please show me a way to calculate the possible ones. For example I know that for the barometric pressure the best equation is the Laplace equation which is an improved version of the barometric formula right?


How the nucleon structure has been identified experimentally?

It is known that nucleons (proton, neutron) are composed of partons (quarks, etc.). How was this identified experimentally? In particular, how it has been identified that nucleons comprise of more than one constituent?


Would it be possible to develop special relativity without knowing about light?

My understanding of special relativity is that it is fundamentally based on the constancy of the speed of electromagnetic radiation - that this speed is a physical law (or derivable from physical laws and facts about the vacuum) and not a contingent fact. Does that mean that, if humanity did not know about light (or electromagnetic radiation, generally), would we then be unable to come up with special relativity? To put it another way, imagine a species of intelligent life insensitive to electromagnetic radiation - no eyes etc. Would they still have a reason to produce the theory of relativity, or would they likely have created a physics framework which doesn't involve the speed of light so centrally?


The origin of femto, atto and zepto SI prefixes

Do you know why the SI prefixes: femto, atto, zepto have been accepted by Scientific Community, if this triad of metric units, is neither greek nor latin?


Imaginary time?

If we begin with the equation for time dilation: $$ \Delta t^\prime = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}} $$ Now, for a particle moving in space with no potential energy, $$ E = \frac{1}{2} mv^2 $$ So $$ v^2 = \frac{2E}{m} $$ Then $$ \Delta t^\prime = \frac{\Delta t}{\sqrt{1-\frac{2E}{mc^2}}} $$ $$ = \frac{\Delta t}{\sqrt{1-2}} $$ $$ = \frac{\Delta t}{\sqrt{-1}} $$ Explain please?


Pauli matrices as measurement operators

I am trying to understand a paper on the Bell test experiments. I understand that if we wanted to measure the spin of a spin-1/2 particle in state $\psi$ along the z-axis we would apply the operator $$\sigma_z= \begin{pmatrix} 1&0\\\0&-1 \end{pmatrix}$$ and to measure along the x-axis we would use $$ \sigma_x = \begin{pmatrix} 0&1 \\\ 1&0 \end{pmatrix}.$$ But if, given an ensemble of spin-1/2 particles we wanted to measure spin along either the x-axis or z-axis in random alteration, how would we construct the relevant measurement operator? What about if we wanted to measure, again choosing randomly, spin either along the line 45 degrees between the positive x and z axes, or along the line 45 degrees between the negative x and positive z axes? What is the operator in this case? The paper I am reading assumes the reader is already familiar with such details.


Why is the Higgs boson created so infrequently at the CERN collider?

Although each collision at CERN has the available energy to create a Higgs boson each time, the vast majority of the time it does not create one. I suspect that part of the reason for this is that most of the energy goes into creating mostly lighter, secondary particles which carry a lot of kinetic energy away. As a result, this would leave very little energy left to create more massive Higgs bosons. So I started thinking and realized that there is more that is going on (e.g. cross sections sizes). I have purposely left this question open-ended to just see what people are thinking since I know very little about this. Therefore, as the title of this question puts it: why is the Higgs boson not created more frequently at the CERN collider? Aside: I guess I could of asked this question about some X-particle instead of the Higgs boson. Either way, my question would still be the same.


Looking backwards in time at yourself

If a person on Earth today is looking at a star, say, 10 billion light years away, is it possible that some of the atoms he is looking at will eventually go on to make him?


Neutrinos and anti-neutrinos in the Standard Model

In standard model neutrinos and the left handed electron forms SU(2) doublet.   1. What about the anti-neutrinos in the standard model? Do they also form some doublet?    2. If neutrinos have tiny masses will it not imply indirectly and conclusively that right-handed neutrinos must exist in nature? **EDIT :** _Neutrinos will have Majorana mass term if they are Majorana fermion. Is that right? Now, if neutrinos are Majorana fermions, will they have definite handedness? For example, does $\nu_M=\begin{pmatrix}\nu_L\\\ i\sigma^2\nu_L^*\end{pmatrix}$ have definite handedness? Therefore, doesn't it imply that if neutrinos are massive then a right-handed component of it $\begin{pmatrix} 0\\\ i\sigma^2\nu_L^*\end{pmatrix}$ must exist? Although we are not using $\nu_R$ to construct this column, does it imply $\nu_M$ do not have a right handed component? It is the column $\nu_M$ which we should call a neutrino. Then it has both the components. However, one can say that a purely right-handed neutrino need not exist if the neutrino is a Majorana fermion. Therefore, it seems that if neutrinos are massive a right handed component of it must exist (be it a Dirac particle as well as a Majorana particle). Correct me if I am wrong._


Do the proton PDFs change much with Q?

Specifically, the second moments, do they change much from say 100 to 1000 GeV? Why or why not?


What effects will the heliosheath have on the Voyager spacecraft?

I wanted to know what effects the heliosheath will have on the Voyager 1 & 2 craft. I know they used the slingshot method to travel through our solar system; so, with being way out in the heliosheath, how fast are the Voyagers going considering:   * the amount of fuel still available,   * and having no objects to slingshot around anymore?


Telescope for 6 year old and dad new to astronomy

What's a good telescope for my 6 year old? We're both brand new to telescopes in general, but love space, stars, planets, cosmology, etc. Happy to spend up to $200 to start. Maybe more for a good reason... **_UPDATE_** : Thank you guys for all the really great answers. I'm about to purchase the Celestron AstroMaster 114 EQ. Seems like the best value for the money based on the feedback below.


Using energy from sun magnetic field

Knowing               * earth spin and translation movement around sun,     * sun have a big magnetic field,     * a variable flux can generate a induced voltage (and current) into the circuit      Would it be posible to receive in a coil that energy? Perhaps it's just enough to detecting the earth movement relative to magnetic field, or if it were strong enough even for use the energy. It doesn't seem to be so strong because if it where perhaps every coil in earth without a current generated by this flux variation.


The formation of shockwave inside pellets in inertial confinement fusion

In the explanations i have read on the internet about ICF they all explain that the top layer of the pellet or the ablator is heated till it explodes and this produces a reaction force which compress the D+T fuel. They then say shock waves are formed due to the collapsing fuel which compress it even further and eventually ignite the fuel. My question is why and when are the shock waves formed?


How to find focal length of a single convex lens?

I need help finding the focal length of a single convex lens. The radius of curvature is 200mm. left side is air and the glass has a index of 1.5. I search on google but there was only formulas for think lens, convex and converge lends. I need the formula for a single convex lens. Any idea? I know the ray transfer matrix. $$\left(\begin{matrix} 1 & 0 \\\ \frac{n-n'}{nR} & \frac{n'}{n} \end{matrix}\right)$$


Electric heating rod

I usually heat my bathing water with electric heating rod, I always thought that the base of rod is an insulator so that it can develop high heat. But when I tried touching water while rod was dipped in it, I got current, damn! Why that happened? I am talking about this type of rod :![enter image description here](http://i.stack.imgur.com/nS3jk.png)


Rays in Symmetric Resonator

I'm having some trouble figuring out how to get started on this question: If I have a symmetric resonator with two concave mirrors of radii $R$ separated by a certain distance, after how many round trips through the resonator could I expect the ray to retrace its path? It seems to me that this implies some sort of periodic function should exist that models the behavior of a ray in such circumstances.


The temperature at the surface of Sol is apparently well above 5000C; I'm assuming the layers beneath the surface may be even hotter. At school we learnt heating a metal beyond a certain temperature specific to each metal would demagnetize the magnet. How does Sol's magnetic field continue to exist at such high temperatures?


Derivation of the full generator of the lorentz transformations

Let us study the subgroup of the Poincare group that leaves the point $x=0$ invariant, that is the Lorentz group. The action of an infinitesimal Lorentz transformation on a field $\Phi(0)$ is $L_{\mu \nu}\Phi(0) = S_{\mu \nu}\Phi(0)$. By use of the commutation relations of the Poincare group, we translate the generator $L_{\mu \nu}$ to a nonzero value of $x$: $$e^{ix^{\rho}P_{\rho}} L_{\mu \nu} e^{-ix^{\sigma}P_{\sigma}} = S_{\mu \nu} - x_{\mu}P_{\nu} + x_{\nu}P_{\mu}\,\,\,\,\,\,\,\,\,(1),$$ where the RHS is computed using the Baker-Campbell-Hausdorff formula to the second term. Then we can write the action of the generators $$P_{\mu} \Phi(x) = -i\partial_{\mu}\Phi(x) \,\,\,\,\text{and}\,\,\,\,L_{\mu \nu}\Phi(x) = i(x_{\mu}\partial_{\nu} - x_{\nu}\partial_{\mu})\Phi(x) + S_{\mu \nu}\Phi(x)\,\,\,\,\,(2)$$ Could someone explain the contents of Eqn (1) and how (2) is deduced from it? I think there is an unitary transformation on the LHS of Eqn (1), but I don't understand what the equation is doing and why the contents are what they are in the exponents of the exp terms. Many thanks for any clarity.


Did a racer on a "restless" course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)

Consider two participants, $A$ and $B$ moving uniformly wrt. each other; first having approached each other, then having met and passed each other, and eventually having separated from each other such that their mutual speed wrt. each other is given quantitatively as $\beta ~ c$ with some particular positive real number value $\beta < 1$. (Evidently, therefore, $A$ and $B$ had not been at rest to each other.) After having observed $B$'s passage, $A$ has been preparing starting blocks;   and at duration $\mathop{\Delta}\limits_{\stackrel{\normalsize{\text{until}}}{\small{\text{start}}}} \\! \tau_A$ after having observed $B$'s passage, $A$ issued a starting signal to a racer ($R$) upon which the racer took off from $A$'s starting blocks. After having observed $A$'s passage (i.e. soon enough after that), $B$ has been preparing a finish line;   and at duration $\mathop{\Delta}\limits_{\stackrel{\normalsize{\text{until}}}{\small{\text{finish}}}} \\! \tau_B$ after having observed $A$'s passage, $B$'s finish line ribbon was passed by $R$. The ratio of those durations is given as some particular positive real number value: $r := \mathop{\Delta}\limits_{\stackrel{\normalsize{\text{until}}}{\small{\text{finish}}}} \\! \tau_B ~ / ~ \mathop{\Delta}\limits_{\stackrel{\normalsize{\text{until}}}{\small{\text{start}}}} \\! \tau_A$ which (of course) satisfies $r > \sqrt{\frac{1 + \beta}{1 - \beta}}$. **Question:** Can an "average velocity" of racer $R$ with respect to the race course, as far as it is specified by $A$'s starting blocks and $B$'s finish line be defined,   such that its magnitude could be expressed in terms of the given quantities $\beta ~ c$, $\beta$ and $r$   (and where its direction is obviously "from $A$ to $B$") ?


branes-collision -> big-bang -> 2nd Universe?

If the cosmological model of the Ekpyrotic universe is correct should there not always a 2nd universe be created on the 2nd brane? Pictures and articles I found speak only of one universe. ![branes-collision](http://i.stack.imgur.com/Pzm82.jpg) Why is that? Did I miss something?


How does the light waves make up to my eye?

I was thinking about a situation were there is a group of people in a room. While all those people are observing something(none of them are blind) the light ways need to reach their respective eyes. But, how does the light ways make to one's eyes without being disturbed by the other light waves moving towards the eyes of others? Even there are a million people in a room one can see everything clear. How is it possible?


Axioms behind entropy!

The concept of entropy is very ubiquitous, we learn about its uses starting from Information Theory (Shannon entropy) up to its basic definition in statistical mechanics in terms of number of micro-states. Limiting the discussion to physics, when studying a physical system, can be a box filled with an ideal gas, a melt of polymers or the state of rods/molecules in a liquid crystalline system, in all such scenarios, there are specific entropic terms that we define in describing the evolution of the system (by including it in the free energy expression). From a statistical mechanics point of view, we use Boltzmann's definition of: $$S=k_B \ln\Omega$$ where $\Omega$, can be the partition function in an ensemble, or simply the number of microstates within a given macrostate. But of course we almost never use this exact form of entropy in studying real systems, as it is impossible to count the microstates by any means. Instead we define entropic terms based on macroscopic variables of a system, like the followings (examples among the usual ones):   * For a perfect gas one can write the entropy per atom of N atoms in volume V as: $$S_{\rm ideal}=k_B \ln\left(a\frac{V}{N}\right)$$ with $a$ a constant.    * In soft condensed matter, studying liquid crystallinity, often we define an orientational entropy, describing the entropy lost when molecules are oriented. In its most general form defined as: $$S_{\rm orient}=-k_B \int f(\theta)\ln f(\theta)d\Omega$$ where $f(\theta)$ is an orientation distribution function and $d\Omega$ a small solid angle.   * In polymers physics, often there are entropic terms attributed to homogeneity in number density distribution of monomers $n(i)$ along the chains (often called Lifshitz entropy), described as $$S_{\rm homogeneity}\propto -\left(\nabla \sqrt{n(i)}\right)^2$$ which is just the gradient of a density distribution. In all such cases, it is relatively straightforward to see as to why we refer to such state functions as "entropic", as they all boil down to describing a certain kind of disorder in the system, and how their inclusion would effect the equilibrium state of a system. But the underlying question is about, how we give physical and mathematical bounds to these entropic terms must be consistent, hence there must be a set of axioms that a function must fulfill in order to qualify as an entropic term. **To elaborate on the question:** On the one hand, from a physical point of view, the entropy is always set to reach its maximum at the most disordered state, and a minimum for the most ordered case (i.e. 1 possible micro-state, complete certainty over its state). For example in liquid crystals again, we want $S_{\rm orient}$ to be at maximum in the orientationally disordered state, the isotropic, and for it to be at minimum in the completely ordered states, like the smectic. On the other hand, mathematically we require that:   1. $S$ to be continuous at every $\Omega$   2. To be extensive with system size   3. Differentiable (does it have to be always?)   4. Path independent: state function   5. ...what else? Clearly if we don't know how to bound such functions based on a set of axioms, we cannot know if they will make sense physically and mathematically. (Helpful thought scenario: Imagine studying a system of particles in suspension, where their positional distribution comes with a certain periodicity, if we are to attribute an entropic term to the state of periodicity of the system, what conditions should such state function satisfy.)   * What are the axioms to be satisfied by a state function for it to qualify as entropy?


How to prove that a spacetime is maximally symmetric?

In Carroll's book on general relativity, I found the following remark: > In two dimensions, finding that $R$ is a constant suffices to prove that the > space is maximally symmetric [...] In higher dimensions you have to work > harder Here, $R$ is the Ricci scalar. This begs the following questions:   * How _does_ ones prove that a $d$-dimensional spacetime is maximally symmetric?    * If the general case is very complicated, then is there a simpler way to obtain the result in four dimensions?


Identify the weight of operator under conformal transformation

I have a stupid (homework tag may be suitable =_=) question about the problem 2.7 in Polchinski's string theory volume 1. Why the weight of operator $:e^{ik\cdot X}:$ is $(\frac{\alpha'k^2}{4}, \frac{\alpha' k^2}{4})$? From the solution manual http://arxiv.org/abs/0812.4408 and a similar procedure in the answers of this post Identity of Operator Product Expansion (OPE), $$ T(z):e^{ik\cdot X}: \sim \frac{ \alpha' k^2}{4z^2} :e^{ik\cdot X(0,0)}: +\frac{1}{z} i k_{\mu} : \partial X^{\mu} (0) e^{ik\cdot X(0,0)}: (1)$$ (skip the antiholomorphic part.) How Eq.(1) is related to the weight $(\frac{\alpha'k^2}{4}, \frac{\alpha' k^2}{4})$ without the contribution of $1/z^2$? For my understanding, consider $T(z) :\mathrm{operator}:$ is to use the Ward identity (2.3.11) in Polchinski's string theory $$\mathrm{Res}_{z \rightarrow z_0} j(z)A(z_0, \bar z_0) + \overline{\mathrm{Res}}_{\bar z \rightarrow \bar z_0} \tilde{j}(z)A(z_0, \bar z_0) = \frac{1}{i \epsilon} \delta A(z_0, \bar z_0)$$ with $j(z)=iv(z)T(z)$ and $\tilde{j}(z)=iv(z)\tilde{T}(z)$ thus we relate $T(z) :\mathrm{operator}:$ to $\delta :\mathrm{operator}:$ For example, the operator $X^{\mu}$, we got $\sim \frac{1}{z} \partial X^{\mu}(0)$. For conformal transformation $z \rightarrow \zeta z$, $\partial$ increase weight 1, $1/z$ decrease 1, thus the weight is (0,0). However, how Eq. (1) is related to $(\frac{\alpha'k^2}{4}, \frac{\alpha' k^2}{4})$ without the contribution of $1/z^2$?


Why do solar panels not have focusing mirrors?

Most of the solar panels that I have seen do not have any mirrors, etc., but usually solar cookers have mirrors. What is the reason for solar panels not having focusing mirrors?


Formula for polarized "light" transmission through close filters?

I'm still trying to understand photons (or polarized electromagnetic radiation).   This question is similar to one of my previous questions, but different:   Consider the famous demonstration of crossed polarizers.   Two polarizers at right angles block all light. Insert a polarizer at 45 degree angle between the two crossed polarizers, and 25% of the light is transmitted (neglecting absorption). Now perform the experiment with "light" of wavelength on the order of centimeters to meters, ie radar or radio band.   (Let's just choose 10 centimeters to be specific.)   Use parallel wire filters as the polarizing filters.   Now place the two crossed filters a meter apart, say at postions 0 and 100 centimeters.   Consider inserting the 45 degree filter in the middle at position 50 centimeters.   We should get 25% transmission. What happens if we move the 45 degree filter from position 0 to position 100?   I expect zero percent transmission at positions 0 and 100.   I expect 25% transmission at all positions between 10 and 90, but I can't prove it.   I expect transmission intermediate betwween 0% and 25% as the filter moves from position 0 to position 10, and the mirror image as the filter moves from position 90 to position 100.   What formula describes this transition?   More generally what formula describes the transmission as a function of position all the way from 0 to 100?


right hand rule using left hand?

im kind of confused, the right hand rule is used to know where the direction the magnetic field created by the current is going . is there any particular reason why the right hand is used to determine the direction of the Magnetic field?


1. Why do we treat time as parameter in non relativistic QM instead of treating position as parameter?   2. What is actually a parameter?    3. I have read this Wikipedia page, but could not understand why should we take only time as parameter?


Odd number of second class constraints (!)

For my thesis, I have calculated the constraints for a system using Dirac method of constraint analysis. The problem is I got odd number of second class constraints (!), which gives me unusual numbers of degrees of freedom in phase space. I might have made some mistakes. Is there any other method beside Dirac, to analyze the constraints of the system?


What would a length contracted object look like?

I have multiple sub-questions but they are related.   * What would the object look like if in were passing by?    * What would a star look like if we were traveling near $c$? Would the perspective be a a large blue-shifted disk??


How does a supersymmetry transformatio relates bosons to fermions?

A superfield is defined as a function of of a spacetime coordinate and two spinor coordinates, $\Phi (x, \theta, \theta^*)$. A supersymmetric transformation is then generated by the operator $\partial_{a}+i\sigma^{\mu}_{a\dot{c}}\theta^{\dot{c}}\partial_{\mu}$. What does it mean when we say this transformation relates fermionic to bosonic fields? Does the parameter $\theta$ measure the 'bosonic' content of a superfield?


Integrals of Power

This question has two parts: a question about a specific expression and a question about a family of expressions. _In what follows I will omit all constants of integration_. If we integrate force with respect to time we get momentum: $\int F \, \operatorname{d}\\!t = mv$. If an object accelerates along a line then we can find its acceleration at any given point and write force as a function of distance. Doing so, and integrating gives kinetic energy (or work done): $\int F \, \operatorname{d}\\!x = \frac{1}{2}mv^2$. I decided to try the same thing with power using the definition $P = Fv$. When I integrated with respect to time I got, as expected, $\frac{1}{2}mv^2$ which is work done. If an object accelerates along a line then we can measure the power at regular distances and write power as a function of distance. Then we may integrate power with respect to distance: $$\int P \, \operatorname{d}\\!x = \int mav \, \operatorname{d}\\!x = m\int\frac{\operatorname{d}\\!v}{\operatorname{d}\\!t}v\, \operatorname{d}\\!x = m\int\frac{\operatorname{d}\\!x}{\operatorname{d}\\!t}\frac{\operatorname{d}\\!v}{\operatorname{d}\\!x}v\,\operatorname{d}\\!x = m\int v^2\operatorname{d}\\!v = \tfrac{1}{3}mv^3$$ **Here is the first question** : _What is $\frac{1}{3}mv^3$? Does it have a name? How can I understand it conceptually?_ **Here is the second question** : Momentum is $mv$, kinetic energy is $\frac{1}{2}mv^2$, this new quantity $\frac{1}{3}mv^3$ has turned up. It seems that $\frac{1}{k}mv^k$ for positive whole numbers $k$ plays a role. These can all be combined into one power series. Provided $v$ is very small, we have $$mv + \tfrac{1}{2}mv^2 + \tfrac{1}{3}mv^3 + \cdots + \tfrac{1}{k}mv^k + \cdots \equiv m\ln\left(\frac{1}{1-v}\right)$$ _Do the expressions $\frac{1}{k}mv^k$ have any general meaning; and does this logarithmic expression have any use?_ (I realise that the units in the sum are all different.)


Knowing the mass and force acting on a particle, how do we derive the relativistic function for velocity with respect to time?

Use this scenario: An electron gains speed in the Stanford Linear Accelerator (SLA) across 3000 meters, reaching a final velocity of 0.95c due to a constant force pushing on the electron. Given the mass of the electron and the constant force pushing on the electron, what is the function for velocity in terms of time? How do we derive this?


How is charge measured and mass of the electron at the same time?

1. There are few constants that usually come together, $e/m$ also $h/e$. How are they decoupled?    2. If the speed of light is "derived" as Wikipedia states how meter defined and time?


How can the properties of the Earth be determined? And how is the gravitational constant then determined from them? A satellite can determine the G.M.. But, how they are individually determined nowadays ?


Extracting acceleration vector from rotated aircraft

Suppose we have an aircraft with accelerometer measuring accelerations along each axis. It is mounted in a way so it is perpendicular to the plane in all axes (that should be obvious). We also have gyrometer sensor mounted on the plane - it can tell us how is the plane currently aligned (rotated) in air. **1st case:** Suppose we have an aircraft on the ground. Then we can get our desired acceleration component just by reading the accelerometer's Z axis reading (Z axis is pointing from sky to the center of the earth). **2nd case:** The plane is in the air, doing some loops for example - now to get the desired reading as previously (acceleration towards earth) we would need to compensate the Z axis acceleration with current rotation and/or accelerations. So, I have all the data but I don't know what physical/mathematical formulas I should use to extract the acceleration component pointing towards the earth. I would also be glad to see some tips about what literature I should read to be able to do this.


What is the place of an electromagnetic field in the electromagnetic spectrum?

What place should I give an electromagnetic field (produced by a current conducting coil) in the electromagnetic spectrum? What will its wavelength and frequency be?


After normalizing a wavefunction I don't know how to calculate probability on an interval (-0.1 + 0.1)

This is quite large homework where I 1st had to normalize the wavefunction $\psi = Axe^{-x^2/2a}$ and I got a constant $A=\sqrt{2/(a\sqrt{\pi a})}$. How do I calculate the probability now for the interval (-0.1nm , + 0.1nm) if $a=1nm^2$? * * * I first tried to calculate the normalisation factor but the units are weird... is this possible? Is most likely my normalisation factor wrong? $$A=\sqrt{\frac{2}{1nm^2 \cdot \sqrt{\pi\cdot1nm^2}}}=1.06\frac{1}{nm\sqrt{nm}}$$ * * * **EDIT:** After thinking again I decided to include the derivation of the normalisation factor $A$: \begin{align} &\phantom{=}\int\limits_{-\infty}^\infty A^2x^2 \exp\left[\tfrac{-x^2}{a}\right]dx = 1 = A^2\int\limits_{-\infty}^\infty \underbrace{x}_f\cdot \underbrace{x\exp\left[\tfrac{-x^2}{a}\right]}_{dg/dx} dx= \\\ &=A^2\left[\smash{\underbrace{\left.x\left(-\tfrac{a}{2}\right)\exp\left[\tfrac{-x^2}{a}\right]\right|_{-\infty}^{\infty}}_0}-\int\limits_{-\infty}^\infty\left(-\tfrac{a}{2}\right)\exp\left[\tfrac{-x^2}{a}\right]dx\right]=\\\ &=\tfrac{A^2a}{2}\int\limits_{-\infty}^{\infty} \exp\left[\tfrac{-x^2}{a}\right]dx = \tfrac{A^2a}{2}\int\limits_{-\infty}^{\infty} e^{-t^2}\sqrt{a}\,dt = \tfrac{A^2a\sqrt{a}}{2}\int\limits_{-\infty}^{\infty} e^{-t^2}dt = \frac{A^2a\sqrt{a}}{2}\sqrt{\pi} \end{align} Which means that: \begin{align} \frac{A^2a\sqrt{a}}{2}\sqrt{\pi} &= 1\\\ A&=\sqrt{\frac{2}{a\sqrt{a\pi}}} \end{align}


separation between slip rings of DC generator

![enter image description here](http://i.stack.imgur.com/9lj41.gif) this is a DC generator which converts mechanical energy to Direct Current. The commutators in a DC generator are separated (as you can see in the image). It is explained in our book that it prevents the change in direction of current. But i don't understand, even if we don't keep them separated, how will current change direction? In short, what is the function of the gap between the two split rings?


Calculating the gravity gradient of a black hole

Given a black hole of, say, $10^8 \odot$ (solar masses), how can I calculate the gravity gradient ($\frac{\Delta g}{m}$?) at a given distance from it?


Gauss's Law problem

I've been thinking about this for a while, but I'm not sure how to proceed. I understand uniform charge-density problems, but the added non-uniform deal makes me uneasy:   * find the e. field inside a sphere (rad. = R) with nonuniform charge density $\rho = {\rho _0}\frac{r}{R}$ **Edit** I did this: ![](http://s8.postimg.org/qpv82s4xx/img.jpg). Can someone confirm if this is correct?


Hermitian conjugation in Radial Quantization

I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\dagger = t$ and $x^\dagger = x$. We then Wick rotate the theory to Euclidean time $t \to - i \tau$ and therefore $\tau^\dagger = - \tau$. Next, we define the complex coordinates on the Euclidean plane $w = x + i \tau$ and ${\bar w} = x - i \tau$. The Hermitian conjugate of acts on this coordinate as $w^\dagger = w$ and ${\bar w}^\dagger = {\bar w}$. On the radial plane with $z = e^{- i w}$ we get $$ z^\dagger = e^{i w^\dagger} = e^{i w} = \frac{1}{z} $$ However, Francesco explicitly says (pg. 152, above equation (6.4)), that $z^\dagger = \frac{1}{z^*}$. Where am I going wrong? Can anyone explain this?


Where are the L3, L4, and L5 of a hyperbolic orbit?

Do the L3, L4, L5 points exist in hyperbolic orbits? If yes, then where do they lie?


Functional field integral in condensed matter field theory (Altland)

This is the action for the 1+1 dimensional interacting electron system; $$S_{cl}[\theta , \phi]= \frac{1}{2\pi} \int dxd\tau \left(g^{-1}v(\partial_x \theta)^2 + gv(\partial_x \phi)^2 + 2i\partial_{\tau} \theta \partial_x \phi \right).$$ I want to integrate out the Gaussian field $\phi$. This book says that it is just an "elementary" Gaussian integration. So, I tried some modification to this action; $$S_{cl}[\theta , \phi]= \frac{1}{2\pi} \int dxd\tau \left(g^{-1}v(\partial_x \theta)^2 + (\sqrt{gv}\partial_x \phi + \frac{i}{\sqrt{gv}}\partial_{\tau} \theta)^2 + \frac{1}{gv}(\partial_{\tau} \theta)^2 \right).$$ For this action, partition function is given by $$\int D\theta D\phi \exp[-S_{cl}].$$ Maybe, the second term in the action is related to Gaussian integral. But, I don't know how to calculate it. How can I calculate this?


Dimensional reduction and Schwarzschild solution

It is well-known that, given a Killing vector $\xi_\mu$ for a metric, this can be dimensionally reduced. This is exactly the case of the Schwarzschild solution due to the rotational symmetry. This procedure is quite general and one can always reformulate a general 2d (or 3d) model, as the Jackiw- Teitelboim model, of general relativity and obtain solutions for the 4d case with the given symmetry properties.A reference about this is this. The effect, as always happens in dimensional reduction, is to introduce another field into the gravity action. So far, I was not able to see the derivation of the reduced Schwarzschild metric from the dimensionally reduced Einstein equations. This is generally given for granted. So, I would appreciate to see such a derivation or to get some tutorial about this. Thanks.


Can a Celestial body have a mountain rise all the way to Synchronous Orbit?

On Earth, the tallest mountain is Mauna Loa. In the solar system, to date, the tallest mountain is Mons Olympus. Both fall far short of GSO. Mars gravity is 0.376 of Earth gravity. If Mauna Loa were on Mars it would have an altitude of ~80000 ft. If Mons Olympus were on Earth it would similarly be no more than ~27000 ft. Yet these are almost comparable altitudes. Is it possible for a mountain to rise as high as GSO? Are all known tallest peaks on planets in the Solar System similarly limited?


What are the forces used in this weightlifting example?

You have a person doing a bench press. Their hands are equidistant apart and they are centered on the bar - each hand about 1.5 feet from center of the bar. The bar weighs 45 pounds. In example 1 there are two 45 pound weights on each side at 2.5-3 feet each from the center (lets say each weight is 3 inches wide). In example 2 there are nine 10 pound weights on each side starting at 2.5 feet again and then ending 27 inches later. What are the forces that cause example to to feel heavier and is there any way to calculate it.


returning to physics research?

What the the most realistic and practical ways for a PhD in physics who has left the fold to return to a career of research in physics? The traditional employer of choice, universities won't hire such people. What alternative career choices are there for someone without an impressive track record so far? Are there alternative forms of support, or part time careers earning just enough money to get by while not interfering with research?


Some sort of conservation equation

As far as I know, in General Relativity, an expression of the kind $\nabla_{\mu} X = 0$ states that, associated to $X$, there exist a charge which is conserved. The first example that comes to mind is the conservation of energy-momentum $\nabla_{\mu} T^{\mu\nu} = 0$, as a consequence of the contracted Bianchi identities and assuming Einstein's field equations. My question regards the meaning of the following expression: $\nabla_{\mu} \left( R^{\mu\nu} K_{\nu} \right) = 0$ where $R^{\mu\nu}$ is the covariant Ricci tensor and $K_{\nu}$ is a **generic** vector field. Does this equation mean something as it stands? Or do I have to impose a metric (and hence a form of the Ricci tensor) in order to find more informations? The fact is that I don't understand what kind of conserved quantity could be related to the vector $R^{\mu\nu} K_{\nu}$. The equation comes from a rather peculiar way to obtain a modified theory of gravity. The reference is here, but the original source is there I admit that the procedure is not entirely clear to me, but basically it seems to appeal to a "gauge theory"-like approach in order to obtain modified gravitational field equations. It's kind of lengthy, but ultimately they consider commutators of covariant derivatives of successive orders to obtain relevant quantities, such as the Riemann tensor (first-order commutator) and matter current (second-order commutator). For the latter, the analogy with gauge theories is necessary. Finally they get the modified gravitational field equations, eq.(16) in the first reference. The very nature of the approach makes it necessary to use a vector field $K_{\nu}$ in order to carry on the calculations, but the character of such a vector field is never specified (there's no need to, in fact). Only after eq.(16) they comment that Einstein's case is recovered if $\nabla_{\mu}K_{\nu}=0$. Going back to the equation that I posted... in order to have some hints about the meaning of it all, starting from eq.(16) in the paper, I just impose the vacuum condition $T^{\mu\nu}=0$ and see what happens: $\left(\nabla_{\mu}R^{\mu\nu}\right) K_{\nu} + R^{\mu\nu} \left( \nabla_{\mu} K_{\nu} \right) = 0$ and that's it. I'm not at all convinced about the comments they make about the vector field (they call it "substratum") but the overall procedure doesn't seem entirely bad.


Diagonal matrix in k-space

I'm having some trouble with an integration I hope you guys can help me with. I have that: ${{\mathbf{v}}_{i}}\left( \mathbf{k} \right)=\frac{\hbar {{\mathbf{k}}_{i}}}{m}$ and ${{\mathbf{v}}_{j}}\left( \mathbf{k} \right)=\frac{\hbar {{\mathbf{k}}_{j}}}{m}$ Now according to my teacher when I multiply these two, and integrate the angular part ($\,d\theta \,d\varphi$) over $d{\mathbf{k}}$, with $d\mathbf{k}={{k}^{2}}\,dk\sin \left( \theta \right)\,d\theta \,d\varphi$, I should end up with: ${{\mathbf{v}}_{ij}}{{\left( \mathbf{k} \right)}^{2}}=\frac{{4\pi{\hbar }^{2}}}{3{{m}^{2}}}{{k}^{2}}$ My question is why?I see that when there is no angle dependency I get the $4\pi$, and nothing touches $\hbar$, $m$ and $k$. But where to the $1/3$ come from? That's probably my main concern. So anyone with a hint or something ? Thanks in advance.


Why are black mosquito nets so much less visible than white ones?

I have just been installing a couple of mosquito nets for windows in our apartment. The material is polyester, fibers are approximately 0.2mm thick, and the hexagonal "holes" apprixmately 1.5mm in diameter. Just like the packaging says ("buy black ones if you want to have a better view"), the black ones are almost invisible from a little distance (~2 meters, its a little bit darker, almost in effect like an ND photo filter). From the same distance, the white ones are clearly visible, with good eyes you can also spot the "holes" (the effect looks almost like if you put the black point in photoshop into a much whiter area, making the contrast much less). Why is that so? Is there a physical effect that objectively reduces the "image information" that reaches the human eye? I would appreciate an answer that -- if applicable -- contains a short average human explanation, and additional one that goes a bit more in-depth into the "hobbyist physics" level


Is this 2D structure triclinic?

![enter image description here](http://i.stack.imgur.com/v8Uhd.jpg) The only rotation axis obvious to me is rotation by 360 degrees, the identity. Vertical mirror planes I've been dicing and cutting it through several planes and I still see none. Yet, the structure looks fairly symmetric and I feel like I'm missing something. No center of inversion either. Not too terribly sure about horizontal mirror planes. I suppose that one bisecting the plane horizontally would be there but I am not entirely sure if this is appropriate. Right now I'm saying this structure belongs to the C1 point group.


True or false? Particle physics

1. It is not possible to prove the point of origin of a photon   2. It is not possible to prove the point of origin of a free electron   3. It is not possible to prove that protons or neutrons exist inside a nucleus Can anyone confirm?


Gas Circulation Using Pressure Difference

Dear all, see attached picture Please, is it possible to have the gas recirculated from the gas phase to the liquid as described in the diagram assuming the gas is not soluble in the water. These are the conditions. 1\. The valve remained closed, 2\. The liquid level didn’t change 3\. P1 increases. Could P1 get to a level where it would force itself through the pipe and go through the liquid. If P1 increases, what happens to P2. ![enter image description here](http://i.stack.imgur.com/CHf0S.jpg)


Electric field inside and outside a metallic hollow sphere

1) It is known that inside a metallic hollow sphere it will not experience outside electric field because of the charge separation of electrons and holes at the surface of sphere and creating an equal and opposite field. So it is better to stay inside the car during lightning. Why stay in the car during thunderstorms? But isn't there a maximum value of electric field the metallic surface can resist because at some point (higher value) of electric field the movement of electrons will get stopped because it can't generate more electron holes pairs and which will limit the further change in increment of electric field to zero. Since lighting produce such a high value of electric field is it safe to stay inside a car during lightning? _assuming no heat generated during lightning_ 2) Similar case about charge inside a earthed metallic hollow sphere. If I keep some charge inside the metallic hollow sphere, outside field is zero and inside field is dependent upon the charge. If I increase the charge inside the metallic hollow sphere to such a level which is more than the total electron charge on the surface of sphere then the earthing has no effect because it has already neutralized the maximum amount of holes (+ve) by accepting the electrons from earth. So the field will escape outside.


Trace of stress tensor vanishes ==> Weyl invariant

You often see in textbooks the statement that ${T^\mu}_\mu = 0$ implies Weyl invariance or conformal invariance. The proof goes like $\delta S \sim \int \sqrt{g} T^{\mu\nu} \delta g_{\mu\nu} \sim \int \sqrt{g} {T^\mu}_\mu$ where I have varied the action with respect to the metric and assumed $\delta g_{\mu\nu} \propto g_{\mu\nu}$ (i.e. a Weyl transformation). This does not seem to be completely general because I can imagine a Lagrangian containing matter fields with non-trivial conformal weights. Then the full variation under Weyl tranformation contains a term proprotional to the matter equation of motion. So I would conclude that the correct statement is more like ${T^\mu}_\mu = 0, \quad \& \quad \frac{\delta S}{\delta \phi} = 0\implies \textrm{Weyl invariant}$ Is it true that Weyl invariance only holds when the matter fields are on-shell or am I missing something? Thanks.


Optimal velocity in rain

**_Problem statement:_**   > A sphere is rolling in rain from point A to point B. The vertical speed of > rain is V and horizontal speed of rain is v, as shown in the picture. Angle > between horizontal component of rain's velocity and sphere's velocity is > $\varphi$. What is the optimal speed of sphere so that it would be as dry as > possible?   >   > ![Problem picture](http://i.stack.imgur.com/sljvY.png) So here's what I tried to do:   The velocity of rain in respect to the sphere is $\vec{v} = \vec{v_s} - \vec{v_r}; v=\sqrt{v_s^2 + v_r^2} = \sqrt{v_s^2+(\sqrt{V^2+v^2})^2}=\sqrt{v_s^2+V^2+v^2}$. I thought that maybe if I found the derivative of this function in respect to $v_s$ and equated it to 0, that could be the answer. But I don't know how to find the derivative of such function and the solution doesn't seem right to me, though I couldn't come up with anything better.. Any help appreciated!


How do I solve for the phase constant given the amplitude and the angular frequency?

> A piston (with mass M) in a car engine is in vertical simple harmonic motion > with amplitude A. The engine is running at a period T. Suppose a small piece > of metal with mass m were to break loose from the surface of the piston when > it (the piston) is at the lowest point z and velocity is v. At what position > in meters does the piece of metal lose contact with the piston? Also the maximum velocity and acceleration are given. The text book gives these equations: $x(t) = Acos(wt+φ)$ $v(t) = -wAsin(wt+φ)$ $a(t) = -w^2Acos(wt+φ)$ So I propose that the best way to figure this out would be to solve for the time at which acceleration is at its max, then plug that time into $x(t)$. But I must first solve for the phase constant. Is this correct? If so, how do I find the phase constant?


Voltage of open circuit

> A battery with emf $\varepsilon$ and internal resistance $r$ is connected > with a resistor $R$ in the following open circuit. What is the voltage > $V_{ab}=V_a-V_b$? > > ![enter image description here](http://i.stack.imgur.com/8RVNX.png) The answer is $- \varepsilon$. " _No current. There is no voltage change across R and r._ ". But I don't really understand why ... I was thinking intuitively it should be $0$? Then thinking of how to get 0, I was thinking ... $V_a = - \varepsilon$ since its on the negative terminal, the $V_b = + \varepsilon$ since its on the positive terminal. But $V_a - V_b = -2 \varepsilon$ ... how do I make sense of this?


Inserting metal into parallel plate capacitor

> The plates of an isolated parallel plate capacitor with a capacitance C > carry a charge Q. The plate separation is d. Initially, the space between > the plates contains only air. Then, an isolated metal sheet of thickness > 0.5d is inserted between, but not touching, the plates. How does the > potential difference between the plates change as a result of inserting the > metal sheet? The answer is: The potential difference will decrease. How do I get this result, it seems right but how do I reason it out? $$V = Ed$$ There was mention about " _E is the same outside the conductor. But the field inside is 0._ ". But in the formula, E is the field outside conductor right? If so it remains the same? Then $d = 0.5d$?


Where the angular momentum has gone?

I came across an interesting problem when I prepared for the preliminary exam on electromagnetism. Below is the problem in its original words: > A metallic sphere of mass, $m$, and radius, $a$, carries a net charge, $Q$, > and is magnetized with uniform magnetization, $M$, in the $z$-direction. > > (a) Determine the total angular momentum associated with the electromagnetic > field. > > (b) The magnetization of the sphere is now reduced to zero. Assuming that no > external mechanical forces act on it (PS: this is possible. For instance, > just heat up the sphere and the magnetization will be eliminated ), > determine the angular velocity and the sense of rotation of the sphere when > $M=0$. > > (c) Describe the origin of the torque that causes the sphere to rotate. For part (a), it is straight forward to calculate electric and magnetic field E, H inside and outside of the sphere. Generated by the net charge $Q$ uniformly distributed on the surface, the electric field $E$ is isotropic outside the sphere and vanishes inside. The magnetic field H is generated by magnetization $M$ and can be calculated using the "magnetic scalar potential" formalism since there is no free current. The result is that H is also uniform inside the sphere and takes the dipole form outside the sphere. The total angular momentum $J$ is obtained by integrating over the region where a local angular momentum density of the EM field exists, i.e. outside of the sphere. It is in the $z$-direction and I find it to be $$J= \frac{2 Q M a^{2}}{9 \epsilon_0 c^{2}}$$ For part (b) and (c), it is interesting to ask why the sphere would rotate after magnetization $M$ is erased. What I think up is that as $M$ decreases to zero, a electric field with non-zero curl is generated, as implied by Faraday's law. This electric field is in the azimuthal direction, and will act on the surface charge $Q$ to provide a torque. However, a straightforward calculation can show that the angular momentum this torque can transfer is $$J'= \frac{Q M a^{2}}{9 \epsilon_0 c^{2}}$$ Obviously a factor of 2 is missing, so this only accounts for half of the angular momentum stored originally in the EM field. So I am perplexed by where the other half has gone? I come up with 2 possibilities:   1. since the charged sphere begins to rotate, there is also non-zero angular momentum stored in EM field after magnetization is removed. But it could hardly account for the missing amount, since in that case the electric field $E$ is proportion to $Q$, the magnetic field $H$ to $Q^2$ ( one $Q$ from the charge and a second $Q$ from the angular velocity). Hence one expects the angular momentum stored is proportional to $Q^3$, which is very different from $J$'s linear dependence on $Q$.   2. maybe in the process of magnetization removal, EM radiation carries away some angular momentum. But this possibility has to be validated quantitatively before it can convince people. What I feel is that if the process of removal is infinitely slow, then the effect of radiation can be neglected. Then what is the correct answer? I provide quantitative result for $J$ and $J'$ in the hope that someone can check them and point out that I have made a wrong calculation lol. But I am really eager to hear from any explanation that sounds real ... Electromagnetism is as interesting as it was 3 years ago, but I just don't remember as much as I did then ...


What did Marie Curie do for atomic theory?

There appears to be a distinct lack of agreement in the physics community on what exactly Marie Curie did for atomic theory. Many journals state that Curie was responsible for shifting scientific opinion from the idea that the atom was solid and indivisible to an understanding of subatomic particles. However, JJ Thompson had already discovered the electron prior to Curie's work, and Rutherford proved the nature of atomic structure - so what did Marie Curie really do for atomic theory?


How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?

I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. Here, it says you have two options and that one option is to construct simultaneous eigenket $\vert j_1j_2;m_1m_2\rangle$ of $J_1^2, J_2^2, J_{1z}, J_{2z}$ since the four operators commute with each other. I understand that $J_1^2$ and $J_{1z}$ commute, but I'm not sure how $J_1^2$ and $J_{2z}$ can commute intuitively. "commute" means that one can measure both at once right? But total angular momentum of spin 1 and angular momentum of spin 2 are independent. Where am I wrong here?


Does mass equal angular momentum?

At the wikipedia pages for angular momentum ($L$) and moment of inertia ($I$) we find the equations: $$L=I \omega$$ $$I=m r^2$$ where $m$ is mass and $r$ is the distance between said mass and barycenter and $\omega$ is angular velocity. Now rename the angular velocity as follows: $\omega=\frac{d\theta}{dt}$. Note that the area of an ellipse sector swept out from a focus is given by: $$A(\theta)=\frac{1}{2}\int_0^{\theta} r^2 \:\: ds$$ And so: $$\frac{dA}{d\theta}=\frac{r^2}{2}$$ Note that for a mass in an elliptical orbit this area is proportional to time. Therefore, we can write: $$\frac{d\theta}{dt}\sim\frac{d\theta}{dA}\sim\frac{1}{r^2}$$ Substituting this back into the first two equations at the top of this post gives: $$L\sim m$$


Wave function of hydrogen atom including spin of nucleus

How do I write the wave function of hydrogen atom taking into consideration of nucleus spin? For example consider $2S_{\frac{1}{2}}$ state with nucleus spin $I$, then wave function $\psi=\langle2S_{\frac{1}{2}},F,F_{3}|$ where $F$ is the total angular momentum of hydrogen atom $F=J+I$ and $F_{3}$ is the projection of it along z axis. Now what will be the explicit form of the wave function? Thanks in advance.


What did Hawking mean? 'Time started at the big bang'. Book suggestions please

**After writing down this question, I have come to realize that. What I really want is _reading materials_ on the questions below.** Before the big bang there was no such thing as 'time' (Steven Hawking on why the pope was wrong when he stated that physics and cristian beleves could go together as someone 'must have started the big bang'). Hawking stated that according to his theories time started at the big bang. Before the big bang, no time, hence no creator. Hawking actually said this in a speach, in memorial to Gallileo when the pope made official appologices on behalf of the church for banning Gallileo. I read this years ago so I might be wrong in some detail. **I would like something to read to understand what Hawking meant when he said that there could be no creator since time 'started' at big bang and did not exist before that**. I found this paper but I am not sure I understand what it means. Concentric circles in WMAP data may provide evidence of violent pre- Big-Bang activity. I think I have to read other things before actually digging in to that paper since I find it pretty hard. **What can I read as an introduction before reading this paper. Frankly, I don't understand it.** From a physics phd student I was informed that many physics researchers actually are unsure that the big bang really existed. Since they don't believe that everything could have existed in a singular point. Do they? (Might be a tricky question to answer). When you spread out time it turns out that time can not be spread out. I.e. it does not work like a rubberband. Or perhaps, When you compare time (maybe as a result of jouries at extreme speed where the two objects then have different ideas of how long time it has taken) does not follow form a bell curve statistically. Why not? The uncertainty evaluated at t = 0 also gives strange results but I am not sure it is related. _Perhaps it is perhaps not_. **I know a lot of calculus, a lot of linear algebra and basic quantum mechanics.**


Synchronising the Earth's rotation via mass redistribution

How much material would have to be moved per year from mountain-tops to valleys in order to keep the Earth's rotation synchronised with UTC, thus removing the need for leap seconds to be periodically added? Would it be a feasible project to undertake in order to resolve the current disagreement about the future of the leap second?