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It has been asked and answered before how to derive the Maxwell equations from the electromagnetic Lagrangian which in the vacuum is: $$\mathcal{L}=-\frac{1}{4\mu_0} F_{\mu\nu}F^{\mu\nu}$$ As in the linked question and answers, the correct approach is to write $\mathcal{L}$ in terms of the vector potential $A_\mu$ , and then apply the variational calculus. In this question, my concern is about ho…

Let’s consider a wire of resistivity $\rho$ , length $L$ , and cross-sectional area $A$ , carrying a time-varying current $I(t)$ such that $\frac{dI}{dt} = k$ , where $k$ is a constant. This wire is connected to a parallel-plate capacitor of capacitance $C$ , forming a standard charging capacitor setup. We apply Ampère–Maxwell’s Law in integral form: $$ \oint \vec{B} \cdot d\vec{l} = \mu_0 \left(…

Let me start off the bat by saying that I know that the derivation of $E\propto m$ in special relativity must be somewhat involved - please forgive me the title of this question. Nevertheless, I like the intuition of the following argument and I would like to know if it can be put on firmer ground. A central conceit of special relativity is the unification of space and time into space time. Given…

When we "cut" an ordinary path integral, we obtain a state in the position representation. That is, if we fix some initial position $x_i$ , then the path integral $$\int_{x_i}^{x_f}Dx e^{-S}$$ is just a complex function of the final position $x_f$ - i.e. it's just a wavefunction $\psi(x_f)$ . The phase space path integral is given by $$\int DxDp e^{i\int(p\dot{q}-H)dt}.$$ Usually, this path integ…

I often read that there is no known mechanism that prevents a charged black hole, if the charge is high enough compared to its irreducible mass, from forming a naked singularity. But I don't get how that is possible. Imagine a hollow sphere of charge $q$ , with an “infinite” radius and “infinitesimal” mass. Imagine we start shrinking it. To do that we need to do work. That work will create mass, …

I am a physics senior wondering which math topics and sections I should revise for thermo. So far, I am revising derivatives and integrals; what else should I revise?

Really simple question, but I'm confused with it because of contradictions between sources (absence of clarity online vs lecture notes (says nuclear) and practise questions provided (uses atomic) ) In the reaction energy: $$Q=E_{R,i} - E_{R,f} = K_f - K_i$$ Does one use the atomic masses or the mass of the nucleus (like what is needed for binding energy?) when solving for this? Thanks very much i…

I am trying to understand how $\sigma_b=\vec P \cdot \hat n$ . In wikipedia I found the correct derivation but I don't understand some things. There it is said: The bound surface charge is the charge piled up at the surface of the time dielectric, given by the dipole moment perpendicular to the surface: $q_b=\frac {\vec d \cdot \hat n}{|\vec s|}$ where $\vec s$ is the separation between the point…

Gaussian plume models are often used to model atmospheric dispersion because they are simple and computationally efficient. When not constrained by the ground or by inversion layers, the Gaussian plume equation has the following form: $$\chi=\frac1{2\cdot\pi\cdot\sigma_y(x)\cdot\sigma_z(x)\cdot u}\cdot\exp\left(-\frac{y^2}{2\cdot\sigma_y^2(x)}\right)\cdot\exp\left(-\frac{\left(z-h\right)^2}{2\cdo…

In this YouTube video water ripples start out with a wavelength of, say, 10 cm at the outer edge and they end up with a wavelength of about 30-40 cm towards the end of the video. Why does this happen, in simple, plain English? I can see here wave equations that produce the same result in simulations, but the equations don't give me a "feel" for why this happens. Intuitively, I'd expect the opposi…

I am in AP Physics E&M and have learned how to find electric field strengths using Gaussian surfaces for static charges, however I cannot understand why the field produced by an insulating plate is half of that produced by a conducting one. Also, in the example, we are only considering surface charge density, so in theory, it shouldn't matter whether or not the material is a conductor or insulato…

I am a current GeoPhysics graduate student studying how high-voltage / high-current electricity can be used to heat up an underground reservoir. I have designed a basic experiment, where I will drill two vertical wells into a bedrock containing water / electrolyte, and place electrodes (anode and cathode) into each well. I will then apply a voltage difference (alternating current) across the wel…

Wikipedia says, "In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional space-time. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in space-time." Does this mean that four-momentum is the type of vector resulting from three spatial dimensions being placed, as a whole, into a greater level of m…

Suppose an electron (mass $m$ , charge $e$ ) in the xy-plane with $B=(0,0,B)$ (The classical EOM result in circular orbit). Using the Bohr-Sommerfeld quantization rule we can find that $E_n = (n+1/2)\hbar\omega_c$ where $\omega_c=eB/m$ - the cyclotron frequency. The next exercise asks me to show that the quantization rule can also be expressed in terms of the flux $\Phi$ enclosed by the cyclotron…

I am trying to work out how I would find the rate of heat transfer through two pipes made of the same material that are joined at their ends but which have different cross-sectional areas and lengths. One end of the combination is held at $T_1$ and the other at $T_3$ but I don't know the intermediate temperature $T_2$ where the two pipes are joined which is a problem since the thermal conductivit…

Deflection of charged particles in a magnetic field can be derived by equating centripetal force and magnetic force, resulting in the equation r = mv/qB = p/qB. Accordingly, if an electron and proton enter a magnetic field with equal momentum, then the above equation implies that they will deflect with equal radii. However, due to the far lesser mass of the electron, would not the centripetal for…

I am reading Polchinski Volume 1, and I am stuck at equation for the momentum operator in free scalar field theory in $2D$ euclidian space which is given as : $$p^\mu=\frac{1}{2\pi\ i}\oint_C \left( dz\ j^\mu-d\overline{z}\ \tilde{j^\mu} \right)=\sqrt\frac{2}{\alpha'}\alpha^\mu_0=\sqrt{\frac{2}{\alpha'}}\tilde{\alpha^\mu_0} \tag{2.7.3} $$ where $j^\mu$ and $\tilde{j^\mu}$ are the holomorphic and …

This is basically the same question as Why resolve some forces into components and not others? However, I don't understand how the preferance of coordinate systems affect the resolution of forces. In the diagram above, if we resolve the forces with respect to the horizontal axes, that is the one of the left, we get \begin{equation} F_N=\dfrac{mg}{\cos \theta} \end{equation} whereas if we resolve …

I've been thinking about whether gravity could be an emergent phenomenon rather than a fundamental interaction. If gravity is emergent, I assume that at some deeper level there would be more fundamental degrees of freedom whose collective behavior produces the spacetime geometry described by general relativity. My main question is: what observable or experimentally testable predictions could dist…

I am trying to understand how the magnetic moment is invariant in slowly changing magnetic fields. There is a proof in the textbook I am using, but I'm stuck on how $-e\int\frac{\partial B}{\partial t}dS$ becomes $\pi r_{L}|e|\dot{B}$ and how the final $\delta(W_{\perp}/B) = 0$ is obtained. I was thinking of doing some sort of Taylor expansion on the rate of change of B and ignore higher-order te…