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This gauge invariant vector potential increases without bound as long as there exists a static electric field. Indeed, even when the electric field is removed, there appears to be no mechanism by which the gauge invariant vector potential disappears. Static electric field has zero transversal component; entire field is longitudinal. The unbounded ...

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From the point of view of electromagnetism I can see it the following non-mathematical way. $curl$ or $rot$ differential operators calculates magnitude of "vorticity" of the field, i.e. how much it is "spinning" or changing in rotation. Now, "vorticity" of the "vortex" is how much this "vortex" spinning again. Take, for example, long spring and connect its ...

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It's funny that all the answers so far forgot the simple and elegant following criterion : quantum mechanics appears when $$\dfrac{\hbar\omega}{k_{B}T} > 1$$ with $\hbar\approx6,63.10^{-34}\text{J}\cdot\text{s}$ the Planck constant and $k_{B}\approx1,38.10^{-23}\text{J}\cdot\text{K}^{-1}$ the Boltzmann constant, $\omega$ and $T$ being the (angular) ...

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Weinberg is right. The issue here is with the usual interpretation of the wavefunction as an amplitude density. This implies being able to localize the particle in an arbitrarily small region. However, it is not possible to localize photons (or any massless particles with spin, for that matter). The reason for this is the careful definition of what ...

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Wave function for $N$ particles is a scalar function of $3N$ spatial coordinates. Wave function for $N$ photons, if named so at all, should thus also be a scalar function of $3N$ spatial coordinates. EM field, on the other hand, is a vector function of 3 spatial coordinates. Any chosen wave function is non-unique; there are many different wave functions, ...

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I believe Weinberg is trying to make a distinction between two viewpoints of quantum particles, one historical (although we still use it when thinking about and teaching non-relativistic quantum mechanics) and one modern. In non-relativistic quantum mechanics, we typically start by assuming the existence of "particles" familiar to us from classical ...

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The integral and differential versions are equivalent, so it sounds like your text simply doesn't know how to use the differential version in as general a way as your text knows how to use the integral version. For instance, you do not need to have partial derivatives to define the divergence and/or curl, but if you have the partial derivatives, then there ...

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Dirac delta is the most common distribution in physics. It is defined by an equation: $\int \limits_{-\infty}^\infty \delta(x) f(x) = f(0)$ One could say that this could be "intuitively: a function, which is $\infty$ at $0$, and $0$ everywhere else, but this makes no sense, since such integral would be $0$ (not $f(0)$), as integral over the set of zero ...

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Since you speak about point charge, let us check the Gauss law (to start with something well known)? $\nabla \cdot E = \frac{\rho}{\epsilon_0}$ We know that divergence has something to do with flow across some closed surface (sphere surface is the best) around the $\rho$ - the same in integral form is: $\oint\limits_{\delta\Omega} E \cdot dS = ... -2 using V.E=p/e0 in integral form that is gauss law you get your first answer E going to zero as r go to infinity. 4 You are correct that the vanishing of the field does not follow from Lorentz' force law and Maxwell's equations alone. An additional physical argument is needed: If you didn't have$E\to 0$as$r\to\infty$, you would have non-vanishing force$F = qE \neq 0$at infinity. That is physically non-sensical because it would mean that a charge influences charges ... 18 Maxwellian electrodynamics fails when quantum mechanical phenomena are involved, in the same way that Newtonian mechanics needs to be replaced in that regime by quantum mechanics. Maxwell's equations don't really "fail", as there is still an equivalent version in QM, it's just the mechanics itself that changes. Let me elaborate on that one for a bit. In ... 2 Anna V is wrong when she says Maxwell's equations are inconsistent with black body radiation. In quantum mechanics, even if you ignore radiation there is a charge density which you can calculate (in principle) from Schroedinger's Equation. In a warm body, this charge density is fluctuating by random thermal motion. If you track the time evolution of the ... 3 Classical electromagnetic waves emerge from the underlying quantum electrodynamic description in a smooth and consistent manner. The quantum framework means that the classical waves are built out of photons, and the only time one has to worry about more detail than what Maxwell's equations provide is at the level of particle physics and wherever quantization ... 0 Whenever electric field or polarization changes in time, the displacement current is present. If electric current in wire that makes up the inductor changes in time, electric field will change as well and there will be displacement current. 0 If you move a magnet, you create an electrical field around the magnet perpendicular to its direction of motion. As the magnet accelerates, the electrical field evolves, and generates a magnetic field. I don't know the specifics, but I'm guessing that if you spin the magnet, this magnetic field will act counter to the spin of the magnet, and slow it down. 0 It is easy to show that the differential and integral forms of Maxwell's equations are equivalent using Gauss's and Stokes's theorems. Correct, they are equivalent (assume no GR, and no QM) in the sense that if the integral versions hold for any surface/loop then the differential versions hold for any point, and if the differential versions hold for every ... 1 Did physicists immediately realize Newtonian mechanics was incorrect after special relativity was published? Of course not. Physicists did not immediately realize that Einstein's description of electromagnetism (the second part of his 1905 paper on special relativity) was correct after special relativity was published. I can't think of a single ... 2 There are two questions.. 1. What is the concept of magnetic monopole here? 2. Why Maxwell's eqns remain unmodified? They are not elementary particles as anticipated by P.M.Dirac. But the concept comes from the non-zero divergence of magnetization field. In electrostatics when you have$\nabla\cdot \mathbf{P}\$ (polarization vector) not equal to zero, you ...

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The term has the dimension of the current density and may be non zero in absence of electric current. It completes the possible sources of the magnetic field.

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