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Maxwell's equation can be given in the form $$\text dF = 0$$ $$\text d\star F + J = 0$$ where $F$ is a 2-form and $J$ an $n-1$-form (a current density) which in principle can be generalised to any manifold (for physical reasons one might want to consider pseudo-Riemannian manifolds with signature $(+,-,\cdots,-)$). In the four dimensional theory one usually ...


7

You can generalize Maxwell's equations to an arbitrary number of dimensions by using either the tensor or differential form version, as the vector formalism does not help too much (For instance, in two dimensions, the magnetic field is a (pseudo) scalar field, not a vector field). The equations are then : $\partial_\alpha F^{\alpha\beta} = \mu_0 J^\beta$ ...


1

I think your limiting procedure should work. For the homogeneous Maxwell equations, when we restrict to a surface of constant $x$, the pull back of the field strength looks something like $$\iota^* F = E^\perp_i dt \wedge dx^i + B_x dy \wedge dz,$$ where $E^\perp$ is the component of $E$ in the $yz$-plane. Now your timelike pillbox integral will have ...


3

You need to watch what you mean by the ambiguous term "derive", which can mean either "was derived historically" (i.e. was motivated by or is a derivative of, in the non-mathematical sense) or "is derived logically/mathematically". Historically, I think you are correct that $\boldsymbol{\nabla}\cdot ...


0

Well I dont know if we can prove it but there is a much more elegant way of formulating EM which may be helpful here. As you may know there are two potentials on EM: the scalar potential $\phi$ and the vector potential $\vec{A}$, from which $\vec{E}(t,x)$ and $\vec{B}(t,x)$ are derived. From this two objects and following symmetry considerations you can ...


3

Maxwell derived his equations from 1) charge conservation law; 2) Coulomb's law; 3) Bio--Savart--Laplace law; 4) Faraday's law of induction. The equation $\boldsymbol{\nabla}\cdot \textbf{E}(\textbf{r})=\frac{\rho(\textbf{r})}{\epsilon_0}$ was indeed derived from Coulomb's law and in its differential form is written using Gauss--Ostrogradskiy theorem. ...


3

No we cannot prove it; Maxwell postulated that it would hold dynamically because it made the most sense for it to do so as he pondered the displacement current problem. As you likely know, Maxwell pondered the inconsistency between Ampère's law for magnetostatics and the charge continuity equation. Ampère's law for magnetostatics reads $\nabla\times ...


1

The derivation assumes the wire is a perfect conductor, and also that it is negligibly thin. If it had some resistivity, then you're right, there would be an electric field in the wire, but even in that case the electric flux $\int \vec{E}\cdot\text{d}\vec{a}$ would be negligible, and so would its time derivative.


2

There is never actually an electric field in a conductor in the electrostatic sense. An E field is always generated perpendicular to a charged surface (the wire). For any wire carrying current, the electric field tends to radiate outward from the wire. The magnetic field will be circulating around the wire such that the Poynting vector, $ \vec S = \vec E ...


0

For the flat Amperian Loop, The current flowing through the wire that pierces the surface of the loop) is I. However, there is no field piercing the surface of the loop. Now, why is there is no field piercing the loop:- Of course the field between the plates of the capacitor no way pierces the surface of the loop "Isn't there a field inside the wire, ...


3

It's not altogether clear what you're asking, but I'm guessing you're doubting the "standard" set: $$\nabla\cdot\vec{D} = \rho$$ $$\nabla\cdot\vec{B} = 0$$ $$\nabla \times \vec{E} = -\partial_t\vec{B}$$ $$\nabla \times \vec{H} = \vec{J} + \partial_t\vec{D}$$ These are the set you use in linear, isotropic inhomogeneous mediums. So, for example, from the ...



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