Linked Questions

70
votes
7answers
9k views

Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and $\vec{B}=(B_x,B_y,B_z)$...
10
votes
1answer
2k views

Recovering all of Maxwell's equations from the variational principle

Whether you can get the first couple of Maxwell equations from a variational principle? In the second volume of the Landau theoretical physics said that it is impossible.
5
votes
2answers
396 views

Comparison of covariant form of Maxwell equations with Einstein's GR

We know, the the vector form of Maxwell equations \begin{align} \vec\nabla\cdot\vec{E} &= 4\pi\rho \label{Diff I}\\ \vec\nabla\times\vec{B} &= \dfrac{4\pi}{c} \vec{j}+\dfrac{1}{c}\dfrac{\...
3
votes
2answers
282 views

Why is this “the” functional of Laplace's equation?

Halfway through a discussion of the finite element method for solutions to Laplace's equation, Sadiku (2000) drops in a formulation of the work functional for an electric field: Algebraically and ...
10
votes
1answer
356 views

How much of Maxwell's equations is recoverable from the zero divergence of the stress-energy tensor?

As a motivating example, consider the static electromagnetic field defined by $\textbf{E}=(\text{const})x\hat{\textbf{y}}$, $\textbf{B}=0$. The stress-energy tensor for this field is $T=\operatorname{...
0
votes
1answer
883 views

Deriving Maxwell Equations in their covariant form

Mawell Equations, in a particular unit system, are: \begin{eqnarray} \nabla \cdot \vec{E} &=& \rho &(1)\\ \nabla \times \vec{B} &=& \frac{\partial \vec{E}}{\partial t} + \vec{J}&...
0
votes
1answer
525 views

About Homogeneous Maxwell equation from EM Lagrangian

I have studied some of the relevant Q&A here. Everything is quite satisfactory. But is there any way to prove homogeneous part of 4 Maxwell equation from Lagrangian formalism, i.e constructing the ...
1
vote
2answers
187 views

Does Gauss-Faraday law play any role in the quantization of the electromagnetic field?

The Gauss-Faraday law, in the covariant form, reads $ \epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta,\alpha} = 0, $ while the vacuum field equation is $ \partial_\mu F^{\mu\nu} = 0. $ When it ...
2
votes
1answer
236 views

What is the mass in $\vec{\nabla}\cdot \vec{B}_g=\text{something}$ called if it could be non-zero?

Maxwell's equation for gravity has $$\vec{\nabla}\cdot \vec{B}_g~=~0,$$ see Gravitoelectromagnetism in analogy with the electrodynamics. What is the mass called that needs to make these equations ...
2
votes
1answer
191 views

Homogenuous Maxwell Equations in the Language of Differential Forms

I understand that if I define electric field to be $E=E_i dx^i$, magnetic field to be $B=B_1 dx^2 \wedge dx^3 + B_2 dx^3 \wedge dx^1 + B_3 dx^1 \wedge dx^2 $, and field strength to be $F= dx^0 \wedge ...