electromagnetic potential

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< Classical electrodynamics(Redirected from Classical electrodynamics/electromagnetic potentials)

$\boldsymbol{\nabla} \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$ ,	(1)
$\boldsymbol{\nabla} \cdot \mathbf{B} = 0$ ,	(2)
$\boldsymbol{\nabla} \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$ ,	(3)
$\boldsymbol{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}$ .	(4)

Let us assume that the fields vanish sufficiently rapidly at infinity. Then, $\nabla \cdot \mathbf{B} = 0$ combined with Helmholtz's theorem guarantees that we can find some $\mathbf{A}\,$ such that

$\mathbf{B} = \boldsymbol{\nabla}\times\mathbf{A}\,$ .

Eqn. (3) then becomes

$\boldsymbol{\nabla} \times \left(\mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}\right) = 0\,$ ,

meaning that we can some $\Phi\,$ such that

$\mathbf{E} = -\boldsymbol{\nabla} \Phi - \frac{\partial\mathbf{A}}{\partial t}\,$ .

In terms of the potentials,

$\nabla^2 \Phi + \frac{\partial}{\partial t} (\boldsymbol{\nabla}\cdot\mathbf{A}) = -\frac{\rho}{\epsilon_0}\,$ ,

along with

$\boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{A}) - \nabla^2 \mathbf{A} = \mu_0 \mathbf{J} +\mu_0 \epsilon_0 \boldsymbol{\nabla} \frac{\partial \Phi}{\partial t} + \mu_0 \epsilon_0\frac{\partial^2\mathbf{A}}{\partial t^2}\,$ ,

which can be rewritten as

$\boldsymbol{\nabla} \left( \mu_0 \epsilon_0 \frac{\partial \Phi}{\partial t} + \boldsymbol{\nabla} \cdot \mathbf{A}\right) + \left( \mu_0 \epsilon_0\frac{\partial^2\mathbf{A}}{\partial t^2} - \nabla^2 \mathbf{A} \right) = \mu_0 \mathbf{J}\,$ .

back to Maxwell's equations

on to gauge transformation

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Classical_electrodynamics/electromagnetic_potential"

electromagnetic potential

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