Potential formulation with magnetic charge

[accountkiller]

Homework Statement

Develop the potential formulation for electrodynamics with magnetic charge. Use the Lorenz Gauge. You will need two scalar potentials (V_e and V_m) and two vector potentials (A_e and A_m). Write Maxwell's equations in terms of the potentials, write the electric and magnetic fields in terms of the potentials, and write the potentials in terms of the charge and current densities.

Homework Equations

Maxwell's equations for magnetic charge:
∇*E=ρ_e/ε₀
∇*B=μ₀ρ_m
∇xE=-μ₀J_m - dB/dt
∇xB=μ₀J_e + μ₀ε₀ dE/dt

Lorenz gauge:
∇*A=-μ₀ε₀ dV/dt

Fields in terms of potentials for electric charge:
E=-∇V - dA/dt
B=∇xA

The Attempt at a Solution

I needed to modify Gauss's law of magnetism and Faraday's law to include magnetic charge & current, which is done above in the equations. Next, I'd need to define B and E to satisfy Maxwell's equations. Well, how do I start defining E & B? I can't use the equations given for E and B above, can I? Those are assuming only electric charge right?

Most of my classmates are having trouble with this simply because we aren't sure how to start; the idea of a magnetic charge isn't the most intuitive thing for us to wrap our heads around. Any tips on how to begin would be greatly appreciated.

 

[vanhees71]

The key is the Helmholtz theorem of vector calculus, according to which any vector field $\vec{V}$ can be decomposed into a potential field (gradient of a scalar field) and a solenoidal field (curl of another vector field), each determined (up to a "gauge transformation") by it's surces and its vortices:
$$\vec{V}=-\vec{\nabla} \Phi + \vec{\nabla} \times \vec{A}.$$
Taking the divergence you get
$$\vec{\nabla} \cdot \vec{V}=-\Delta \Phi \stackrel{!}{=} \rho$$
and
$$\vec{\nabla} \times \vec{V} = \vec{\nabla} \cdot (\vec{\nabla} \cdot \vec{A} ) -\Delta \vec{A} \stackrel{!}{=}\vec{W}.$$
The sources $\rho$ and $\vec{W}$ are supposed to be given.

The starting point for the analysis of the gauge invariance is that $\vec{A}$ is only determined up to a scalar field, i.e., for any scalar field $\chi$
$$\vec{A}'=\vec{A}-\vec{\nabla} \chi$$
gives the same solenoidal part as $\vec{A}$.

At presence magnetic monopoles (i.e., magnetic charge and currend distributions in addition to the usual electric charge and current distributions) you make the corresponding decomposition for both $\vec{E}$ and $\vec{B}$ and analyse the equations following from the given Maxwell equations, including the additional magnetic sources.

 

[accountkiller]

Thank you so much for your reply! One of my classmates suggested using the Helmholtz Theorem but I wasn't sure how it explicitly worked out (as the appendix on it in my textbook is quite short). You make the steps much clearer; I think I can get it now. I really, really appreciate your help!