(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If the current density is time independent and divergence free, show that the Maxwell Equations separate into independent equations for [tex]\vec{E}[/tex] and [tex]\vec{B}[/tex].

2. Relevant equations

Maxwell's equations

3. The attempt at a solution

The only Maxwell equation with [tex]\vec{j}[/tex] in it is the Maxwell-Ampere law, so that seems like the right place to start. By taking the partial derivative with respect to time of this equation and using the fact [tex]\vec{j}[/tex] is time independent, Faraday's Law and Gauss' Law for [tex]\vec{B}[/tex] I can get a wave equation for [tex]\vec{B}[/tex].

What is confusing me is how to use the fact [tex]\vec{j}[/tex] is divergence free. If I take the divergence of the Maxwell-Ampere equation I get:

∇⋅∇x[tex]\vec{B}[/tex] = εµ (∂∇⋅[tex]\vec{E}[/tex]/∂t) + µ ∇[tex]\vec{j}[/tex]

The LHS = 0 (vector identity), ∇[tex]\vec{j}[/tex] = 0 as given, and then using Gauss' Law for [tex]\vec{E}[/tex] I simply get:

(∂ρ)/(∂t) = 0

But this isn't surprising as the equation of charge conservation would have given me this anyway. How do I get the equation for [tex]\vec{E}[/tex]? Thanks.

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# Homework Help: Maxwell Equations when current density is time independent and divergence free

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