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Show that for charge density ρ(z, t) and current density [0,0, J(z, t)] all Maxwell's equations are satisfied by the purely electric field 𝐄=[0,0, E(z, t)], where (∂E)/(∂z)=ρ/ ϵ0, (∂E)/(∂t)=-J / ϵ0 . Deduce that to any time independent solution ρ=f^'(z), J=0, E=(I / ϵ0) f(z) there corresponds a solution ρ=f^'(z) g(t), J=-f(z) g^'(t), E=(1 / ε0) f(z) g(t), where the function g is arbitrary. Verify that J=v ρ=v ρ0 e^-k|z-v t|, E=(ρ0)/(ϵ0 k) (|z-v t|)/(z-v t)(I-e^-k|z-v t|), where k(>0), v and

Show that for charge density ρ(z, t) and current density [0,0, J(z, t)] al…

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