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## Homework Statement

The planes x= ±a are charged to uniform surface density ±σ respectively.

Find the charge and current densities in a frame moving with velocity (0,v,0) - [done]

Find also the electromagnetic field in the moving frame by solving the problem in the moving frame

## Homework Equations

Note that my lecturer reconfigures the problem (wlog) to a more familiar scenario from lectures/other problems, i.e. (with y-axis pointing up) the planes are at y=±a, and there is a frame [tex]\,\Sigma\,'[/tex] moving with velocity [tex]\,v\hat{x}[/tex].

We obtain

[tex]\rho = \sigma \delta(y-a) - \sigma \delta(y+a) [/tex]

where [tex]\rho[/tex] is charge density.

[tex] \vec{j} = \rho \vec{v} = \sigma v \left(\delta(y-a) - \delta(y+a)\right) \hat{x} [/tex]

where [tex]\vec{j}[/tex] is current density.

Also required is [tex]\nabla.\vec{D} = \rho \quad \mbox{where} \;\; \vec{D} = \epsilon_0 \vec{E}[/tex]

## The Attempt at a Solution

Since [tex](c \rho , \vec{j})[/tex] is a 4-vector, using the Lorentz transformation matrix, we can derive

[tex]\rho\,' = \frac{\rho}{\gamma}[/tex]

(usual defn of gamma)

[tex]j_1\,' = 0[/tex]

Hence

[tex]\rho\,' = \sigma\,'\left(\delta(y' - a) - \delta(y' + a)\right)[/tex]

[tex]\sigma\,' = \frac{\sigma}{\gamma}[/tex]

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For the next bit, we start with one of Maxwell's equations in the [tex]\,\Sigma\,'[/tex] frame: [tex]\nabla'.\vec{E\,'} = \frac{\rho\,'}{\epsilon_0}[/tex]

Now the solutions say we should integrate over two discs containing the planes y = ±a , and then use the divergence theorem to find

[tex]\vec{E\,'} = -\frac{\sigma\,'}{\epsilon_0} \hat{y} \quad \mbox{where} \;\; -a < y < a \; \mbox{, else 0}[/tex]

I can handle integration over a sphere/cylinder, but am not sure what to do with a disc, and also what happens on the RHS (which is a sum of delta fns, from derived equation for [tex]\rho\,'[/tex]).

Thanks.