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Suppose you have a configuration with two infinite rods parallel to the $z$ axis, each with uniform charge density $\lambda$. Suppose the rods move. In general, the electromagnetic field they generate satisfies Maxwell's equations. If the charge density on the rods doubles, what happens to $E$ and $B$ in general? I am inclined to think that $E$ doubles. Does $B$ double too?

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I do think everything will double, as Maxwell's equations are linear. Take the solution for the first case, which includes the source term (the two rods); multiply fields and charge density times two: you still satisfy Maxwell's equations.

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