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In empty space there is an infinite cylinder, with its axis parallel to z axis and radius a, filled with an eletric field of equation

[tex]\vec{E}(t) = E_0 e^{\beta t} \hat{z}[/tex]

Now I put a rectangular wire on the plane yz out of the cylinder of side l and b (l lies on the y axis) and the question is: which is the current on the wire?

I try to attack the problem in the straightforward way. So I try to solve Maxwell's fourth equation

[tex]\vec{\nabla} \times \vec{B} = \frac{1}{c} \partial_t \vec{E}[/tex]

and initially I thought that I had to compute the flux through the wire of the varying magnetic field I get.....but I realize that this is impossible because this equation is only valid inside the cylinder, where I have a varying electric field, and not outside where there is only empty space!!!

Could anyone help...please?!?