A cylinder of electric field varying in time

[alle.fabbri]

Hi, I have this problem:
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
$$\vec{E}(t) = E_0 e^{\beta t} \hat{z}$$
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
$$\vec{\nabla} \times \vec{B} = \frac{1}{c} \partial_t \vec{E}$$
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?!?

 

[pam]

Use Ampere's law (with the displacement current) to get B(t) outside the cylinder.

 

[alle.fabbri]

What do you mean with Amperè's Law?? Maybe the fourth Maxwell's Equation?? This one?
$$\vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \frac{1}{c^2} \frac{\partial \vec{E}}{\partial t}$$
And if I do?? The displacement current is zero outside the cylinder...

 

[pam]

Using Stoke's theorem, you can derive Ampere's law:
$$\oint{\bf dr}\cdot{\bf B}=\frac{1}{c^2}\int{\bf dS}\cdot\partial_t{\bf E}$$.

 

[alle.fabbri]

Ok...I get what you mean. So I choose a cylinder of radius r>a and height z, evaluate the flux of $$\frac{\partial \vec{E}}{\partial t}$$ and thanks to Ampere's Law this is equal to the path integral of B over a circle of radius r in the at height z parallel to xy plane. May I say that B is always tangential so the path integral is easy to evaluate?? If yes why?? I thought in analogy with the magnetic field generated by a wire...

Thank you

Ciao!

 

[pam]

It is just like a wire, but it doesn't matter.
For the loop you have only B tangential enters the flux.