EM plane wave incident upon infinite sheet of current

[aftershock]

Homework Statement

What are the electric and magnetic fields due to an infinite sheet of surface current [itex]\vec{K}[/itex] = [itex]\hat{j}[/itex]K_o in the plane x=0? The plane is electrically neutral and K_o is a constant. Plane electromagnetic waves are APPROACHING the plane x =0 from either side, incident normally with the magnetic field given by -[itex]\hat{k}[/itex]B_ocos(kx+ωt) for x>0 and [itex]\hat{k}[/itex]B_ocos(kx-ωt) for x<0

Find the electric field [itex]\vec{E}[/itex]([itex]\vec{r}[/itex],t) on either side of the plane x=0

Find the current (surface current) in the plane x=0?

Homework Equations

The Attempt at a Solution

There is no electric field caused by the plane since it is electrically neutral. Using ampere's law I determined the magnetic field to be 2μk[itex]\hat{z}[/itex] (negative on the positive side of the plane).

To find the electric field on both sides I'm assuming I just consider the electric field of the EM wave and use the E/c = B relation. This is easy enough so far but then it asks me to find the surface current in the plane. The surface current is given in the problem, so I'm assuming the waves affect the surface current to give me a new value? Is this correct, and if so how do I go about it?

 

[mark726]

Yes, you are correct in assuming that the waves will affect the surface current in the plane. To find the new value of the surface current, you can use the continuity equation, which states that the change in current density is equal to the negative of the time derivative of the charge density. In this case, the charge density is zero since the plane is electrically neutral. Therefore, the change in current density is equal to the time derivative of the surface current.

To find the time derivative of the surface current, you can use Faraday's law, which relates the time derivative of the magnetic field to the curl of the electric field. Since the magnetic field is known from the given information, you can solve for the curl of the electric field and then take its time derivative to find the change in current density. This will give you the new value of the surface current.

Alternatively, you can also use the boundary conditions for the electric and magnetic fields at the interface between the two media (in this case, the interface between the plane and the surrounding space) to find the change in current density and then solve for the new value of the surface current.