EM wave reflection at conducting surface with boundary conditions

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Homework Statement

EM Waves propagating through vacuum towards the x-direction approaches a conducting surface at x = 0.

Part (a): Prove the following relations. Find the boundary conditions for $$E$$ and $$B$$ if charge density $$σ$$ and current density $$s$$ exists on the surface.

Part (b): Find $$B$$ in terms of $$E_0$$

Part (c): Show the reflected wave has the following form:

Part (d) Find magnetic field of reflected wave and $$s$$

Homework Equations

The Attempt at a Solution

Part(a)

Part(b)

Part(c)

This is the part that disturbs me. I assumed the free current s = 0 and σ = 0. Usually $$k$$ would be perpendicular to the surface, making derivations easier.

I actually got perpendicular reflected E-field = 0, which is wrong..
But, I got parallel reflected E-field = - incoming parallel E-field, which is right

I'm confused - If it's a perfect conductor it should have no free charge or currents, but the question says it does. Then again, in order to derive the reflected e-field, we need to assume they are 0.

 

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Edit: Question here

 

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