EM: Uniform plane wave incident on normal boundry

[Alxb577]

Homework Statement

Given a uniform plane wave in air as:
E_i=40cos(wt- Bz)a_x +30sin(wt- Bz)a_y
(a) Find H_i
(b) If the wave encounters a perfectly conducting plate normal to the z axis at z = 0, find
the reflected wave E_r and H_r.
(c) What are the total E and H fields for z < 0?

Homework Equations

[1] direction of H is the cross product of the direction of propagation with the direction of the E wave.

The Attempt at a Solution

[/B]
This problem should be really easy for me but I'm not getting the boundary conditions I feel like I should have at z=0.

The E wave is propagating in the + z direction with +x and +y components.

Because of equation [1] the corresponding H wave will have corresponding +y and -x components. {Note I don't really care about the magnitude for now}.

Because the wave hits a perfect conducting boundary, the reflection coefficient is -1.

Now the reflected E wave would be propagating in the - z direction with -x and -y components.

From equation [1] the H wave will have corresponding +y and -x components.

Since none of the waves are transmitted and both the E and H are tangential to the boundary at z=0, The sum of the incident and reflective wave of both the E and H wave must be zero, correct? I don't believe there is a surface current or anything like that either.

If you look at the components of the E incident +x , +y and reflected -x , -y the boundary conditions are satisfied at z = 0; Now looking at the H incident +y , -x and reflected +y , -x they are not satisfied.

If there is something dumb I'm doing please let me know, I've spent way too long on this problem.

 

[rude man]

(b)

Homework Statement

Given a uniform plane wave in air as:
E_i=40cos(wt- Bz)a_x +30sin(wt- Bz)a_y
(a) Find H_i
(b) If the wave encounters a perfectly conducting plate normal to the z axis at z = 0, find
the reflected wave E_r and H_r.
(c) What are the total E and H fields for z < 0?

Homework Equations

[1] direction of H is the cross product of the direction of propagation with the direction of the E wave.

No, the direction of propagation is the cross product of the E and H fields: P = E x H. Still, you got the right polarizations for the incident E and H fields.

Since none of the waves are transmitted and both the E and H are tangential to the boundary at z=0, The sum of the incident and reflective wave of both the E and H wave must be zero, correct? I don't believe there is a surface current or anything like that either.

I assume you mean at the boundary. Certainly the sum of incident and reflected waves is not everywhere zero outside the conductor!
The boundary condition is for E_tangential = 0 but not for H_tangential. Once you have the reflected E components the H components are solved for by Maxwell's del x E = - ∂B/∂t. When you do that you will find that the H components for each incident wave double at the interface, rather than go to zero.

There is actually infinite current density AT the interface; penetration though is zero. One of those infinity times zero deals!

(a) Use the Poynting vector to determine the spatial direction of H_i. Temporally you know there is what phase shift between the two E vectors? And for each E of the two vectors what is the phase shift between it and its companion H vector in a non-conducting medium?