# EM: Uniform plane wave incident on normal boundry

## 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

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

## The Attempt at a Solution

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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  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  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
Homework Helper
Gold Member
(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

 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 Etangential = 0 but not for Htangential. 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 Hi. 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?

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