A uniform plane wave-I need serious help

[unstoppable]

Homework Statement

A uniform plane wave traveling in free space is incident obliquely at angle theta onto the surface of a prefectly conduction plane(a mirror). The resultant magnetic field vector is disovered to be zero at a distance s in front of the mirror.
(a) What is the lowest possible frequency omega of the wave if the incident wave polarization is TM?
(b) Why can the incident wave polarization not be TE?

Homework Equations

The fields in the dielectrics are
E=Eoexp(-jk1n*r)-Eoexp(+jk1 n*r)=-2jEosin(k1n*r)
H=Hexp(-jk1n*r)+Hoexp(jk1n*r)=2Hocos(k1 n*r)

E(r,t)=2*Eosin(k1n*r)sin(omega*t)
H(r,t)=2*Hocos(k1n*r)cos(omega*t)

NOTE:the n*r denotes the dot product of the position and the normal vector

The Attempt at a Solution

I made omega*t=theta and then I used Euler's equation e^j*theta, however I don't think what I'm doing is correct!

 

[Jhero]

Thank you for your question. I would like to provide you with a thorough and accurate answer.

(a) To determine the lowest possible frequency omega of the wave, we can use the relationship between the electric and magnetic fields in a plane wave, given by the equations you have provided. In order for the magnetic field to be zero at a distance s from the mirror, the cosine term in the equation for H must also be zero. This means that cos(k1n*r) = 0, which occurs when k1n*r = (2n+1)*pi/2, where n is an integer.

Substituting this into the equation for H, we get H = 0. This means that the lowest possible frequency omega for the TM-polarized wave is when the electric field is at its maximum value, which occurs when sin(k1n*r) = 1. Therefore, the lowest possible frequency is when k1n*r = n*pi, where n is an integer.

Using the relationship between the wave number k1 and the frequency omega, given by k1 = omega/c, we can rearrange the equation to get omega = n*pi*c/r. Therefore, the lowest possible frequency is given by omega = pi*c/r, where r is the distance from the incident wave to the mirror.

(b) The incident wave polarization cannot be TE because in this case, the electric field would be perpendicular to the plane of incidence, and the magnetic field would be parallel to the plane of incidence. This means that the electric field would not be able to induce a magnetic field in the perfectly conducting mirror, resulting in a zero magnetic field at a distance s from the mirror.

I hope this explanation helps to clarify the concept for you. If you have any further questions, please don't hesitate to ask.