Find the positions of all nodal and anti-nodal planes of

Question

In this problem we study the formation of standing electromagnetic waves using two perfectly reflecting conducting (infinite) planes, as shown in the picture above. Assume vacuum between the two conducting planes and that the distance between them is L. Consider two sinusoidal plane electromagnetic waves, one travelling in the positive x-direction described by the following equations

E_1y (x ,t ) = E₀ sin(kx - ωt )

B_1z (x ,t ) = B₀ sin(kx - ωt )

and the other travelling in the negative x-direction described by the following equations

E_2y (x ,t ) = E₀sin(kx + ωt )

B_2z (x ,t ) = -B₀sin(kx + ωt ) 

where k = 2π / λ is the wave-vector and ω the angular frequency of the waves in vacuum.

For the following questions express your answers in terms of given quantities of the problem and constants of nature.

A. Find the total electric field E y (x , t ) and the total magnetic field Bz (x , t ) in the region between the two conducting planes.

B. Find the positions of all nodal and anti-nodal planes of the electric field. The concept of nodal and anti-nodal plane is similar to the concept of node and anti-node respectively as used e.g. for a one-dimensional standing wave on a string. Which factor/factors determines/determine the total number of nodal or anti-nodal planes of the electric field formed between the two conducting planes?

C. Find the positions of all nodal and anti-nodal planes of the magnetic field. Is there any correspondence between nodal and anti-nodal planes of the electric and magnetic field? Justify your answer.

D. What happens to the total magnetic field component at the times when the total electric field component is zero everywhere? Justify your answer. What can be said about the phase relationship between the total magnetic field and the total electric field in a standing electromagnetic wave? How is that different from the phase relationship between the electric and magnetic field component in a travelling electromagnetic wave?

E. Suppose that we place a very small positive charge q on the x-axis at a distance d = 50.25 λ ( 0 < d < L ) from one of the conducting planes without initial kinetic energy. Briefly describe its subsequent motion under the influence of the standing electromagnetic wave. Suppose that we place the same charge again on the x-axis but at a distance d = 150 λ ( 0 < d < L ) from one of the conducting planes without initial kinetic energy.