ps6 problem2.

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
E1y (x ,t ) = E 0 sin(kx − ωt )
(1)
B1z (x ,t ) = B0 sin(kx − ωt )
(2)
and the other travelling in the negative x-direction described by the following equations
E2y (x ,t ) = E 0 sin(kx + ωt )
(3)
B2z (x ,t ) = −B0 sin(kx + ωt )

(4)

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.
2 points
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?
2.5 points
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.
2.5 points
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?
Justify your answers.
1.5 points
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.
Again describe its subsequent motion. Justify your answers.
1.5 points