33.13:
Standing Electromagnetic Waves
Recall the standing waves formed in a tight string.
Standing waves are the combination of two waves that move in the opposite direction but have the same amplitude and frequency.
Consider a perfect conducting sheet placed in the y–z plane. A linearly polarized electromagnetic wave strikes the sheet, while traveling in the negative x-direction.
The incident wave induces the current on the surface of the conductor, resulting in additional electric fields, producing a reflected wave with the same frequency and amplitude as the incident wave.
The superposition of these two waves generates the standing wave. Simplifying these expressions provide the points on the wave where the magnitudes of the electric field and the magnetic field are zero. These are called the nodes or nodal planes.
An antinode is a point where the amplitude goes from the maximum positive to the maximum negative displacement.
The nodes of the electric field coincide with the antinodes of the magnetic fields, and vice versa. Hence, they are 90-degrees out of phase at each point.
33.13:
Standing Electromagnetic Waves
Electromagnetic waves can be reflected; the surface of a conductor or a dielectric can act as a reflector. As electric and magnetic fields obey the superposition principle, so do electromagnetic waves. The superposition of an incident wave and a reflected electromagnetic wave produces a standing wave analogous to the standing waves created on a stretched string.
Suppose a sheet of a perfect conductor is placed in the yz-plane, and a linearly polarized electromagnetic wave traveling in the negative x-direction strikes it. Since the electric field cannot have a component parallel to a perfect conductor's surface, the electric field must be zero everywhere in the yz-plane. The electric field of the incident electromagnetic wave is not zero at all times in the yz-plane, but this incident wave induces oscillating currents on the conductor's surface, giving rise to an additional electric field. The net electric field, which is the vector sum of this field and the incident field, is zero everywhere inside and on the conductor's surface. The currents induced on the conductor's surface also produce a reflected wave that travels out from the plane. The superposition principle states that the total field at any point is the vector sum of the electric fields of the incident and reflected waves, similar to the magnetic field. The superposition of incident and reflected waves generate standing waves.
Simplifying these expressions provides the points on the wave where the electric field and magnetic field magnitudes are zero. These are called the nodes or nodal planes. Midway between any two adjacent nodal planes are the planes of maximum amplitude; these are the antinodal planes.
The total electric field is a sine function, and the total magnetic field is a cosine function. Therefore, the sinusoidal variations of the two fields are out of phase at each point. The electric field nodes coincide with the antinodes of the magnetic fields and vice versa. Hence, they are 90° out of phase at each point. This is in contrast to a wave traveling in one direction, for which the sinusoidal variations of the electric and magnetic fields are in phase at any particular point.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Section 32.5; pages 1069–1070.