33.4:
Electromagnetic Wave Equation
Consider the differential form of the four Maxwell's equations in a vacuum, which mutually relate the electric and magnetic fields.
Separate equations for the electric and magnetic fields can be found by applying the curl to both sides of the third and fourth equations, using geometrical identities, the first and second equations, the rule of partial derivatives, and the third and fourth equations.
Simplification reveals that the electric and magnetic fields follow the same equation. It implies that each Cartesian component satisfies the wave equation. So, the electromagnetic fields are three-dimensional waves.
They propagate in a vacuum with a speed determined by natural constants.
Assume the wave traverses in the z-direction. The general solutions for the fields are written.
To these solutions, the first and second Maxwell's equations are applied. Simplification reveals that the fields' z components are zero. So, electromagnetic waves are transverse waves.
Maxwell's third equation applied to the solutions gives relations between their x and y components, which can be clubbed in vector form. So, the traveling electric and magnetic fields are mutually perpendicular.
33.4:
Electromagnetic Wave Equation
Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations: What happens to the electric and magnetic fields when there are no source charges?
Consider the differential forms of the four Maxwell's equations in a vacuum. A natural symmetry is observed in the first-order partial differential equations except for the natural constants.
Mathematical explorations revealed that each electric and magnetic field component follows the same wave equation. The speed is determined by a combination of the natural constants: the permittivity of the vacuum and the permeability of the vacuum. Substituting their numbers, Maxwell discovered that this speed is very close to the experimental value of the speed of light.
The result is startling given the independence of the observations: one from the theory of electrodynamics, which emerged from studying charges and currents in the laboratory, and the other an experimental determination of light's speed.
Connecting the dots, Maxwell hypothesized that light is nothing but electromagnetic waves. The hypothesis was later experimentally verified. This course of history is an essential lesson on the importance of mathematics in explaining nature. Moreover, electric and magnetic fields are not purely mathematical constructs; they can travel without any material present, which gives them a physical meaning.
The general solutions to the wave equations are known. Since electromagnetic fields follow the principle of superposition, more general solutions of the wave equations can be written as linear superpositions of these solutions.
Maxwell's equations apply additional constraints on the general solutions. They reveal other remarkable facts: the electric and magnetic fields are in phase with each other, are mutually perpendicular, and are both perpendicular to the wave's propagation direction. Thus, electromagnetic waves are transverse waves.
Suggested Reading
- Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Upper Saddle River, New Jersey 07458; section 9.2; pages 375–379.
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 32.2; page 1058.