Electromagnetic Radiation - Derivation From Electromagnetic Theory

Derivation From Electromagnetic Theory

Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. Inspection of Maxwell's equations without sources (charges or currents) results in, along with the possibility of nothing happening, nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwell's equations in free space:

where

is a vector differential operator (see Del).

One solution,

is trivial.

For a more useful solution, we utilize vector identities, which work for any vector, as follows:

To see how we can use this, take the curl of equation (2):

Evaluating the left hand side:

where we simplified the above by using equation (1).

Evaluate the right hand side:

Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely

Applying a similar pattern results in similar differential equation for the magnetic field:

These differential equations are equivalent to the wave equation:

where

c₀ is the speed of the wave in free space and

f describes a displacement

Or more simply:

where is d'Alembertian:

Notice that, in the case of the electric and magnetic fields, the speed is:

This is the speed of light in vacuum. Maxwell's equations have unified the vacuum permittivity, the vacuum permeability, and the speed of light itself, c₀. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism.

But these are only two equations and we started with four, so there is still more information pertaining to these waves hidden within Maxwell's equations. Let's consider a generic vector wave for the electric field.

Here, is the constant amplitude, is any second differentiable function, is a unit vector in the direction of propagation, and is a position vector. We observe that is a generic solution to the wave equation. In other words

for a generic wave traveling in the direction.

This form will satisfy the wave equation, but will it satisfy all of Maxwell's equations, and with what corresponding magnetic field?

The first of Maxwell's equations implies that electric field is orthogonal to the direction the wave propagates.

The second of Maxwell's equations yields the magnetic field. The remaining equations will be satisfied by this choice of .

Not only are the electric and magnetic field waves in the far-field traveling at the speed of light, but they always have a special restricted orientation and proportional magnitudes, which can be seen immediately from the Poynting vector. The electric field, magnetic field, and direction of wave propagation are all orthogonal, and the wave propagates in the same direction as . Also, E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are always in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spacial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time, resulting in the same phase shift for both fields in each mathematical operation.

From the viewpoint of an electromagnetic wave traveling forward, the electric field might be oscillating up and down, while the magnetic field oscillates right and left; but this picture can be rotated with the electric field oscillating right and left and the magnetic field oscillating down and up. This is a different solution that is traveling in the same direction. This arbitrariness in the orientation with respect to propagation direction is known as polarization. On a quantum level, it is described as photon polarization. The direction of the polarization is defined as the direction of the electric field.

More general forms of the second-order wave equations given above are available, allowing for both non-vacuum propagation media and sources. A great many competing derivations exist, all with varying levels of approximation and intended applications. One very general example is a form of the electric field equation, which was factorized into a pair of explicitly directional wave equations, and then efficiently reduced into a single uni-directional wave equation by means of a simple slow-evolution approximation.