Electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum.
is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇2 is the Laplace operator.
In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, James Clerk Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force.
In Part VI of his 1864 paper titled Electromagnetic Theory of Light,[2] Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light.
He commented: The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.
[3]Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.
These are the general Maxwell's equations specialized to the case with charge and current both set to zero.
then the first term on the right in the identity vanishes and we obtain the wave equations:
The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.
is the Ricci curvature tensor and the semicolon indicates covariant differentiation.
The generalization of the Lorenz gauge condition in curved spacetime is assumed:
Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum.
for virtually any well-behaved function g of dimensionless argument φ, where ω is the angular frequency (in radians per second), and k = (kx, ky, kz) is the wave vector (in radians per meter).
Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic.
In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space.
As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.
In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:
The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:
There are also circularly polarized solutions in which the fields rotate about the normal vector.
Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids.
This is the basis for the Fourier transform method for the solution of differential equations.
The sinusoidal solution to the electromagnetic wave equation takes the form
The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.
A generic electromagnetic field with frequency ω can be written as a sum of solutions to these two equations.
However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ ⋅ E = ∇ ⋅ B = 0), and therefore require additional restrictions on the coefficients.
The multipole expansion circumvents this difficulty by expanding not E or B, but r ⋅ E or r ⋅ B into spherical harmonics.
These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇2 (r ⋅ F) = r ⋅ (∇2 F).
are the corresponding magnetic multipole fields, and aE(l, m) and aM(l, m) are the coefficients of the expansion.
where hl(1,2)(x) are the spherical Hankel functions, El(1,2) and Bl(1,2) are determined by boundary conditions, and
The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay.
