Get Electromagnetic Wave Equation essential facts below. View Videos or join the Electromagnetic Wave Equation discussion. Add Electromagnetic Wave Equation to your PopFlock.com topic list for future reference or share this resource on social media.
In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, 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, 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.
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.
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:
These are the general Maxwell's equations specialized to the case with charge and current both set to zero.
Taking the curl of the curl equations gives:
Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.
Solutions to the homogeneous electromagnetic wave equation
The general solution to the electromagnetic wave equation is a linear superposition of waves of the form
for virtually any well-behaved function g of dimensionless argument ?, where ? 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:
where k is the wavenumber and ? is the wavelength. The variable c can only be used in this equation when the electromagnetic wave is in a vacuum.
Monochromatic, sinusoidal steady-state
The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:
Then planar traveling wave solutions of the wave equations are
where r = (x, y, z) is the position vector (in meters).
These solutions represent planar waves traveling in the direction of the normal vector n. If we define the z direction as the direction of n. and the x direction as the direction of E, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation
Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.
This solution is the linearly polarized solution of the wave equations. 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
Assuming monochromatic fields varying in time as , if one uses Maxwell's Equations to eliminate B, the electromagnetic wave equation reduces to the Helmholtz Equation for E:
with k = ω/c as given above. Alternatively, one can eliminate E in favor of B to obtain:
A generic electromagnetic field with frequency ? can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. 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 · (?2F). The resulting expressions for a generic electromagnetic field are:
where and are the electric multipole fields of order (l, m), and and are the corresponding magnetic multipole fields, and aE(l, m) and aM(l, m) are the coefficients of the expansion. The multipole fields are given by
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. In these applications, one is often interested in the power radiated in the far-field. In this regions, the E and B fields asymptote to
The angular distribution of the time-averaged radiated power is then given by
Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)