Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*


version of 1 September 1996

Clerk Maxwell codified the laws of electricity and magnetism in four partial differential equations, which in in rationalized mks units read:

   Curl H = partial D / partial t + J       (extension of Ampère's law)
   Curl E = - partial B  / partial t       (Faraday's law of induction)
   Div B = 0      (Gauss's law for magnetism, in the absence of monopoles)
   Div D = rho       ( Gauss-Coulomb law)

Here, H is the magnetic force and B = Mu H is the magnetic induction; E is the electric intensity and D = Epsilon E is the electric induction; J is the current density and rho is the charge density. In free space, rho = 0 and J = 0, while the value of Mu Epsilon turns out to equal 1/c2, where c is the speed of light.

Let us calculate the second time derivative of B:
   B_{tt} = -(1/ Mu Epsilon) Curl Curl B.
There happens to be a vector identity for the curl of a curl:
   Curl( Curl(v)) = -Grad^2 v + Grad(Div v),
where the Laplacian of a vector v = (v1, v2,v3) is defined by its action on each component:
   (- Grad^2 v_1, - Grad^2 v_2, - Grad^2 v_3)
Because of the absence of magnetic monopoles, the vector identity as applied to B becomes
   Curl( Curl(B)) = - Grad^2 B,
which means that
   B_{tt} = Grad^2 B/(Epsilon Mu)..

Clerk Maxwell recognized that this equation was the wave equation in three dimensions, for each component of the magnetic field B. A very similar analysis shows that each component of E satisfies the same wave equation. Of course, Clerk Maxwell's equations imply that the electric and magnetic fields in a wave are coupled; if we assume a traveling wave, the fields E and B turn out to oscillate exactly out of phase.

After deducing the possibility of these electromagnetic waves, Clerk Maxwell was interested in studying the properties of these waves to see whether they could be experimentally verified. Imagine his excitement in 1854(?) when he calculated the characteristic speed of the waves with the best known values of Mu and Epsilon for a vacuum and discovered that it was quite close to the observed value of the speed of light. This was surely one of the crowning moments in the history of physics. In Clerk Maxwell's own words,


(W.D. Niven, ed., The Scientific Papers of J. Clerk Maxwell, Cambridge, 1890, vol. I, p. 500).

Compare with the derivation of the wave equation for the vibrating string

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