Basic Electromagnetic Theory: Field Theory Foundations and Structure by James Babington

Author:James Babington [James Babington]
Language: eng
Format: epub, mobi
ISBN: 978-1-942270-74-4
Publisher: Mercury Learning and Information
Published: 2017-02-09T05:00:00+00:00

5.2.1 Solutions in Spherical Coordinates

Turning now to spherical coordinates one can try to build up a general solution using spherical waves, the equivalent of plane waves in the previous discussion. There is a subtlety now that since in spherical coordinates the metric is non-trivial, we would need to ensure that we are using the correct form of the Laplacian operator to act of the electric vector field. The Laplacian operator given in Equation (2.66) is for a scalar field. We will assume for now that a scalar field will be sufficient but it is worth checking under what conditions this is valid or is a well defined approximation.

The Helmholtz operator in spherical coordinates acting on a scalar field ε(x) is given by

(5.31)

Assuming again the separation of variables method so that the solution in spherical coordinates take the form one can start the reduction off again (we have included a factor of k in the radial solution so that its argument is dimensionless). The simplest term in this Laplacian is the partial derivative and so this is the place to start with

(5.32)

In spherical coordinates, any solution must be invariant under a full rotation of the coordinate so that leaves any solution invariant. This forces the two conditions that m2 > 0, so that the solutions are not exponentially growing or decaying, and that so that the solutions are periodic. It is a straightforward differential equation solve, with the two solutions

(5.33)

Using this to simplify Equation (5.31) results in

(5.34)

Since each term involve factors of r2, one needs to ensure that there is a resulting differential equation in r has no other factors involving the other two coordinates in it. So the two terms that have the θ factors in should be set equal to some constant

(5.35)

where the constant has been chosen to be the value −l(l + 1) for later convenience. Indeed, given the earlier discussion about eigenvalues, one may well be tempted to suspect that the l given here is something to do with this - and it certainly is. It is therefore helpful to rewrite Equations (5.34) and (5.35) in the following form

(5.36)

(5.37)

(Exercise: verify this is true.)

The problem has now been reduced to finding solutions for two separate second order differential equations. These two equations occur so often in theoretical physics that they have been named. Equation (5.36) is known as the spherical Bessel equation, while Equation (5.37) is known as the associated Legendre equation. A standard piece of mathematical analysis to solve these equations is to make a power series substitution into the differential equations and then deduce recurrence relations amongst the coefficients. Any standard textbook on mathematical methods describe this procedure, so we shall not pursue this avenue further (see for example [15]). The radial part of the solution then consists of the spherical Bessel functions jl(kr) (which are regular at the origin) and the spherical Neumann functions nl(kr) (which diverge at the origin). The angular part of the solution consists of the associated Legendre polynomials Pl,m(cos θ). The general solution then looks like

(5.

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