Next: Solution by Separation of Up: Maxwell's Equations and a Previous: Wave Equation from Maxwell's

Wave Equation in Cylindrical Coordinates

As pointed out above, the divergence of is zero, so the wave equation reduces to

 (8)

The particular geometry I am interested in is the initial condition of a toroidal magnetic flux loop, which is to say, a magnetic field loop situated on a plane, concentrated between a minor and major radius. The magnetic field is always toroidal (i.e., in the direction). Because of this symmetry, I will use cylindrical coordinates (,,). In this arrangement the flux loop is placed in the plane, with uniform magnetic field in the direction. This further simplifies the equations because, at least initially, and are zero. Also, we can assume that derivatives with respect to are zero because the field is azimuthally uniform.

Standard calculus and physics textbooks contain the formulae for differential operators expressed in cylindrical coordinates. The curl becomes

 (9)

The evaluation of this formula appears at first glance to be very difficult, but because of the symmetry in the initial conditions, we can set all and components, and all terms to zero. This leaves only two terms remaining after performing the curl operation twice:

 (10)

Substituting back into the modified wave equation (8), I arrive at this differential equation,

 (11)

This is a very fortuitous result because the direction of the gradient term is in the azimuthal direction. I can conclude that the magnetic field will always be in the azimuthal direction, and I can ignore and for all times with no loss of information. From this point I will use to refer to . Also, it is worth noting that the above equation is very similar to, but not the same as, the divergence equation for the electric field in cylindrical coordinates. The 's appear in different places, so the solutions will not be exactly the same, however I will show that ultimately the radial portion of the equation is solved by a standard Bessel function.

Next: Solution by Separation of Up: Maxwell's Equations and a Previous: Wave Equation from Maxwell's
Craig Markwardt 2001-07-30