Abstract:
The electromagnetic field in a cone of arbitrary slant
height with a symmetrically placed time harmonic ring source is
studied. Through the use of the modified Helmholtz equation as
an intermediate, we obtain the solution of the semi-infinite
cone directly from the finite cone. To demonstrate the need
for the modified Helmholtz equation a simple example is used in
which the solution is known. The Green function is derived
from a well known summation formula involving the eigenvalues
and eigenfunctions, which are determined from the roots of
certain Legendre and Bessel functions.
The results obtained here for the semi-infinite cone are
compared with those obtained by Buchholz [4], and the special
case Ѳ0 = π/2 is compared with a double ring in free space. In
both cases the results are in agreement. Once the results have been obtained for the time harmonic
case, they are generalized by Laplace transform theory to arbitrary
time dependency. This is accomplished by finding the field for a
"Dirac" pulse at time t and integrating the product of the pulse
function and the "Dirac" pulse field with respect to time.
