Abstract
Hodge–deRham theory is applied to Maxwell’s equations for a transverse electromagnetic wave with a given wave-front surface S. It is shown that the E and B fields, considered as tangent fields to S, are harmonic in the sense of Hodge theory. If S is a spheroid it is known that the space of harmonic fields on S has dimension zero, and hence transverse fields with spheroidal wave fronts do not exist. The same result holds, but for a different reason, if S is a noncircular cylinder or a surface of revolution, and it is conjectured that smooth, singularity-free, transverse solutions to Maxwell’s equations exist only if S is a plane or a circular cylinder.
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