The Helmholtz Theorem and Scalar Potential Expansion

Abstract

The classical Helmholtz theorem which decomposes a given vector field to curl-free and divergence-free components and presents the field in terms of a scalar and a vector potential is reformulated so that the divergence-free part is further decomposed in two parts with respect to either one or two given unit vectors. It is shown that these decompositions follow in a straightforward way from certain operator identities. The field is represented in terms of three scalar potential functions, two of which can be related to Hertzian potentials and TE/TM decomposition when decomposing time-harmonic electromagnetic field vectors outside the source region. Applying the decomposition to time-harmonic sources as well as the fields, equations between scalar source and field potentials can be formulated which gives an alternative method of solving electromagnetic problems.