Derivation
We consider all quantities in the frequency domain, and so assume a time-dependency that is suppressed throughout.
Begin with the Maxwell equations relating the electric and magnetic field an assume linear, homogeneous media with permeability and permittivity and, respectively:
Following the third equation involving the divergence of H
by vector calculus we can write any divergenceless vector as the curl of another vector, hence
where A is called the magnetic vector potential. Substituting this into the above we get
and any curl-free vector can be written as the gradient of a scalar, hence
where is the electric scalar potential. These relationships now allow us to write
which can be rewritten by vector identity as
As we have only specified the curl of A, we are free to define the divergence, and choose the following:
which is called the Lorenz gauge condition. The previous expression for A now reduces to
which is the vector Helmholtz equation. The solution of this equation for A is
where is the three-dimensional homogeneous Green's function given by
We can now write what is called the electric field integral equation (EFIE), relating the electric field E to the vector potential A
We can further represent the EFIE in the dyadic form as
where here is the dyadic homogeneous Green's Function given by
Read more about this topic: Electric-field Integral Equation