The fields from a line element can be solve with the help of the Hertz
Vector [Marion and Heald, 1980]. In order to solve Maxwell's
equations we define, in empty space, the vector function 
[
Marion and Heald, 1980] that is related to the current density 
and the charge density 
as
Note that 
solves the continuity equation trivially, and
furthermore, it can be used to define another vector function, namely the
Hertz vector 
as
where the fields are then defined as
The time-Fourier transformed Maxwell's equations, with 
and 
can be solved with the help of the Hertz vector 
,
where the line element has orientation 
and length L, and is
parametrized by 
. Values with the hat 
indicate unit vectors, variables in bold indicate vectors, 
is the
frequency, 
. The time dependence can be found by inverting
the above equation.