Normal electron currents radiate when time varying. The simplest example is a filamentary current on wire, but we include surface and volumetric current densities as well. We analyze them by using the magnetic vector potential. Far-field electric fields are proportional to the magnetic vector potential A:

  (1)

We determine the magnetic field from

  (2)

and realizing the cross product of the electric field with the magnetic field points in he direction of power flow, the Poynting vector. Since the electric field direction efines polarization, we usually ignore the magnetic field. We derive the magnetic ector potential from a retarded volume integral over the current density J:

  (3)

where r is the field measurement point radius vector, r the source-point radius vector,  the permeability (  ), and k, the wave number, is 2π/λ. s written, Eq. (3) calculates the potential A everywhere: near and far field. The vector potential can be written in terms of a free-space Green’s function:

  (4)

Radiation Approximation   When we are interested only in the far-field response of an antenna, we can simplify the integral [Eq. (3)]. An antenna must be large in terms of wavelengths before it can radiate efficiently with gain, but at great distances it still appears as a point source. Consider the radiation from two different parts of an antenna. Far away from the antenna, the ratio of the two distances to the different parts will be nearly 1. The phase shift from each part will go through many cycles before reaching the observation point, and when adding the response from each part, we need only the difference in phase shift. In the radiation approximation we pick a reference point on the antenna and use the distance from that point to the far-field observation point for amplitudes, 1/R, for all parts of the antenna. The direction of radiation defines a plane through the reference point. This plane is defined by the radius normal vector, given in rectangular coordinates by

We compute the phase difference to the far-field point by dropping a normal to the reference plane from each point on the antenna. This distance multiplied by k, the propagation constant, is the phase difference. Given a point on the antenna r, the phase difference is . When we substitute these ideas into Eq. (3), the equation becomes

  (5)

In rectangular coordinates  becomes

We can combine k and ˆr to form a k-space vector:

and the phase constant becomes k · r. Currents in filaments (wires) simplify Eq. (5) to a single line integral. Magnetic vector potentials and electric fields are in the same directions as the wires that limit the directions of current. For example, filamentary current along the z-axis produces z-directed electric fields. Spherical waves (far field) have only ˆ θ and ˆφ components found from the projection of Ez onto those axes. Filamentary currents on the z-axis produce only z-directed electric fields with a null from ˆ θ · ˆz = −sin θ at θ = 0. In turn, x- or y-directed currents produce electric fields depending on the scalar products (projections) of the ˆx and ˆy unit vectors onto the ˆ θ and ˆφ vectors in the far field: