Q3Electromagnetics Waves

Setting up the potential wave equation: in the Lorenz gauge, the magnetic vector potential A satisfies an inhomogeneous wave equation directly driven by the current source J:

For a source oscillating at a single frequency ω (phasor form, ∂/∂t → jω), this becomes the inhomogeneous Helmholtz equation ∇²A + k²A = −μJ, where k = ω√(με).

Free-space Green's function and the retarded potential: the wave equation is solved using the free-space Green's function for a point source, which for the scalar Helmholtz operator is G(R) = e^{−jkR}/(4πR), where R = |r − r'| is the distance from source point r' to observation point r. This Green's function embodies the physical requirement of causality: a disturbance created at the source at time t' is felt at the observation point only after the finite time delay R/c required for the field to propagate there at the speed of light (the 'retarded' time t − R/c), and the e^{−jkR} phase factor (equivalently, the time delay R/c in the time domain) together with the 1/R amplitude decay together constitute an outward-travelling spherical wave from each source point. Superposing (integrating) this Green's function response over the entire current distribution gives the full retarded vector potential:

or, restoring explicit time dependence (time-domain retarded potential):

Extracting the radiated fields: once A(r) is known throughout space (found by evaluating the above integral over the known antenna current distribution J), the magnetic and electric fields follow from the standard potential relations:

Far-field simplification: for observation points far from the source (r ≫ dimensions of the current distribution, and r ≫ λ), R in the phase factor e^{−jkR} is approximated as R ≈ r − r'·r̂ (retaining the leading direction-dependent phase term, since even small path-length differences matter through the exponential), while R in the slowly-varying denominator is approximated simply as R ≈ r (since amplitude is insensitive to small distance corrections); this standard far-field approximation reduces the exact retarded-potential integral to a much simpler radiation integral A(r,θ,φ) ≈ (μe^{−jkr}/4πr)∫J(r')e^{jkr̂·r'}dv', from which only the transverse (θ, φ) components of A survive as radiating fields (the radial component of A does not contribute to E, H in the far zone), directly yielding the antenna's far-field radiation pattern, radiated power, and all associated antenna parameters (gain, directivity, radiation resistance) discussed elsewhere in this paper — this retarded-potential/Green's-function method is thus the fundamental mathematical bridge connecting a specified antenna current distribution to its observable radiated field.

Why the vector potential route is used instead of solving for E and H directly: attempting to solve Maxwell's equations directly for E and H in the presence of an arbitrary source current is considerably harder than solving for the auxiliary potential A, because A satisfies a single, simple inhomogeneous scalar-like (component-wise) wave equation driven directly by J, whereas E and H individually satisfy coupled inhomogeneous wave equations involving both J and its spatial derivatives (∇ρ, ∇×J). Introducing A (and, if needed, the scalar potential Φ, related to A through the Lorenz gauge condition ∇·A + με∂Φ/∂t = 0) decouples the problem into first finding the comparatively simple potential from the known source, and only then differentiating to obtain the physically observable fields — this two-step potential method is the standard technique used throughout antenna theory precisely because the radiation integral for A remains tractable even for realistically complicated current distributions (dipoles, loops, arrays, apertures) where a direct field-based solution would be intractable.