Retarded potentials — delayed EM potentials from time-varying sources

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Antenna models — reading order

Introduction

Maxwell’s equations in free space support waves at speed \(c\). So for time-varying sources the potentials cannot be the instantaneous Coulomb or Biot–Savart integrals; they must be evaluated at a retarded time. We write those retarded potentials and then specialise to a single-frequency (phasor) source.

Motive

Radiation, interference, and array factors all assume that what leaves the antenna arrives later with a phase \(e^{-jkR}\). We want that phase to come from retardation of the potentials, not from inserting it by hand in the far field.

Technical development

In the Lorenz gauge the retarded scalar and vector potentials are

\[\Phi(\mathbf{r},t)=\frac{1}{4\pi\varepsilon}\int\frac{\rho(\mathbf{r}’,t-R/c)}{R}\,dV’, \quad \mathbf{A}(\mathbf{r},t)=\frac{\mu}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}’,t-R/c)}{R}\,dV’,\]

with \(R=|\mathbf{r}-\mathbf{r}’|\). For a phasor convention \(e^{j\omega t}\), retardation becomes multiplication by \(e^{-jkR}\) with \(k=\omega/c\), yielding the familiar Helmholtz integrals used in antenna texts.

How this achieves the motive

The Green’s functions we use later are special cases of these integrals. The spherical wave \(e^{-jkR}/R\) is just retardation sitting on top of the static \(1/R\) kernel.

Sample simulation

import numpy as np

c = 299_792_458.0
f = 1e9
k = 2 * np.pi * f / c
R = np.array([0.5, 1.0, 2.0, 10.0])  # metres
 gre = np.exp(-1j * k * R) / R
print("k =", k, "1/m")
print("e^{-jkR}/R =", gre)

Sample results

The printed values shrink as \(1/R\) and rotate in phase as \(-kR\). That is the same amplitude and phase behaviour you will use in far-field array work.

Next topic

When is the observer “far” enough for pattern work? See the Fraunhofer boundary.

References

[1] Jackson. [2] Orfanidis, EWA.

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