From Maxwell’s equations to antenna array – part 1

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Antenna models — reading order

Introduction

Antenna theory is Maxwell’s equations plus boundary conditions and a far-field approximation. Before retarded potentials and array factors, write the four laws carefully—especially Ampère’s law with Maxwell’s displacement current. A typesetting slip (\(\vec{D}/\partial t\) instead of \(\partial\vec{D}/\partial t\)) has confused more than one student, so we will watch that term.

Motive

Every far-field formula later in this series rests on these equations. Put the differential and integral forms side by side, and check that displacement current restores charge conservation.

Technical development

\[\begin{aligned} \nabla\cdot\vec{D} &= \rho &\oint_S \vec{D}\cdot d\vec{S} &= \int_V \rho\, dV \\ \nabla\cdot\vec{B} &= 0 &\oint_S \vec{B}\cdot d\vec{S} &= 0 \\ \nabla\times\vec{E} &= -\frac{\partial\vec{B}}{\partial t} &\oint_C \vec{E}\cdot d\vec{l} &= -\frac{\partial}{\partial t}\int_S \vec{B}\cdot d\vec{S} \\ \nabla\times\vec{H} &= \vec{J}+\frac{\partial\vec{D}}{\partial t} &\oint_C \vec{H}\cdot d\vec{l} &= \int_S \vec{J}\cdot d\vec{S}+\frac{\partial}{\partial t}\int_S \vec{D}\cdot d\vec{S} \end{aligned}\]

Gauss’s law for \(\vec{D}\) recovers Coulomb’s field for a point charge. Gauss’s law for \(\vec{B}\) forbids magnetic monopoles. Faraday–Lenz induction carries the minus sign of Lenz’s law. Ampère–Maxwell adds \(\partial\vec{D}/\partial t\) so that the continuity equation \(\nabla\cdot\vec{J}+\partial_t\rho=0\) holds identically when combined with Gauss’s law.

How this achieves the motive

Take the divergence of Ampère–Maxwell and use \(\nabla\cdot\vec{D}=\rho\): you recover charge conservation. That is why Maxwell added displacement current—and why the partial derivative must act on \(\vec{D}\), not sit as a stray denominator.

Sample simulation (consistency check)

# Continuity implied by Maxwell (conceptual printout)
# div(curl H) = 0 = div(J) + d(div D)/dt = div(J) + d(rho)/dt
print("div(curl H) = 0  =>  div(J) + d(rho)/dt = 0  (charge conservation)")
print("Displacement current term is dD/dt, not D/dt.")

Sample results

There is no plot here—only a logical identity. If your notes still show \(\vec{D}/\partial t\), correct them before moving on.

Next topic

Static potentials are not enough for radiation. Next we introduce retarded potentials, where cause at the source is felt later at the observer.

References

[1] Feynman Lectures Vol. II, Ch. 18. [2] J. D. Jackson, Classical Electrodynamics.

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