>>16841239
1. Maxwell’s equations

On Minkowski spacetime [math]( M = \mathbb{R}^{1,3} )[/math] with coordinates [math]( (t, x, y, z) )[/math],
let [math]( F )[/math] denote the electromagnetic field 2-form and [math]( J )[/math] the current 3-form.
Maxwell’s equations are
[eqn]
dF = 0, \qquad d\star F = \mu_0 J,
[/eqn]
where [math]( \star )[/math] is the Hodge dual associated with the Minkowski metric.
Since [math]( dF = 0 )[/math], we can introduce the potential 1-form [math]( A )[/math] such that
[eqn]
F = dA.
[/eqn]
In the Lorenz gauge [math]( d\star A = 0 )[/math], Maxwell’s equations reduce to the wave equation
[eqn]
\Box A = \mu_0 J,
[/eqn]
where [math]( \Box = d\star d\star + \star d\star d )[/math] is the spacetime d’Alembertian.
In coordinates, [math]( \Box = \frac{1}{c^2}\partial_t^2 - \nabla^2 )[/math].

2. Geometry of the square antenna

Consider a thin, conducting square patch located at [math]( z = 0 )[/math] with side length [math]( a )[/math]:
[eqn]
S = \{ (x, y, 0) \mid |x| \le \tfrac{a}{2},\ |y| \le \tfrac{a}{2} \}.
[/eqn]

Let the antenna carry a tangential surface current density [math]( \mathbf{K}(x, y, t) )[/math]
that oscillates harmonically with angular frequency [math]( \omega )[/math]:
[eqn]
\mathbf{K}(x, y, t) = \Re \left\{ \tilde{\mathbf{K}}(x, y)\, e^{-i\omega t} \right\}.
[/eqn]