>>16841239
>>16841281
Part 2

3. 3-form for a surface current

A tangential surface current at [math]( z = 0 )[/math] can be represented by the 3-form
[eqn]
J = \Re \left\{
\left( \tilde{K}_x\, dy \wedge dz + \tilde{K}_y\, dz \wedge dx \right)
\wedge e^{-i\omega t}\, dt \; \delta(z)\, \chi_S(x, y)
\right\},
[/eqn]
where [math]( \chi_S(x, y) )[/math] is the indicator function of the square [math]( S )[/math], and
[math]( \delta(z) )[/math] confines the current to the plane [math]( z = 0 )[/math].

4. Solution for the potential

In the Lorenz gauge, the potential satisfies
[eqn]
(\nabla^2 + k^2)\, \tilde{\mathbf{A}}(\mathbf{r}) = -\mu_0\, \tilde{\mathbf{J}}(\mathbf{r}), \qquad k = \frac{\omega}{c}.
[/eqn]
The spatial Green’s function of the Helmholtz operator is
[eqn]
G(\mathbf{r}, \mathbf{r}') = \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|}.
[/eqn]
Hence, the complex vector potential is
[eqn]
\tilde{\mathbf{A}}(\mathbf{r}) = \mu_0
\int_S \tilde{\mathbf{K}}(\mathbf{r}')
\frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|}\, dS',
[/eqn]
and the field strength 2-form is [math]( F = dA )[\math].

QED

check mate