- #1

EmilyRuck

- 136

- 6

By manipulating Maxwell's equation, with the potential vector [itex]\mathbf{A}[/itex] and the Lorentz' gauge, one can obtain the following vector wave equation:

[itex]∇^2 \mathbf{A}(\mathbf{r}) + k^2 \mathbf{A}(\mathbf{r}) = -\mu \mathbf{J}(\mathbf{r})[/itex]

The first step for the solution is to consider a point source [itex]- \delta (\mathbf{r} - \mathbf{r}')[/itex]. But what about its meaning in a vector form?

A vector is a 1-dimensional object in the space; a point is a 0-dimensional object, according to linear algebra. How can we relate a 0-dimensional object to a vector context?

With such a source, the unknown function is a Green function and the previous equation becomes

[itex]∇^2 G(\mathbf{r}, \mathbf{r}') + k^2 G(\mathbf{r}, \mathbf{r}') = - \delta (\mathbf{r} - \mathbf{r}')[/itex]

but now we don't have a vector function [itex]\mathbf{A}(\mathbf{r})[/itex] any more, but just a scalar function [itex]G(\mathbf{r}, \mathbf{r}')[/itex]. Why?

This is like saying that, when the source is a point, the vector potential [itex]\mathbf{A}(\mathbf{r})[/itex] becomes

*scalar*...?!

Thank you for having read,

Emily