Is it possible to solve these partial differential equations directly, relating to Antenna Theory; [tex]∇^2 E - μ_0 ε_0 \frac{∂^2E}{∂t^2} = -μ_0 \frac{∂J}{∂t}.[/tex] AND [tex]∇^2 B - μ_0 ε_0 \frac{∂^2B}{∂t^2} = -μ_0 ∇ x J.[/tex] I don't like the idea of having to make up fields that don't exist in order to make the math work. The x is a cross product not a variable or multiplication.
Here, let me help with that:[tex]\nabla^2 \vec B - \mu_0 \epsilon_0 \frac{\partial^2\vec B}{\partial t^2} = -\mu_0 \nabla \times \vec J.[/tex]... better? (Hit "quote" to see how I did that.) There may be some geometries where the equations can be solved directly, I've not heard of any for antennas. Follows that you have to use a Trick. Welcome to realmaths.