# Time generalized Biot-Savart and Coulomb laws: Jefimenko's Equations, boundary-conditions, reflection, transmission

I am attempting to use the time generalised Biot-Savart and Coulomb laws (sometimes called the Jefimenko's equatios) to calculate the electric and magnetic field of a wire loop in a homogeneous sphere with the relative permitivity $$\epsilon_r = 10$$. The space between the loop and sphere is assumed to be free space ($$\epsilon_r = 1$$). The exact geometry is shown in the figure below.

For the calculation I use the following equations:

$$\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi} \int ( \frac{\vec{J(t_r)}}{R^2} + \frac{1}{Rc}\frac{d\vec{J(t_r)}}{dt} ) \times \vec{R}_u d\vec{r}'$$

$$\vec{E}(\vec{r},t) = -\frac{1}{4\pi\epsilon_0\epsilon_rc^2} \int \frac{1}{R}\frac{d\vec{J(t_r)}}{dt} d\vec{r}'$$

Here is $$\vec{R}_u$$ the unit vector in the direction of $$\vec{R}$$ (see figure), $$c = c_0 / \sqrt(\epsilon_r)$$ is the speed of light and $$t_r = t - R/c$$ is the retarded time. The current distrubution $$\vec{J}(\vec{r},t_r)$$ varies with $$100\,$$MHz and is known.

The Jefimenko equations are only intended for free space or for a homogeneous medium with constant permittivity (see Jefimenko 1992). These conditions are not given here because two media are present. Nevertheless, these equations give results that agree well with the literature. Smaller deviations can be observed in area B (see figure). I suspect that these deviations are dominated by field components caused by reflection at the boundary between the sphere and space. As far as I know, reflection and transmission are not taken into account by the Jefimenko equations.

My questions are now:

Does anyone know how I can calculate the reflected field components that are not taken into account by the Jefimenko equations?

Has anyone already dealt with a similar issue? I would be very grateful for further reading recommendations or papers.

I think that the problem can be solved with a superposition of the form:

$$\vec{B}_{tot} = \vec{B}_{inc} + \vec{B}_{scat}$$

The incidence field $$\vec{B}_{inc}$$ is given by the Jefimenko equation. The scattered/reflected field $$\vec{B}_{scat}$$ should be proportional to the incidence field $$\vec{B}_{inc}$$. I think the core question can then be rephrased to:

$$\vec{B}_{scat} = f(\,\vec{r}, t_r ) \cdot \vec{B}_{inc}$$

What is $$f(\vec{r},t_r)$$ = ?

What do you think? What would be an Ansatz to calculate $$f(\vec{r},t_r)$$ or $$\vec{B}_{scat}$$?

I would be very grateful for tips, approaches and reading recommendations.

Thanks a lot!