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Let's consider the two Jefimenko Equations:

$$E(r,t) = \frac{1}{4πϵ_0}∭_V[\frac{e_{r-r'}}{|r-r'|^2} ρ(r',t_r' )+ \frac{1}{c} \frac{e_{r-r'}}{|r-r'|} \frac{∂ρ(r',t_r')}{∂t} - \frac{1}{c^2} \frac{1}{|r-r' |} \frac{∂J(r',t_r' )}{∂t}] d^3 r'$$

$$B(r,t) = \frac{μ_0}{4π} ∭_V [\frac{1}{|r-r'|^2} J(r',t_r')+ \frac{1}{c} \frac{1}{|r-r' |} \frac{∂J(r',t_r' )}{∂t}] × e_{r-r'} d^3 r'$$

This model describes the E and B fields as consequences of the sources $\rho$ and $J$ (causes).

It is really difficult for me to understand the precise meaning of $\rho$ and $J$ in these equations:

  1. Q1: Are they impressed charge/current, or induced ones?

Example about charge. Let's place an infinitesimal voltage source on a certain system made of some dielectrics and conductors. Sunch a voltage sources just creates a separation of charges. Is $\rho$ such charge distribution? Or is it a "generic" $\rho$ which, as function of position, both embodies the source charge density and the induced charge density on surrounding materials?

Example with current. Similar situation, but now a current source is placed on the system. Same doubt: is $J$ the whole currents in both the source and sourrounding materials?

  1. Q2: If $\rho$ and $J$ are charge and current densities everywhere (both voltage/current sources but also induced charge/currents in surrounding materials), how can we find E and B if the induced ones are unknown (determined by E and B)? Equivalently, how can we solve a scattering problem if part of the sources are unknown ?

  2. Q3: If $\rho$ and $J$ are charge and current densities just on the "known" sources, how can we say that "we know them"? They may be charge and current of those sources when isolated, in free space, but when placed in a system with other materials, their charge and current distributions will be affected by the surrounding structure. Do we approximate them as the same as in free space? If yes, where does it come such an assumption in Jefimenko equations?

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In Ignatowski's equations the $\rho(\mathbf r' , t')$ and the $\mathbf J(\mathbf r' , t')$ are the charges and currents that at time $t$ and position $\mathbf r' $ induce the fields at position $\mathbf r $ at a later time $t=t'+\tfrac{1}{c}|\mathbf r -\mathbf r'|$, and because of the retardation it does not matter whether impressed or induced those charges and currents are.

In scattering, for example, you can have an incident plane wave hitting two metal objects and the wave on scattering from one will scatter again on the other, etc. The incident wave induces a current on both objects, and in turn those currents are now sources for the primary scattered and then multiply scattered waves, ad infinitum. This is analogous to a transmission line terminated on either end in a mismatched load: waves are reflected back and forth between the two terminations but because of the single mode nature of the transmission line it is possible to set up a steady state analysis that much simplifies.

When these sources are induced they are not known but what is known is that the surface currents and charges plus the induced waves plus the incident wave together satisfy at every instant a set of continuity equations. As to how to handle that question in detail see, for example, chapter 5 in Silver: "Microwave Antenna Theory And Design".

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