# Understanding sources (charge and current densities) in Jefimenko Equations

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?

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.