Retarded charge density (Homework 3-323)

** 4. Consider the retarded-time charge density ρ(r’, t_r) appearing in the retarded potential Griffiths 10.26. Does the volume integral $∫∫∫ ρ (r’, t_r )d^3x$ represent the total charge of the system? If not, explain why.

Griffiths equation 10.26 for the retarded potential is

$V(r,t)=1/(4πϵ_0 ) ∫ ρ(r’,t_r )/R_{rond} *dτ’$

It is true that the integration is made over the entire charge distribution volume, but it is considered that all this charge distribution is taken at the same moment of time $t_r$ (retarded time). If the charge distribution is very large, the time $t_r=t-(R_{rond}/c)$    should be variable with distance $R_{rond}$ from infinitesimal charge to observation point where V is computed. Only if the retarded time is variable the real true charge distribution is taken into account when computing V. Otherwise, since the charge distribution varies with time it is possible taking just one value for $t_r$ we can leave apart a small part of the total charge (that fluctuates with time). The further the infinitesimal charge is from observation point, the more retarded it is.