Question

Consider a point charge q moving arbitrar ily along a trajectory described by vector function of time r (t). The velocity of the charge is thus V(t)- di,(t)/dt. Suppose Q and Q'represent points on the trajectory where the charge is at time t and was at an earlier time t'. Let R(t) F r,(t) be the vector from the charge to the fixed point P as shown in the figure of particle re volume element de r" a) Prove the velocity of the charge is equation R(tr) dt c) Determine the retarded potentials V(r, t) and A(r,t) b) Show that the retarded time satisfies the

Answer 1

Given that: - A point charge q moving arbitrarily along a trajectory described by vector function of time r_q^vector (t). The velocity of the charge is thus v^vector (t) = dr_q^vector (t)/dt Q & Q^' represent points on the trajectory where the charge is at time t & was at an earlier time t^'. Let R^vector (t) = r^vector - r_q^vector (t) (a) The velocity of the charge is v^vector (t) = -dR^vector/dt t_r = t - r/c t_r = t - R(t_r)/c t_r = t - r/c \r\n (b) The natural generation for nm static source is v(r, t) = 1/mu pi epsilon_0 integral p(r^', ts)/r dT^' A(r, t) = mu_0/4 pi integral J(r^', tr)/r dT^' Integrals are evaluated the retarded time these are called retarded potentials. We check that, there potentials satisfy laplace equation. To calculate the total charge of a configuration, we have to integrate, p were the entire distribution at one instant of time, but here the retardation t_r