Question

9. Suppose we have some charge density ρ(t,x) in space and a current density J(t, x), and suppose both quantities are known to be zero outside of a sphere of some radius R. Use the conservation equation to show that the total charge Q(t) dxdydz ρ(t, x) Jall of space is (a) finite and (b) conserved, ie, IQI 〈 x and Q-0.*

Answer 1

Q(t) = integral_all the space dx dy dz p(t, x) in polar coordinate, dx dy dz = p^2 sin theta dr d theta d phi Q(t) = integral_all the space r^2 sin theta dc d theta d phi p(t, p cos theta)d phi integral can be done, integral d phi = 2 pi Q(t) = 2 pi integral r^2 sin theta p (t, r cos theta) d theta dr since p (t, x) is zero outside a sphere a sphere of radius R, the limit of the integral are finite. Q(t) = 2 pi integral^R_0 r^2 intgeral^pi_0 sin theta p (t, r cos theta) d theta dr so, this integral can not be plusminus infinity, the plusminus sign is due to the sign of p (t, x). so, Vertical-bar Q (t) Vertical-bar < infinity b) taking time derivative of Q(t) d/dt Q(t) = integral_all the space dx dy dz partial differential p(t, x)/partial differential t