A collection of N charges lies inside a volume V. With respect to a fixed origin, the angular momentum of the charges and the electromagnetic fields they produce within V is

(a) Let S be the surface which bounds V. The text used a general conservation law to establish that

Derive this expression using the expression for L stated just above, the Maxwell equations, and the
Lorentz force law.
(b) Suppose that the N charges generate a time-dependent electric dipole moment p(t). Compute dL/dt when V is chosen as a spherical volume (centered at the origin) so large that only the radiation fields due to p(t) have any significant magnitude on S. Does the answer surprise you?
(c) Now use the exact fields associated with p(t) and recalculate dL/dt keeping only those terms which survive in the limit when the sphere radius goes to infinity. Show that

where [p]ret = p(t − r/c).

(d) Carry out the integral and confirm that

Transcribed Image Text:

N to fd³r d³r {rx (Ex B)} + Σr₁ × Pi. i=1 V L = 60 = co