Electromagnetic Radiation Exercises

 

Radiation due to a point charge, q, is most directly determined by using retarded potentials


φ(
r, t) = (1/4πε0) [qc/(|rr'|c − (r  − r')
·v)] ,
and
A(r, t) = (v/c2) φ(r, t) ,
which are the Lienard-Wiechert potentials for moving point charges.

To determine whether or not we will have radiation from a particular moving charge, we look at the Poynting vector,
S = (1/μ0)(E × B). When we integrate the Poynting vector over a surface, we get the power delivered through that area:

P
(r) = ∫
S·n da\;.
In order to have radiation, there must be some power delivered to infinity, and therefore P(r) must not vanish as r ∞.  Since the surface area goes like r2, for radiating systems the Poynting vector must go like 1/r2 otherwise the power would vanish at large r.  The electric and magnetic fields, therefore, must each fall off like 1/r to have radiation.  Since static electric and magnetic fields go like 1/r2 at best, the Poynting vector goes like 1/r4, and hence point charges associated with these kinds of fields do not radiate.  To study radiation, we look for electric and magnetic fields that go like 1/r.

Qualitatively, one can think about the electric and magnetic fields that are created by point charges in the following way: there are the fields that stay attached to the point charge and there are fields that are thrown off by the point charge.  The fields that are thrown off and make it to infinity, are characterized as radiation.
 
© Mario Belloni and Wolfgang Christian (2006).