A localized electric charge distribution produces an electrostatic field, E = . Into this field is placed

A localized electric charge distribution produces an electrostatic field, E = –Ñф. Into this field is placed a small localized time-independent current density J(x), which generates a magnetic field H.

(a) Show that the momentum of these electromagnetic fields, (6.117), can be transformed to

P_field = 1/c² ∫ ФJ d³x

Provided the product ФН falls off rapidly enough at large distances. How rapidly is "rapidly enough"?

(b) Assuming that the current distribution is localized to a region small compared to the scale of variation of the electric field, expand the electrostatic potential in a Taylor series and show that

P_field = 1/c² E(0) x

where E(0) is the electric field at the current distribution and m is the magnetic moment, (5.54), caused by the current.

(c) Suppose the current distribution is placed instead in a uniform electric field E₀ (filling all space). Show that, no matter how complicated is the localized J, the result in part a is augmented by a surface integral contribution from infinity equal to minus one-third of the result of part b, yielding

P_field = 2/3c² E₀ x m

Compare this result with that obtained by working directly with (6.117) and the considerations at the end of Section 5.6.