A current distribution J(x,t) localized near the origin varies slowly in time.
(a) Use the Jefimenko expressions (6.55) and (6.56) for the retarded fields to evaluate the quasi-static fields far from the current distribution. Assuming that there are no electric multipole moments and retaining only the magnetic dipole contributions, show that the magnetic and electric fields at the point (x=r^r,t) to first order in an expansion in successive time derivatives are
B=4πμ0r31(1+cr∂t∂)[3(m(t−r/c)⋅r^)r^−m(t−r/c)]E=4πμ0r21r^×∂t∂m(t−r/c)
(b) The construction and excitation of an infinite, straight, right circular solenoid of radius a, with N turns per unit length, are such that its current I(t) is the same everywhere along its length and is changed very slowly in time. Show that the fields far from the solenoid are approximately
B≈0+O(∂t2∂2I)E≈−2μ0Na2∂t∂I(t−ρ/c)ϕ^
where ρ is the perpendicular distance from the axis, provided max(∣dI/dt/I∣)≪c/ρ. A long solenoid with a time-varying current has an electric field outside it, in contrast to the static situation. Verify that Faraday's law is satisfied.