An infinite wire runs along the z axis; it carries a current I (z) that is a function of z (but not of t), and a charge density λ(t) that is a function of t (but not of z).

(a) By examining the charge flowing into a segment dz in a time dt, show that dλ/dt = −d I/dz. If we stipulate that λ(0) = 0 and I (0) = 0, show that λ(t) = kt, I (z) = −kz, where k is a constant.

(b) Assume for a moment that the process is quasistatic, so the fields are given by Eqs. 2.9 and 5.38. Show that these are in fact the exact fields, by confirming that all four of Maxwell’s equations are satisfied. (First do it in differential form, for the region s > 0, then in integral form for the appropriate Gaussian cylinder/Amperian loop straddling the axis.)