A long coaxial cable consists of a solid cylindrical conductor of radius a=1.0mm carrying a time-dependent current
Ic(t)=2.0cos(106t)A
outwards, and a thin concentric cylindrical conducting sheath of negligible thickness at radius b=5.0mm which returns the current. The region a<r<b is vacuum.
(a) Taking a cylindrical Gaussian surface of radius r such that a<r<b, derive an expression for the displacement current Id(r,t) crossing the surface during the charging process of the cable.
(b) Show that the total displacement current through any such surface equals the conduction current in the central conductor at every instant.
(c) Explain why the result obtained in part (b) is essential for ensuring continuity of the magnetic field in the region between the two conductors.