A coaxial cable consists of two very thin-walled conducting cylinders of radii r₁ and r₂ (Figure). Current I goes in one direction down the inner cylinder and in the opposite direction in the outer cylinder.

(a) Use Ampere's law to find B. Show that B = 0 except in the region between the conductors.

(b) Show that the magnetic energy density in the region between the cylinders is

(c) Find the magnetic energy in a cylindrical shell volume element of length ? and volume dV = ?2?r dr, and integrate your result to show that the total magnetic energy in the volume of length ? is

(d) Use the result in part (c) and Um = ?LI² to show that the self-inductance per unit length is

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