Consider a circular cylindrical solenoid of finite length L, radius a, with N turns of wire carrying current I0. The current may be approximated by a surface current density K⃗ = N I 0 / L φˆ .
(a) Calculate the magnetic field on the axis of the cylinder halfway between the end. (Hint: Subdivide the solenoid into infinitesimal current rings dI = Kdz′ and perform an integral.)
(b) Calculate the magnetic field on the axis of the cylinder at either end. (c) Show that Bend/Bcenter approaches 1/2 as L/a → ∞.