A reasonably uniform magnetic field over a limited region of space can be produced with the Helmholtz coil, which consists of two parallel coils centered on the same axis. The coils are connected so that they carry the same current I. Each coil has N turns and radius R, which is also the distance between the coils.
(a) Find the magnitude of the magnetic field at any point on the z-axis shown in the accompanying figure. (Use any variable or symbol stated above along with the following as necessary: z and µ0. Assume the origin is located halfway between the coils.)
(b) Show that dB/dz and d 2B/dz2 are both zero at z = 0. (These vanishing derivatives demonstrate that the magnetic field varies only slightly near z = 0.)