Chapter 12, Problem 75AP

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 tire magnetic field at any point on the z-axis shown in the accompanying figure, (b) Show that $dB/dz$ and $\frac{d^{2}B}{d{z}^2}$ are both zero at z = 0. (These

vanishing derivatives demonstrate that the magnetic field varies only slightly near z = 0.)