Determine the magnetic field on the axis as a function of x

Question

In Example 29-8. we found the magnetic field due to a circular wire of radius R, carrying a current I, at a point a distance x away from the center of the ring but along the axis to be

B = μ₀I/2. R²/(R² + x²)^3/2

A pair of such coils placed coaxially a distance R apart makes up a Helmholtz roil, for which the magnetic field everywhere inside is fairly constant (Fig. 30-4).

(a) Determine the magnetic field on the axis as a function of x, with x = 0 marking the location of the left-hand coil. Evaluate the field at x = 0, x = R 4, and x = R/2.

(b) Show that dB_x/dx = 0 d²B_x/dx² = 0 at x = R/2.