The magnetic field on the axis of a circular current loop (Eq. 5.41) is far from uniform...

The magnetic field on the axis of a circular current loop (Eq. 5.41) is far from uniform (it falls off sharply with increasing z). You can produce a more nearly uniform field by using two such loops a distance d apart (Fig. 5.59).

(a) Find the field (B) as a function of z, and show that ∂ B/∂z is zero at the point midway between them (z = 0). (b) If you pick d just right, the second derivative of B will also vanish at the midpoint. This arrangement is known as a Helmholtz coil; it’s a convenient way of producing relatively uniform fields in the laboratory. Determine d such that ∂²B/∂z² = 0 at the midpoint, and find the resulting magnetic field at the center.

Reference figure 5.59