Problem 5.46 The magnetic field on the axis of a circular current loop (Eq. 5.38) 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. 5.60). 

(a) Find the field (B) as a function of z, and show that ˆ‚B/ˆ‚z is zero at the point midway between them (z = 0). Now, 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 producirlg relatively uniform fields in the laboratory. 

(b) Determine d such that ˆ‚²B/ˆ‚z² = 0 at the midpoint, and find the resulting magnetic field at the center.

z =D0