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).
B(z)=\frac{\mu_{0} I}{4 \pi}\left(\frac{\cos \theta}{ᴫ^{2}}\right) 2 \pi R=\frac{\mu_{0} I}{2} \frac{R^{2}}{\left(R^{2}+z^{2}\right)^{3 / 2}} (5.41)
(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 \partial^{2} B / \partial z^{2}=0 at the midpoint, and find the resulting magnetic field at the center. [Answer : 8 \mu_{0} I / 5 \sqrt{5} R ]
