A Helmholtz coil is a pair of identical circular coils that share a common axis and are spaced, usually, a distance apart equal to their radius. The coils are narrow and tightly wound, so they resemble the current loop of Example 26.3. In this configuration the coils produce an approximately uniform magnetic field in the central part of the region between them. In particular, both the first and second derivative of the field are zero on the coils' axis at the point midway between them. To explore the Helmholtz coil, consider two coils like those of Example 26.3, both with radius $R$ and with their axes coinciding with the $x$-axis. One coil is located at $x=-R / 2$ and the other at $x=+R / 2$. Both coils carry current $I$ in the same direction. (a) Adapt Equation $26.9$ for coils of radius $R$ not at the origin, and write an expression for $B(x)$, the net magnetic field on the $x$-axis in the region between the two coils. (b) Show that both the first and second derivatives of the field are zero at the origin. (c) To see the region of uniform field, take $R=1$ and plot the quantity $2 B / \mu_{0} I$ for $-\frac{1}{2}<x<\frac{1}{2}$.