Helmholtz coils (see picture below) consist of a pair of identical circular coils with radius R. Each coil has n turns of wire. Let the x-axis be perpendicular to the loops, and the center of each loop is on the x axis. The upper Helmholtz coil has its center at x=R/2, the other at x = -R/2. Starting from the Biot-Savart formula for the on-axis field from a single loop of n turns: B(x) = mu 0 n/R2 / 2(R2+x2)3/2 this formula is written for a single loop whose center is the origin, x=0). derive the formula for the on-axis field of two loops as shown in the Helmholtz coil geometry. Show that the field in the center at x = 0 is given by: B(x = 0) = (4/5)3/2 mu 0n1 / R. The Helmholtz coil is quite useful because the field is highly uniform in the center. Show that the first derivative of B (with respect to x) vanishes at the origin: dB/dx| x=0 = 0 Show that the Second derivative of B (with respect to x) vanishes at the origin: d2B / dx2|x=0 = 0 Comment: several derivatives in the radial direction also vanish at the origin. To calculate the radial derivatives requires evaluating the Biot-Savart formulation for arbitrary points. On-axis this is relatively easy like we did here, but off-axis it involves elliptic integrals, or numerical evaluation. Second Comment: The same geometry with the current reversed in one coil is known as an anti-Helmholtz coil, where the field at the origin vanishes, but has a highly linear field gradient.