The design of a magneto-optical atom trap requires a magnetic field $B$ that is directly proportional to position $x$ along an axis. Such a field perturbs the absorption of laser light by atoms in the manner needed to spatially confine atoms in the trap. Let us demonstrate that "anti-Helmholtz"" coils will provide the required field $B=C x,$ where $C$ is a constant. Anti-Helmholtz coils consist of two identical circular wire coils, each with radius $R$ and $N$ turns, carrying current $I$ in opposite directions (Fig,
62 ). The coils share a common axis (defined as the $x$ axis with $x=0$ at the midpoint $(0)$ between the coils. Assume that the centers of the coils are separated by a distance equal to the radius $R$ of the coils. (a) Show that the magnetic field at position $x$ along the $x$ axis is given by$B(x)=\frac{4 \mu_{0} N I}{R}\left\{\left[4+\left(1-\frac{2 x}{R}\right)^{2}\right]^{-\frac{3}{2}}-\left[4+\left(1+\frac{2 x}{R}\right)^{2}\right]^{-\frac{3}{2}}\right\}$
(b) For small excursions from the origin where $|x|<R$ , show that the magnetic field is given by $B \approx C x,$ where the constant $C=48 \mu_{0} N I / 25 \sqrt{5} R^{2} .$ (c) For optimal atom
trapping, $d B / d x$ should be about 0.15 $\mathrm{T} / \mathrm{m} .$ Assume an atom trap uses anti-Helmholtz coils with $R=4.0 \mathrm{cm}$ and $N=150 .$ What current should flow through the coils? [Coil separation equal to coil radius, as assumed in this problem, is not a strict requirement for anti-Helmholtz coils.]