1. Helmholtz Coils (a) A circular loop of wire of radius R carries current /. The loop is in the xy plane and centered on the origin. As viewed from the +z axis, the current flows counterclockwise. Find the magnetic field at any point on the z axis. (b) Two identical parallel circular coils carrying currents in the same direction separated by a distance equal to their radius R are called Helmholtz coils. These make a nearly uniform magnetic field near their midpoint. The two coils each have / turns of wire carrying current / in each turn. The coils are in the planes z = R/2 and z = -R/2, respectively, with the center of each coil on the z axis. The origin is therefore midway between their centers. As viewed from the +z axis, the current flows counterclockwise in both coils. Find the magnetic field at any point on the z axis. Sketch a graph of the field from z = -R/2 to z = +R/2. (c) Calculate OB/Oz and 02B./0=2 at the origin. Use your result to explain why the field is particularly uniform near the origin. (d) Show another example of using current-carrying coils to generate a uniform magnetic field in a region of space. Discuss technical/practical advantages or disadvantages between your example and Helmholtz coils introduced here