Chapter 28, Problem 28.67P

CALC Helmholtz Coils. Figure P28.67 is a sectional view of two circular coils with radius a. each wound with N turns of wire carrying a current I, circulating in the same direction in both coils. The coils are separated by a distance a equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them, (a) Derive the expression for the magnitude B of the magnetic field at a point on the axis a distance x to the right of point P, which is midway between the coils, (b) Graph B versus x for x = 0 to x = a/2. Compare this graph to one for the magnetic field due to the right-hand coil alone, (c) From part (a), obtain an expression for the magnitude of the magnetic field at point P. (d) Calculate the magnitude of the magnetic field at P if N = 300 turns, I = 6.00 A, and a = 8.00 cm. (e) Calculate dB/dx and d²B/dx² at P(x = 0). Discuss how your results show that the field is very uniform in the vicinity of P.

Figure P28.67