29. Sources of Magnetic Field

A small solenoid (radius r_a) is inside a larger solenoid (radius r_b > r_a). They are coaxial with n_a and n_b turns per unit length, respectively. The solenoids carry the same current, but in opposite directions. Let r be the radial distance from the common axis of the solenoids. If the magnetic field inside the inner solenoid (r < r_a) is to be in the opposite direction as the field between the solenoids (r_a < r < r_b), but have half the magnitude, determine the required ratio n_b/n_a.

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius R = 10.0 cm and are separated by a distance R = 10.0 cm. Each coil has 85 loops carrying a current I = 2.0 A. Determine the total magnetic field B along the 𝓍 axis (the center line for the two coils) in steps of 0.2 cm from the center of one coil (𝓍 = 0) to the center of the other (𝓍 = R).

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius R = 10.0 cm and are separated by a distance R = 10.0 cm. Each coil has 85 loops carrying a current I = 2.0 A. By what % does B vary from 𝓍 = 5.0 cm to 𝓍 = 6.0 cm?

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius 𝑅 = 10.0 cm and are separated by a distance 𝑅 = 10.0 cm . Each coil has 85 loops carrying a current I = 2.0 A. Graph B as a function of 𝓍.

"(II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 28–40. There is a dc current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop.

(III) A square loop of wire, of side d, carries a current I. (a) Determine the magnetic field B at points on a line (call it the 𝓍 axis) perpendicular to the plane of the square which passes through the center of the square (Fig. 28–56). Express B as a function of 𝓍, the distance from the center of the square. (b) For 𝓍 ≫ d, does the square appear to be a magnetic dipole? If so, what is its dipole moment?

Part of a long, thin insulated straight wire is formed into a single circular loop of radius 𝑅 (Fig. 28–68) and carries a current I. (a) What is the magnitude and direction of the magnetic field at the center of the loop? (b) If the plane of the loop is twisted 90 degrees so that the plane is perpendicular to the straight part of the wire (i.e., in the yz plane) what is the magnitude and direction of the field now at the center of the loop?