EXAMPLE 4 Vector Potential for a Solenoid An electric current I flowing through a solenoid (a tightly wound spiral of wire; see Figure 11) creates a magnetic field B. If we assume that the solenoid is infinitely long, with radius R and the z-axis as the central axis, then if r > R B(r) = Bk ifr < R where r = (x2 + y?)'/2 is the distance to the z-axis, and B is a constant that depends on the current strength I and the spacing of the turns of wire. (a) Show that a vector potential for B is у х RB(-.0) ifr> R A(r) = B (-y,x,0) if r < R (b) Calculate the flux of B through the surface S (with an upward-pointing normal) in Figure 11 whose boundary is a circle of radius r, where r > R. FIGURE 11 The magnetic field of a long solenoid is nearly uniform inside and weak outside. In practice, we treat the solenoid as infinitely long if it is very long in comparison with its radius.

EXAMPLE 4 Vector Potential for a Solenoid An electric current I flowing through a solenoid (a tightly wound spiral of wire; see Figure 11) creates a magnetic field B. If we assume that the solenoid is infinitely long, with radius R and the z-axis as the central axis, then if r > R B(r) = Bk ifr < R where r = (x2 + y?)'/2 is the distance to the z-axis, and B is a constant that depends on the current strength I and the spacing of the turns of wire. (a) Show that a vector potential for B is у х RB(-.0) ifr> R A(r) = B (-y,x,0) if r < R (b) Calculate the flux of B through the surface S (with an upward-pointing normal) in Figure 11 whose boundary is a circle of radius r, where r > R. FIGURE 11 The magnetic field of a long solenoid is nearly uniform inside and weak outside. In practice, we treat the solenoid as infinitely long if it is very long in comparison with its radius.