Suppose you want to define a magnetic scalar potential U (Eq. 5.67) in the vicinity of a...

Suppose you want to define a magnetic scalar potential U (Eq. 5.67) in the vicinity of a current-carrying wire. First of all, you must stay away from the wire itself (there ∇ × B ≠ 0); but that’s not enough. Show, by applying Ampère’s law to a path that starts at a and circles the wire, returning to b (Fig. 5.47), that the scalar potential cannot be single-valued (that is, U(a)≠U(b), even if they represent the same physical point). As an example, find the scalar potential for an infinite

Figure 5.47

straight wire. (To avoid a multivalued potential, you must restrict yourself to simplyconnected regions that remain on one side or the other of every wire, never allowing you to go all the way around.)

equation 5.67

B = −∇U,