Another way to fill in the “missing link” in Fig. 5.48 is to look for a magnetostatic an...

Another way to fill in the “missing link” in Fig. 5.48 is to look for a magnetostatic analog to Eq. 2.21. The obvious candidate would be

(a) Test this formula for the simplest possible case—uniform B (use the origin as your reference point). Is the result consistent with Prob. 5.25? You could cure this problem by throwing in a factor of 1 2 , but the flaw in this equation runs deeper.

(b) Show that ?(B × dl) is not independent of path, by calculating

As far as I know,²⁸ the best one can do along these lines is the pair of equations

[Equation (i) amounts to selecting a radial path for the integral in Eq. 2.21; equation (ii) constitutes a more “symmetrical” solution to Prob. 5.31.]

(c) Use (ii) to find the vector potential for uniform B.

(d) Use (ii) to find the vector potential of an infinite straight wire carrying a steady current I . Does (ii) automatically satisfy ∇ · A = 0?

Prob. 5.25

If B is uniform, show that works. That is, check that ∇ · A = 0 and ∇ × A = B. Is this result unique, or are there other functions with the same divergence and curl?

Prob. 5.31

a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any divergenceless vector field F can be written as the curl of a vector potential A. What you have to do is find A_x , A_y, and A_z such that (i) ∂ A_z/∂y − ∂ A_y/∂_z = Fx ; (ii) ∂ A_x/∂_z − ∂ A_z/∂_x = F_y ; and (iii) ∂ A_y/∂_x − ∂ A_x/∂_y = F_z . Here’s one way to do it: Pick A_x = 0, and solve (ii) and (iii) for A_y and A_z . Note that the “constants of integration” are themselves functions of y and z—they’re constant only with respect to x. Now plug these expressions into (i), and use the fact that ∇ · F = 0 to obtain

(b) By direct differentiation, check that the A you obtained in part (a) satisfies ∇ × A = F. Is A divergenceless? [This was a very asymmetrical construction, and it would be surprising if it were—although we know that there exists a vector whose curl is F and whose divergence is zero.]

(c) As an example, let Calculate A, and confirm that ∇ × A = F. (For further discussion, see Prob. 5.53.)

Reference prob 5.53

Reference Theorem 2

Divergence-less (or “solenoidal”) fields. The following conditions are equivalent: