Consider two current loops (as in Fig) whose orientation in space is fixed, but whose relative separation can be changed. Let O₁ and O₂ be origins in the two loops, fixed relative to each loop, and x₁ and x₂ be coordinates of elements dI₁ and dI₂, respectively, of the loops referred to the respective origins. Let R be the relative coordinate of the origins, directed from loop 2 to loop 1.

(a) Starting from (5.10), the expression for the force between the loops, show that it can be written

F₁₂ = I₁I₂Ñ_RM₁₂(R)

Where M₁₂ is the mutual inductance of the loops,

And it is assumed that the orientation of the loops does not change with R.

(b) Show that the mutual inductance, viewed as a function of R, is a solution of the Laplace equation,

The importance of this result is that the uniqueness of solutions of the Laplace equation allows the exploitation of the properties of such solutions, provided a solution can be found for a particular value of R.

dl, dlz X1 - X2 + R| %3D M12(R)