C.1 Minimization of Energy Result

Let us assume that V( x, y) is the true solution of Laplace's equation. In addition, let us assume that U( x, y) is another function that can be differentiated and is equal to zero on the boundary L ₁ of the region s. Then, the sum of the two solutions which we will call the variation is given by V( x, y) + ?U( x, y) where ? is a small real parameter. The variation will have the same value on the boundary L ₁ as V( x, y). The electrostatic energy of the summation of the two terms is obtained from the expression (2.67)

(C.1)

This functional-energy W( V + ?U) is expanded in powers of the small parameter ?

(C.2)

The third term on the right hand side can be identified from (2.67) as being the energy of the additional functions U. The second term on the right hand side can be transformed using the vector identity (A.6)

(C.3)

where U is a scalar and ? V is a vector. The last term in (C.3) can be written as U ? ² V. Therefore, we finally write (C.2) as

(C.4)

The term ? V u _n = ?V/ ?n. From Figure 4-17a, we have U