3.1 Capacitance of Solid Conductors and Screens

1. Suppose that the total charge of a conductor is Q and its potential is V. Then

(3.1)

and the coefficient C is called the capacitance of the conductor. If ?( t) is the surface charge distribution, then

(3.2)

and

(3.3)

Thus

(3.4)

The function ?( t) can be calculated by the iterative processes given in Section 2.3 and Section 2.4. If ? _n is an approximation to ? then the potential

(3.5)

is not constant on ?. In this case we introduce the averaged potential

(3.6)

If ? _n ? ? in H = L ²( ?) then V _n ? V and

(3.7)

is an approximation to C. The iterative process (2.2) satisfies condition (2.3),

(3.8)

In this case (3.7) can be written as

(3.9)

where ? _n is the n-th approximation to the solution of the problem

(3.10)

and A is defined as usual (see (1.28)). One can construct ? _n by means of the iterative process

(3.11)

Theorem 2.2 and formula (3.9) imply the following theorem,

Theorem 3.1

Let

(3.12)

where S = meas ? and

(3.13)

Then

(3.14)

where c > 0 and 0 < q < 1 depend on the shape of the conductor but do not depend on n. The following inequality holds:

(3.15)

where

(3.16)

Proof. The first statement of...