COMPARISON OF CONDENSERS

111.] Let us ascertain what this relation must be. In Fig. 26 the same electrical arrangement is represented under a simpler form, in which the con- densers consist each of a pair of disks. Under this form the analogy with Wheatstone’s Bridge becomes apparent to the eye. We have to consider the potentials and charges of four conductors. The first consists of the inner coat- ings of P and R, together with the connecting wire. We shall call this con- ductor α, its charge a, and its potential A. The second consists of the outer coatings of P and Q, together with the insulating stand β. We shall call this conductor β, its charge b, and its potential B. The third consists of the in- ner coatings of Q and S and the connecting wire γ. We shall call this γ, its charge c, and its potential C. The fourth consists of the outer coatings of R and S and of the earth with which they are kept connected. We might use

the letters δ, d, and D with reference to this conductor, but as its potential is always zero and its charge equal and opposite to that of the other conductors we shall not require to consider it. The charge of any one of the conductors depends on its own potential to- gether with the potentials of the two adjacent conductors, and also, but in a very slight degree, on that of the opposite conductor. Let the coefficients of induction between the different pairs of the four con- ductors be as in the following scheme,— Fig. 27.

in which ξ and η are very small compared with P, Q, R, and S. The coeffi- cient of capacity of any one of the conductors will exceed the sum of its three coefficients of induction by a quantity which will be small if the capacity of the knobs of the jars and their connecting wires are small compared with the whole capacities of the jars. Let us denote this excess by the symbols α, β, γ, δ, which belong to the conductors. The capacities therefore will be, P + R + α + η, P + Q + β + ξ, Q + S + γ + η, R + S + δ + ξ, and the charges will therefore be,COMPARISON OF CONDENSERS. 107 for α, a = (P + R + α + η)A − P B − RD − ηC, for β, b = (P + Q + β + ξ)B − P A − QC − ξD, for γ, c = (Q + S + γ + η)C − QB − SD − ηA, for δ, d = (R + S + δ + ξ)D − RA − SC − ξB. In the first part of the experiment the potentials of α and γ are A and C respectively, while those of β and δ are zero. Hence, at first, a = (P + R + α + η)A − ηC, b= − P A − QC, c = (Q + S + γ + η)C − ηA.

We need not determine the charge of δ. Now let a communication be made between α and γ, and let us denote the charges and potentials of the conductors after the discharge by accented letters. The potentials of α and γ will become equal; let us call their common potential y, then A′ = C ′ = y. The sum of their charges remains the same, or a′ + c ′ = a + c. The charge of β remains the same as before, or b′ = b, but its potential is no longer zero, but B ′ , and we have to determine the value of B ′ in terms of A and C by eliminating the other quantities entering into the equations.COMPARISON OF CONDENSERS. 108 After discharge, a′ = (P + R + α)y − P B ′ , b′ = (P + Q + β + ξ)B ′ − (P + Q)y, c ′ = (Q + S + γ)y − QB ′ . Hence, the equation a′ + c ′ = a + c becomes (P + R + Q + S + α + γ)y − (P + Q)B ′ = (P + R + α)A + (Q + S + γ)C, and b′ = b becomes (P + Q + β + ξ)B ′ − (P + Q)y = −P A − QC. Eliminating y from these equations, we find B ′ {(P + Q)(R + S) + (P + Q)(α + β + γ + ξ) + (R + S + α + γ)(β + ξ)} = {Q(R + α) − P (S + γ)}(A − C). If, therefore, the electrometer is not disturbed by the discharge, B ′ = 0, and P ∶ Q ∶∶ R + α ∶ S + γ.