Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/410

378 $$R$$ rises from zero to its final value is expressed by a formula of exactly the same kind as that which expresses the value of a current urged by a constant electromotive force through the coil of an electromagnet. Hence we may place a condenser and an electromagnet on two opposite members of Wheatstone's Bridge in such a way that the current through the galvanometer is always zero, even at the instant of making or breaking the battery circuit.

In the figure, let $$P$$, $$Q$$, $$R$$, $$S$$ be the resistances of the four members of Wheatstone's Bridge respectively. Let a coil, whose coefficient of self-induction is $$L$$ be made part of the member $$AH$$, whose resistance is $$Q$$, and let the electrodes of a condenser, whose capacity is $$C$$, be connected by pieces of small resistance with the points $$F$$ and $$Z$$. For the sake of simplicity, we shall assume that there is no current in the galvanometer $$G$$, the electrodes of which are connected to $$F$$ and $$H$$. We have therefore to determine the condition that the potential at $$F$$ may be equal to that at $$H$$. It is only when we wish to estimate the degree of accuracy of the method that we require to calculate the current through the galvanometer when this condition is not fulfilled.

Let $$x$$ be the total quantity of electricity which has passed through the member $$AF$$, and $$z$$ that which has passed through $$FZ$$ at the time $$t$$, then $$x - z$$ will be the charge of the condenser. The electromotive force acting between the electrodes of the condenser is, by Ohm's law, $$R\frac{dz}{dt}$$, so that if the capacity of the condenser is $$C$$,

Let $$y$$ be the total quantity of electricity which has passed through the member $$AH$$, the electromotive force from $$A$$ to $$H$$ must be equal to that from $$A$$ to $$F$$, or

Since there is no current through the galvanometer, the quantity which has passed through $$HZ$$ must be also $$y$$, and we find

Substituting in (2) the value of $$x$$, derived from (1), and com paring with (3), we find as the condition of no current through the galvanometer