Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/436

On the Comparison of Resistances.

345.] If $E$ is the electromotive force of a battery, and $R$ the resistance of the battery and its connexions, including the galvanometer used in measuring the current, and if the strength of the current is $I$ when the battery connexions are closed, and $I_{1},\,I_{2}$ when additional resistances $r_{1},\,r_{2}$ are introduced into the circuit, then, by Ohm's Law,

$E=IR=I_{1}(R+r_{1})=I_{2}(R+r_{2}).$

Eliminating $E,$ the electromotive force of the battery, and $R$ the resistance of the battery and its connexions, we get Ohm's formula

${\frac {r_{1}}{r_{2}}}={\frac {(I-I_{1})I_{2}}{(I-I_{2})I_{1}}}.$

This method requires a measurement of the ratios of $I,\,I_{1}$ and $I_{2},$ and this implies a galvanometer graduated for absolute measurements.

If the resistances $r_{1}$ and $r_{2}$ are equal, then $I_{1}$ and $I_{2}$ are equal, and we can test the equality of currents by a galvanometer which is not capable of determining their ratios.

But this is rather to be taken as an example of a faulty method than as a practical method of determining resistance. The electromotive force $E$ cannot be maintained rigorously constant, and the internal resistance of the battery is also exceedingly variable, so that any methods in which these are assumed to be even for a short time constant are not to be depended on.

346.] The comparison of resistances can be made with extreme

Fig. 30.

accuracy by either of two methods, in which the result is in dependent of variations of $R$ and $E$.