Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/408

 

galvanometer has been obtained, first with the condenser and commutator, and then with a coil of resistance $R_{1}$ in its place, then the quantity ${\frac {T}{2C}}$ will be measured by the resistance of the circuit of which the coil $R_{1}$ forms part, and which is completed by the remainder of the conducting system including the battery. Hence the resistance, $R$, which we have to calculate, is equal to $R_{1}$, that of the resistance coil, together with $R_{2}$, the resistance of the remainder of the system (including the battery), the extremities of the resistance coil being taken as the electrodes of the system.

In the cases of the differential galvanometer and Wheatstone's Bridge it is not necessary to make a second experiment by substituting a resistance coil for the condenser. The value of the resistance required for this purpose may be found by calculation from the other known resistances in the system.

Using the notation of Art. 347, and supposing the condenser and commutator substituted for the conductor $AC$ in Wheatstone's Bridge, and the galvanometer inserted in $OA$, and that the deflexion of the galvanometer is zero, then we know that the resistance of a coil, which placed in $AC$ would give a zero deflexion, is

$b={\frac {c\gamma }{\beta }}=R_{1}$.

(3)

The other part of the resistance, $R_{2}$, is that of the system of conductors $AO$, $OC$, $AB$, $BC$ and $OB$, the points $A$ and $C$ being considered as the electrodes. Hence

$R_{2}={\frac {\beta (c+a)(\gamma +\alpha )+ca(\gamma +\alpha )+\gamma \alpha (c+a)}{(c+a)(\gamma +\alpha )+\beta (c+a+\gamma +\alpha )}}$.

(4)

In this expression $a$ denotes the internal resistance of the battery and its connexions, the value of which cannot be determined with certainty; but by making it small compared with the other resistances, this uncertainty will only slightly affect the value of $R_{2}$.

The value of the capacity of the condenser in electromagnetic measure is

$C={\frac {t}{2\left(R_{1}+R_{1}\right)}}$.

(5)

777.] If the condenser has a large capacity, and the commutator is very rapid in its action, the condenser may not be fully discharged at each reversal. The equation of the electric current during the discharge is

$Q+R_{2}C{\frac {dQ}{dt}}+EC=0$,

(6)

where $Q$ is the charge, $C$ the capacity of the condenser, $R_{2}$ the