Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/418

Next, let the circuit $r_{1}$ be broken for a time $t_{2},$

$E(=E_{2})=E_{1}e^{-{\frac {t_{2}}{T_{2}}}}\quad {\mbox{where}}\quad T_{2}=CR.$

(9)

Finally, let the surfaces of the condenser be connected by means of a wire whose resistance is $r_{3}$ for a time $t_{3},$

$E(=E_{3})=E_{2}e^{-{\frac {t_{3}}{T_{3}}}}\quad {\mbox{where}}\quad T_{3}={\frac {CRr_{3}}{R+r_{3}}}.$

(10)

If $Q_{3}$ is the total discharge through this wire in the time $t_{3},$

$Q_{3}=E_{0}{\frac {CR^{2}}{(R+r_{1})(R+r_{3})}}\left(1-e^{-{\frac {t_{1}}{T_{1}}}}\right)e^{-{\frac {t_{2}}{T_{2}}}}\left(1-e^{-{\frac {t_{3}}{T_{3}}}}\right).$

(11)

In this way we may find the discharge through a wire which is made to connect the surfaces of a condenser after being charged for a time $t_{l},$ and then insulated for a time $t_{2}.$ If the time of charging is sufficient, as it generally is, to develope the whole charge, and if the time of discharge is sufficient for a complete discharge, the discharge is

$Q_{3}=E_{0}{\frac {CR^{2}}{(R+r_{1})(R+r_{3})}}e^{-{\frac {t_{2}}{CR}}}.$

(12)

327.] In a condenser of this kind, first charged in any way, next discharged through a wire of small resistance, and then insulated, no new electrification will appear. In most actual condensers, however, we find that after discharge and insulation a new charge is gradually developed, of the same kind as the original charge, but inferior in intensity. This is called the residual charge. To account for it we must admit that the constitution of the dielectric medium is different from that which we have just described. We shall find, however, that a medium formed of a conglomeration of small pieces of different simple media would possess this property.

Theory of a Composite Dielectric.

328.] We shall suppose, for the sake of simplicity, that the dielectric consists of a number of plane strata of different materials and of area unity, and that the electric forces act in the direction of the normal to the strata.

Let $a_{1},\,a_{2},$ &c. be the thicknesses of the different strata.

$X_{l},\,X_{2},$ &c. the resultant electrical force within each stratum.

$p_{1},\,p_{2},$ &c. the current due to conduction through each stratum.

$f_{1},\,f_{2},$ &c. the electric displacement.

$u_{1},u_{2},$ &c. the total current, due partly to conduction and partly to variation of displacement.