# Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/424

382
[331.
CONDUCTION IN DIELECTRICS.

ance ${\displaystyle R,}$ and let a current be passed along this series from left to right.

Let us first suppose the plates ${\displaystyle B_{0},\,B_{1},\,B_{2},}$ each insulated and free from charge. Then the total quantity of electricity on each of the plates ${\displaystyle B}$ must remain zero, and since the electricity on the plates ${\displaystyle A}$ is in each case equal and opposite to that of the opposed

Fig. 25.

surface they will not be electrified, and no alteration of the current will be observed.

But let the plates ${\displaystyle B}$ be all connected together, or let each be connected with the earth. Then, since the potential of ${\displaystyle A_{1}}$ is positive, while that of the plates ${\displaystyle B}$ is zero, ${\displaystyle A_{1}}$ will be positively electrified and ${\displaystyle B_{1}}$ negatively.

If ${\displaystyle P_{1},\,P_{2},}$ &c. are the potentials of the plates ${\displaystyle A_{1},\,A_{2},}$ &c., and ${\displaystyle C}$ the capacity of each, and if we suppose that a quantity of electricity equal to ${\displaystyle Q_{0}}$ passes through the wire on the left, ${\displaystyle Q_{l}}$ through the connexion ${\displaystyle R_{1},}$ and so on, then the quantity which exists on the plate ${\displaystyle A_{1}}$ is ${\displaystyle Q_{0}-Q_{1},}$ and we have

 ${\displaystyle Q_{0}-Q_{1}=C_{1}P_{1}.}$
 Similarly ${\displaystyle Q_{1}-Q_{2}=C_{2}P_{2},}$

and so on.

But by Ohm's Law we have

 {\displaystyle {\begin{aligned}P_{1}-P_{2}&=R_{1}{\frac {dQ_{1}}{dt}},\\P_{2}-P_{3}&=R_{2}{\frac {dQ_{2}}{dt}}.\end{aligned}}}

If we suppose the values of ${\displaystyle C}$ the same for each plate, and those of ${\displaystyle R}$ the same for each wire, we shall have a series of equations of the form