Page:A Dynamical Theory of the Electromagnetic Field.pdf/38

Hence $f_{1}=A_{1}e^{-{\frac {a_{1}k_{1}}{r_{1}}}t},\ f_{2}=A_{2}e^{-{\frac {a_{2}k_{2}}{r_{2}}}t}$, &c.; and by referring to the values of $e'_{1},e_{2}$ &c., we find

$\left.{\begin{array}{l}A_{1}={\frac {\Psi }{r}}{\frac {r_{1}}{a_{1}k_{1}}}-{\frac {\Psi }{ak}}\\\\A_{2}={\frac {\Psi }{r}}{\frac {r_{2}}{a_{2}k_{2}}}-{\frac {\Psi }{ak}}\\\And \!\!\!c.;\end{array}}\right\}$

(57)

so that we find for the difference of extreme potentials at any time,

$\Psi '=\Psi \left\{\left({\frac {r_{1}}{r}}-{\frac {a_{1}k_{1}}{ak}}\right)e^{-{\frac {a_{1}k_{1}}{r_{1}}}t}+\left({\frac {r_{2}}{r}}-{\frac {a_{2}k_{2}}{ak}}\right)e^{-{\frac {a_{2}k_{2}}{r_{2}}}t}+\And \!\!\!c.\right\}$

(58)

(89) It appears from this result that if all the layers are made of the same substance, $\Psi '$ will be zero always. If they are of different substances, the order in which they are placed is indifferent, and the effect will be the same whether each substance consists of one layer, or is divided into any number of thin layers and arranged in any order among thin layers of the other substances. Any substance, therefore, the parts of which are not mathematically homogeneous, though they may be apparently so, may exhibit phenomena of absorption. Also, since the order of magnitude of the coefficients is the same as that of the indices, the value of $\Psi '$ can never change sign, but must start from zero, become positive, and finally disappear.

(90) Let us next consider the total amount of electricity which would pass from the first surface to the second, if the condenser, after being thoroughly saturated by the current and then discharged, has its extreme surfaces connected by a conductor of resistance R. Let $p$ be the current in this conductor; then, during the discharge,

$\Psi '=p_{1}r_{1}+p_{2}r_{2}+\And \!\!\!c.=pR\,$

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Integrating with respect to the time, and calling $q_{1},q_{2},q$ the quantities of electricity which traverse the different conductors,

$q_{1}r_{1}+q_{2}r_{2}+\And \!\!\!c.=qR$

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The quantities of electricity on the several surfaces will be

${\begin{array}{l}e'_{1}-q-q_{1},\\e_{2}+q_{1}-q_{2}\\\And \!\!\!c.;\end{array}}$

and since at last all these quantities vanish, we find

${\begin{array}{l}q_{1}=e'_{1}-q,\\q_{2}=e'_{1}+e_{2}-q;\end{array}}$

whence

$qR={\frac {\Psi }{r}}\left({\frac {r_{1}^{2}}{a_{1}k_{1}}}+{\frac {r_{2}^{2}}{a_{2}k_{2}}}+\And \!\!\!c.\right)-{\frac {\Psi r}{ak}},$

or

$q={\frac {\Psi }{akrR}}\left\{a_{1}k_{1}a_{2}k_{2}\left({\frac {r_{1}}{a_{1}k_{1}}}-{\frac {r_{2}}{a_{2}k_{2}}}\right)^{2}+a_{2}k_{2}a_{3}k_{3}\left({\frac {r_{2}}{a_{2}k_{2}}}-{\frac {r_{3}}{a_{3}k_{3}}}\right)^{2}+\And \!\!\!c.\right\}$

(61)