A Dynamical Theory of the Electromagnetic Field/Part V

by James Clerk Maxwell

PART V. — THEORY OF CONDENSERS.

Capacity of a Condenser.

(83) The simplest form of condenser consists of a uniform layer of insulating matter bounded by two conducting surfaces, and its capacity is measured by the quantity of electricity on either surface when the difference of potentials is unity.

Let S be the area of either surface, a the thickness of the dielectric, and $k$ its coefficient of electric elasticity; then on one side of the condenser the potential is $\Psi _{1}$ and on the other side $\Psi _{1}+1$, and within its substance

${\frac {d\Psi }{dx}}={\frac {1}{a}}=kf$

(48)

Since ${\tfrac {d\Psi }{dx}}$ and therefore $f$ is zero outside the condenser, the quantity of electricity on its first surface $=-Sf$, and on the second $+Sf$. The capacity of the condenser is therefore $Sf={\tfrac {S}{ak}}$ in electromagnetic measure.

Specific Capacity of Electric Induction (D).

(84) If the dielectric of the condenser be air, then its capacity in electrostatic measure is ${\tfrac {S}{4\pi a}}$ (neglecting corrections arising from the conditions to be fulfilled at the edges). If the dielectric have a capacity whose ratio to that of air is D, then the capacity of the condenser will be ${\tfrac {DS}{4\pi a}}$.

Hence

$D={\frac {k_{0}}{k}}$

(49)

where $k_{0}$ is the value of $k$ in air, which is taken for unity.

Electric Absorption.

(85) When the dielectric of which the condenser is formed is not a perfect insulator, the phenomena of conduction are combined with those of electric displacement. The condenser, when left charged, gradually loses its charge, and in some cases, after being discharged completely, it gradually acquires a new charge of the same sign as the original charge, and this finally disappears. These phenomena have been described by Professor Faraday (Experimental Researches, Series XI.) and by Mr. F. Jenkin (Report of Committee of Board of Trade on Submarine Cables), and may be classed under the name of "Electric Absorption."

(86) We shall take the case of a condenser composed of any number of parallel layers of different materials. If a constant difference of potentials between its extreme surfaces is kept up for a sufficient time till a condition of permanent steady flow of electricity is established, then each bounding surface will have a charge of electricity depending on the nature of the substances on each side of it. If the extreme surfaces be now discharged, these internal charges will gradually be dissipated, and a certain charge may reappear on the extreme surfaces if they are insulated, or, if they are connected by a conductor, a certain quantity of electricity may be urged through the conductor during the reestablishment of equilibrium.

Let the thickness of the several layers of the condenser be $a_{1},a_{2}$, &c.

Let the values of $k$ for these layers be respectively $k_{1},k_{2},k_{3}$, and let

$a_{1}k_{2}+a_{2}k_{2}+etc.=ak\,$

(50)

where $k$ is the "electric elasticity" of air, and $a$ is the thickness of an equivalent condenser of air.

Let the resistances of the layers be respectively $r_{1},r_{2}$, &c, and let $r_{1}+r_{2}+etc.=r$ be the resistance of the whole condenser, to a steady current through it per unit of surface.

Let the electric displacement in each layer be $f_{1},f_{2}$, &c.

Let the electric current in each layer be $p_{1},p_{2}$, &c.

Let the potential on the first surface be $\Psi _{1}$ and the electricity per unit of surface $e_{1}$.

Let the corresponding quantities at the boundary of the first and second surface be $\Psi _{2}$ and $e_{2}$, and so on. Then by equations (G) and (H),

$\left.{\begin{array}{lll}e_{1}=-f_{1},&&{\frac {de_{1}}{dt}}=-p_{1},\\\\e_{2}=f_{1}-f_{2},&&{\frac {de_{2}}{dt}}=p_{1}-p_{2},\\etc.&&etc.\end{array}}\right\}$

(51)

But by equations (E) and (F),

$\left.{\begin{array}{lll}\Psi _{1}-\Psi _{2}&=a_{1}k_{1}f_{1}&=-r_{1}p_{1},\\\Psi _{2}-\Psi _{3}&=a_{2}k_{2}f_{2}&=-r_{2}p_{2},\\etc.&etc.&etc.\end{array}}\right\}$

(52)

After the electromotive force has been kept up for a sufficient time the current becomes the same in each layer, and

$p_{1}=p_{2}=etc.=p={\frac {\Psi }{r}}$

where $\Psi$ is the total difference of potentials between the extreme layers. We have then

$\left.{\begin{array}{lllcl}&f_{1}=-&{\frac {\Psi }{r}}{\frac {r_{1}}{a_{1}k_{1}}},&&f_{2}=-{\frac {\Psi }{r}}{\frac {r_{2}}{a_{2}k_{2}}},\ etc.\\\mathrm {and} \\&e_{1}=&{\frac {\Psi }{r}}{\frac {r_{1}}{a_{1}k_{1}}},&&e_{2}={\frac {\Psi }{r}}\left({\frac {r_{2}}{a_{2}k_{2}}}-{\frac {r_{1}}{ak_{1}}}\right),\ etc.\end{array}}\right\}$

(53)

These are the quantities of electricity on the different surfaces.

(87) Now let the condenser be discharged by connecting the extreme surfaces through a perfect conductor so that their potentials are instantly rendered equal, then the electricity on the extreme surfaces will be altered, but that on the internal surfaces will not have time to escape. The total difference of potentials is now

$\Psi '=a_{1}k_{1}e'_{1}+a_{2}k_{2}\left(e'_{1}+e_{2}\right)+a_{3}k_{3}\left(e'_{1}+e_{2}+e_{3}\right),\ etc.=0$

(54)

whence if $e'_{1}$ is what $e_{1}$ becomes at the instant of discharge,

$e'_{1}={\frac {\Psi }{r}}{\frac {r_{1}}{a_{1}k_{1}}}-{\frac {\Psi }{ak}}=e_{1}-{\frac {\Psi }{ak}}$

(55)

The instantaneous discharge is therefore ${\frac {\Psi }{ak}}$, or the quantity which would be discharged by a condenser of air of the equivalent thickness $a$, and it is unaffected by the want of perfect insulation.

(88) Now let us suppose the connexion between the extreme surfaces broken, and the condenser left to itself, and let us consider the gradual dissipation of the internal charges. Let $\Psi '$ be the difference of potential of the extreme surfaces at any time $t$; then

$\Psi '=a_{1}k_{1}f_{1}+a_{2}k_{2}f_{2}+etc.$

(56)

but

$a_{1}k_{1}f_{1}=-r_{1}{\frac {df_{1}}{dt}}$ $a_{2}k_{2}f_{2}=-r_{2}{\frac {df_{2}}{dt}}$

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,

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

(59)

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

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

(60)

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)

a quantity essentially positive; so that, when the primary electrification is in one direction, the secondary discharge is always in the same direction as the primary discharge^[1].

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1. Since this paper was communicated to the Royal Society, I have seen a paper by M. Gaugain in the Annales de Chimie for 1864, in which he has deduced the phenomena of electric absorption and secondary discharge from the theory of compound condensers.