A Treatise on Electricity and Magnetism/Part II/Chapter VI

by James Clerk Maxwell

CHAPTER VI.

LINEAR ELECTRIC CURRENTS.

On Systems of Linear Conductors.

273.] Any conductor may be treated as a linear conductor if it is arranged so that the current must always pass in the same manner between two portions of its surface which are called its electrodes. For instance, a mass of metal of any form the surface of which is entirely covered with insulating material except at two places, at which the exposed surface of the conductor is in metallic contact with electrodes formed of a perfectly conducting material, may be treated as a linear conductor. For if the current be made to enter at one of these electrodes and escape at the other the lines of flow will be determinate, and the relation between electromotive force, current and resistance will be expressed by Ohm's Law, for the current in every part of the mass will be a linear function of $E$. But if there be more possible electrodes than two, the conductor may have more than one independent current through it, and these may not be conjugate to each other. See Art. 282.

Ohm's Law.

274.] Let $E$ be the electromotive force in a linear conductor from the electrode $A_{1}$ to the electrode $A_{2}$. (See Art. 69.) Let $C$ be the strength of the electric current along the conductor, that is to say, let $C$ units of electricity pass across every section in the direction $A_{1}A_{2}$ in unit of time, and let $R$ be the resistance of the conductor, then the expression of Ohm's Law is

$E=CR.$

(1)

Linear Conductors arranged in Series.

275.] Let $A_{1}$, $A_{2}$ be the electrodes of the first conductor and let the second conductor be placed with one of its electrodes in contact with $A_{2},$ so that the second conductor has for its electrodes $A_{2},\,$ $A_{3}$. The electrodes of the third conductor may be denoted by $A_{3}$ and $A_{4}$.

Let the electromotive force along each of these conductors be denoted by $E_{12},$ $E_{23},$ $E_{34}$, and so on for the other conductors.

Let the resistance of the conductors be

$R_{12},\;\;R_{23},\;\;R_{34},\;\;\mathrm {\&c} .$

Then, since the conductors are arranged in series so that the same current $C$ flows through each, we have by Ohm's Law,

$E_{12}=CR_{12},\;\;E_{23}=CR_{23},\;\;E_{34}=CR_{34}.$

(2)

If $E$ is the resultant electromotive force, and $R$ the resultant resistance of the system, we must have by Ohm's Law,

$E=CR.$

(3)

Now

${\begin{array}{rl}E&=E_{12}+E_{23}+E_{34}\\&\quad \quad \quad \quad {\mbox{the sum of the separate electromotive forces,}}\\&=C(R_{12}+R_{23}+R_{34})\quad {\mbox{by equations (2).}}\end{array}}$

(4)

Comparing this result with (3), we find

$R=R_{12}+R_{23}+R_{34}.$

(5)

Or, the resistance of a series of conductors is the sum of the resistances of the conductors taken separately.

Potential at any Point of the Series.

Let $A$ and $C$ be the electrodes of the series, $B$ a point between them, $a$, $c$, and $b$ the potentials of these points respectively. Let $R_{1}$ be the resistance of the part from $A$ to $B,\,$ $R_{2}$ that of the part from $B$ to $C$, and $R$ that of the whole from $A$ to $C$, then, since

$a-b=R_{1}C,\;\;\quad b-c=R_{2}C,\;\;\quad$ and $\quad \;\;a-c=RC$,

the potential at $B$ is

$b={\frac {R_{2}a+R_{1}c}{R}}$,

(6)

which determines the potential at $B$ when those at $A$ and $C$ are given.

Resistance of a Multiple Conductor.

276.] Let a number of conductors $ABZ,\,$ $ACZ,\,$ $ADZ$ be arranged side by side with their extremities in contact with the same two points $A$ and $Z$. They are then said to be arranged in multiple arc.

Let the resistances of these conductors be $R_{1},\,$ $R_{2},\,$ $R_{3}$ respectively, and the currents $C_{1},C_{2},C_{3},$ and let the resistance of the multiple conductor be $R,$ and the total current $C$. Then, since the potentials at $A$ and $Z$ are the same for all the conductors, they have the same difference, which we may call $E$. We then have

$E=C_{1}R_{1}=C_{2}R_{2}=C_{3}R_{3}=CR,$

but

$C=C_{l}+C_{2}+C_{3},$

whence

${\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+{\frac {1}{R_{3}}}$

(7)

Or, the reciprocal of the resistance of a multiple conductor is the sum of the reciprocals of the component conductors.

If we call the reciprocal of the resistance of a conductor the conductivity of the conductor, then we may say that the conductivity of a multiple conductor is the sum of the conductivities of the component conductors.

Current in any Branch of a Multiple Conductor.

From the equations of the preceding article, it appears that if $C_{1}$ is the current in any branch of the multiple conductor, and $R_{l}$ the resistance of that branch,

$C_{1}=C{\frac {R}{R_{1}}},$

(8)

where $C$ is the total current, and $R$ is the resistance of the multiple conductor as previously determined.

Longitudinal Resistance of Conductors of Uniform Section.

277.] Let the resistance of a cube of a given material to a current parallel to one of its edges be $\rho$, the side of the cube being unit of length, $\rho$ is called the 'specific resistance of that material for unit of volume.'

Consider next a prismatic conductor of the same material whose length is $l,$ and whose section is unity. This is equivalent to $l$ cubes arranged in series. The resistance of the conductor is therefore $l\rho$.

Finally, consider a conductor of length $l$ and uniform section $s$. This is equivalent to s conductors similar to the last arranged in multiple arc. The resistance of this conductor is therefore

$R={\frac {l\rho }{s}}.$

When we know the resistance of a uniform wire we can determine the specific resistance of the material of which it is made if we can measure its length and its section.

The sectional area of small wires is most accurately determined by calculation from the length, weight, and specific gravity of the specimen. The determination of the specific gravity is sometimes inconvenient, and in such cases the resistance of a wire of unit length and unit mass is used as the 'specific resistance per unit of weight.'

If $r$ is this resistance, $l$ the length, and $m$ the mass of a wire, then

$R={\frac {l^{2}r}{m}}.$

On the Dimensions of the Quantities involved in these Equations.

278.] The resistance of a conductor is the ratio of the electromotive force acting on it to the current produced. The conductivity of the conductor is the reciprocal of this quantity, or in other words, the ratio of the current to the electromotive force producing it.

Now we know that in the electrostatic system of measurement the ratio of a quantity of electricity to the potential of the conductor on which it is spread is the capacity of the conductor, and is measured by a line. If the conductor is a sphere placed in an unlimited field, this line is the radius of the sphere. The ratio of a quantity of electricity to an electromotive force is therefore a line, but the ratio of a quantity of electricity to a current is the time during which the current flows to transmit that quantity. Hence the ratio of a current to an electromotive force is that of a line to a time, or in other words, it is a velocity.

The fact that the conductivity of a conductor is expressed in the electrostatic system of measurement by a velocity may be verified by supposing a sphere of radius $r$ charged to potential $V,$ and then connected with the earth by the given conductor. Let the sphere contract, so that as the electricity escapes through the conductor the potential of the sphere is always kept equal to $V.$ Then the charge on the sphere is $rV$ at any instant, and the current is ${\frac {d}{dt}}(rV),$ but, since $V$ is constant, the current is ${\frac {dr}{dt}}V,$ and the electromotive force through the conductor is $V.$

The conductivity of the conductor is the ratio of the current to the electromotive force, or ${\frac {dr}{dt}}$, that is, the velocity with which the radius of the sphere must diminish in order to maintain the potential constant when the charge is allowed to pass to earth through the conductor.

In the electrostatic system, therefore, the conductivity of a conductor is a velocity, and of the dimensions $[LT^{-1}].$

The resistance of the conductor is therefore of the dimensions $[L^{-1}T].$

The specific resistance per unit of volume is of the dimension of $[T],$ and the specific conductivity per unit of volume is of the dimension of $[T^{-1}].$

The numerical magnitude of these coefficients depends only on the unit of time, which is the same in different countries.

The specific resistance per unit of weight is of the dimensions $[L^{-3}MT].$

279.] We shall afterwards find that in the electromagnetic system of measurement the resistance of a conductor is expressed by a velocity, so that in this system the dimensions of the resistance of a conductor are $[LT^{-1}].$

The conductivity of the conductor is of course the reciprocal of this.

The specific resistance per unit of volume in this system is of the dimensions $[L^{2}T^{-1}],$ and the specific resistance per unit of weight is of the dimensions $[L^{-1}T^{-1}M].$

On Linear Systems of Conductors in general.

280.] The most general case of a linear system is that of $n$ points, $A_{1},A_{2},\ldots A_{n},$ connected together in pairs by ${\frac {1}{2}}n(n-1)$ linear conductors. Let the conductivity (or reciprocal of the resistance) of that conductor which connects any pair of points, say $A_{p}$ and $A_{q},$, be called $K_{pq},$ and let the current from $A_{p}$ to $A_{q}$ be $C_{pq}$. Let $P_{p}$ and $P_{q}$ be the electric potentials at the points $A_{p}$ and $A_{q}$ respectively, and let the internal electromotive force, if there be any, along the conductor from $A_{p}$ to $A_{q}$ be $E_{pq}.$

The current from $A_{p}$ to $A_{q}$ is, by Ohm's Law,

$C_{pq}=K_{pq}(P_{p}-P_{q}+E_{pq}).$

(1)

Among these quantities we have the following sets of relations:

The conductivity of a conductor is the same in either direction,

or

$K_{pq}=K_{qp}.$

(2)

The electromotive force and the current are directed quantities, so that

$E_{pq}=-E_{qp},\quad \quad$and $\quad \quad C_{pq}=-C_{qp}.$

(3)

Let $P_{1},P_{2},\ldots P_{n}$ be the potentials at $A_{1},A_{2},\ldots A_{n}$ respectively, and let $Q_{1},Q_{2},\ldots Q_{n}$ be the quantities of electricity which enter the system in unit of time at each of these points respectively. These are necessarily subject to the condition of 'continuity'

$Q_{1}+Q_{2}\ldots +Q_{n}=0,$

(4)

since electricity can neither be indefinitely accumulated nor produced within the system.

The condition of 'continuity' at any point $A_{p}$ is

$Q_{p}=C_{p1}+C_{p2}+\mathrm {\&c.} +C_{pn},$

(5)

Substituting the values of the currents in terms of equation (1), this becomes

${\begin{aligned}Q_{p}=(K_{p1}+K_{p2}+\mathrm {\&c.} +K_{pn})P_{p}-(K_{p1}P_{1}+K_{p2}P_{2}+\mathrm {\&c.} +K_{pn}P_{n})\\+(K_{pq}E_{p1}+\mathrm {\&c.} +K_{pn}E_{pn}).\end{aligned}}$

(6)

The symbol $K_{pp}$ does not occur in this equation. Let us therefore give it the value

$K_{pp}=-(K_{p1}+K_{p2}+\mathrm {\&c.} +K_{pn});$

(7)

that is, let $K_{pp}$ be a quantity equal and opposite to the sum of all the conductivities of the conductors which meet in $A_{p}$. We may then write the condition of continuity for the point $A_{p}$,

${\begin{aligned}K_{p1}P_{1}+K_{p2}+\mathrm {\&c.} +K_{pn}P_{n}\qquad \qquad \\=K_{p1}E_{p1}+\mathrm {\&c.} +K_{pn}E_{pn}-Q_{p}.\end{aligned}}$

(8)

By substituting 1, 2, &c. n for p in this equation we shall obtain n equations of the same kind from which to determine the n potentials $P_{1},P_{2},{\&c.},P_{n}.$

Since, however, there is a necessary condition, (4), connecting the values of $Q,$ there will be only $n-1$ independent equations. These will be sufficient to determine the differences of the potentials of the points, but not to determine the absolute potential of any. This, however, is not required to calculate the currents in the system.

If we denote by $D$ the determinant

$D={\begin{array}{|lll|}K_{11},&K_{12},&\ldots \ldots K_{1(n-1)},\\K_{21},&K_{22},&\ldots \ldots K_{2(n-1)},\\----&----&--------\\K_{(n-1)1},&K_{(n-1)2},&\ldots \ldots K_{(n-1)(n-1)};\end{array}}$

(9)

and by $D_{pq},$ the minor of $K_{pq},$ we find for the value of $P_{p}-P_{n},$

${\begin{aligned}(P_{p}-P_{n})D=(K_{12}E_{12}+\mathrm {\&c.} -Q_{1})D_{p1}+(K_{21}E_{21}+\mathrm {\&c.} -Q_{2})D_{p2}+\mathrm {\&c.} \\+(K_{q1}E_{q1}+\mathrm {\&c.} +K_{qn}E_{qn}-Q_{q})D_{pq}+\mathrm {\&c.} \qquad \end{aligned}}$

(10)

In the same way the excess of the potential of any other point, say $A_{q},$ over that of $A_{n}$ may be determined. We may then determine the current between $A_{p}$ and $A_{q}$ from equation (1), and so solve the problem completely.

281.] We shall now demonstrate a reciprocal property of any two conductors of the system, answering to the reciprocal property we have already demonstrated for statical electricity in Art. 88.

The coefficient of $Q_{q}$ in the expression for $P_{p}$ is ${\frac {D_{pq}}{D}}.$ That of $Q_{p}$ in the expression for $P_{q}$ is ${\frac {D_{qp}}{D}}.$

Now $D_{pq}$ differs from $D_{qp}$ only by the substitution of the symbols such as $K_{qp}$ for $K_{pq}$. But, by equation (2), these two symbols are equal, since the conductivity of a conductor is the same both ways.

Hence

$D_{pq}=D_{qp}.$

(11)

It follows from this that the part of the potential at $A_{p}$ arising from the introduction of a unit current at $A_{q}$ is equal to the part of the potential at $A_{q}$ arising from the introduction of a unit current at $A_{p}.$

We may deduce from this a proposition of a more practical form.

Let $A,\,B,\,C,\,D$ be any four points of the system, and let the effect of a current $Q,$ made to enter the system at $A$ and leave it at $B,$ be to make the potential at $C$ exceed that at $D$ by $P$. Then, if an equal current $Q$ be made to enter the system at $C$ and leave it at $D,$ the potential at $A$ will exceed that at $B$ by the same quantity $P$.

We may also establish a property of a similar kind relating to the effect of the internal electromotive force $E_{rs},$ acting along the conductor which joins the points $A_{r}$ and $A_{s}$ in producing an external electromotive force on the conductor from $A_{p}$ to $A_{q},$ that is to say, a difference of potentials $P_{p}-P_{q}.$ For since

$E_{rs}=-E_{sr},$

the part of the value of $P_{p}$ which depends on this electromotive force is

${\frac {1}{D}}(D_{pr}-D_{ps})E_{rs},$

and the part of the value of $P_{q}$ is

${\frac {1}{D}}(D_{qr}-D_{qs})E_{rs}.$

Therefore the coefficient of $E_{rs}$ in the value of $P_{p}-P-q$ is

${\frac {1}{D}}{D_{pr}+D+{qs}-D_{ps}-D_{qr}}.$

(12)

This is identical with the coefficient of $E_{pq}$ in the value of $P_{r}-P_{s}.$

If therefore an electromotive force $E$ be introduced, acting in the conductor from $A$ to $B,$ and if this causes the potential at $C$ to exceed that at $D$ by $P,$ then the same electromotive force $E$ introduced into the conductor from $C$ to $D$ will cause the potential at $A$ to exceed that at $B$ by the same quantity $P.$

The electromotive force $E$ may be that of a voltaic battery introduced between the points named, care being taken that the resistance of the conductor is the same before and after the introduction of the battery.

282.] If

$D_{pr}+D_{qs}-D_{ps}-D_{qr}=0,$

(13)

the conductor $A_{p}A_{q}$ is said to be conjugate to $A_{r}A_{s},$ and we have seen that this relation is reciprocal.

An electromotive force in one of two conjugate conductors produces no electromotive force or current along the other. We shall find the practical application of this principle in the case of the electric bridge.

The theory of conjugate conductors has been investigated by Kirchhoff, who has stated the conditions of a linear system in the following manner, in which the consideration of the potential is avoided.

(1) (Condition of 'continuity.') At any point of the system the sum of all the currents which flow towards that point is zero.

(2) In any complete circuit formed by the conductors the sum of the electromotive forces taken round the circuit is equal to the sum of the products of the current in each conductor multiplied by the resistance of that conductor.

We obtain this result by adding equations of the form (1) for the complete circuit, when the potentials necessarily disappear.

Heat Generated in the System.

283.] The mechanical equivalent of the quantity of heat generated in a conductor whose resistance is $R$ by a current $C$ in unit of time is, by Art. 242,

$JH=RC^{2}.$.

(14)

We have therefore to determine the sum of such quantities as $RC^{2}$ for all the conductors of the system.

For the conductor from $A_{p}$ to $A_{q}$ the conductivity is $K_{pq},$ and the resistance $R_{pq},$ where

$K_{pq}\cdot R_{pq}=1.$

(15)

The current in this conductor is, according to Ohm's Law,

$C_{pq}=K_{pq}(P_{p}-P_{q}).$

(16)

We shall suppose, however, that the value of the current is not that given by Ohm's Law, but $X_{pq},$ where

$X_{pq}=C_{pq}+Y_{pq},$

(17)

To determine the heat generated in the system we have to find the sum of all the quantities of the form

$R_{pq}X_{pq}^{2},$

or

$JH=\Sigma {R_{pq}C_{pq}^{2}+2R_{pq}C_{pq}Y_{pq}+R_{pq}Y_{pq}^{2}}.$

(18)

Giving $C_{pq}$ its value, and remembering the relation between $K_{pq}$ and $R_{pq},$ this becomes

$\Sigma (P_{p}-P_{q})(C_{pq}+2Y_{pq})+R_{pq}Y_{pq}^{2}.$

(19)

Now since both $C$ and $X$ must satisfy the condition of continuity at $A_{p},$ we have

$Q_{p}=C_{p1}+C_{p2}+\mathrm {\&c.} +C_{pn},$

(20)

$Q_{p}=X_{p1}+X_{p2}+\mathrm {\&c.} +X_{pn},$

(21)

therefore

$0=Y_{p1}+Y_{p2}+\mathrm {\&c.} +Y_{pn}.$

(22)

Adding together therefore all the terms of (19), we find

$\Sigma (R_{pq}X_{pq}^{2})=\Sigma P_{p}Q_{p}+\Sigma R_{pq}Y_{pq}^{2}$

(23)

Now since $R$ is always positive and $Y^{2}$ is essentially positive, the last term of this equation must be essentially positive. Hence the first term is a minimum when $Y$ is zero in every conductor, that is, when the current in every conductor is that given by Ohm's Law.

Hence the following theorem:

284.] In any system of conductors in which there are no internal electromotive forces the heat generated by currents distributed in accordance with Ohm's Law is less than if the currents had been distributed in any other manner consistent with the actual conditions of supply and outflow of the current.

The heat actually generated when Ohm's Law is fulfilled is mechanically equivalent to $\Sigma P_{p}Q_{q},$ that is, to the sum of the products of the quantities of electricity supplied at the different external electrodes, each multiplied by the potential at which it is supplied.