Kirchhoff's Theory

The theory of signalling by cable originated in a correspondence between Stokes and Thomson in 1854. In the case of long submarine lines, the speed of signalling is so much limited by the electrostatic factor that electro-magnetic induction has no sensible effect; and it was accordingly neglected in the investigation.

In view of other applications of the analysis, however, we shall suppose that the cable has a self-induction L per unit length, and that R denotes the ohmic resistance, and C the capacity per unit length, V the electric potential at a distance x from one terminal, and i the current at this place.

Ohm’s law, as modified for inductance, is expressed by the equation

moreover, since the rate of accumulation of charge in unit length at x is - ∂i/∂x, and since this increases the potential at the rate - (1/C)∂i/∂x, we have

Eliminating i between these two equations, we have

which is known as the equation of telegraphy.[65]

Thomson, in one of his letters[66] to Stokes in 1854, obtained this equation in the form which applies to Atlantic cables, i.e., with the term in L neglected. In this form it is the same as Fourier’s equation for the linear propagation of heat: so that the known solutions of Fourier’s theory may be used in a new interpretation. If we substitute we obtain

and therefore a typical elementary solution of the equation is

The form of this solution shows that if a regular harmonic variation of potential is applied at one end of a cable, the phase is propagated with a velocity which is proportional to the square root of the frequency of the oscillations: since therefore the different harmonics are propagated with different velocities, it is evident that no definite “velocity of transmission” is to be expected for ordinary signals. If a potential is suddenly applied at one end of the cable, a certain time elapses before the current at the other end attains a definite percentage of its maximum value; but it may easily be shown[67] that this retardation is proportional to the square of the length of the cable, so that the apparent velocity of propagation would be less, the greater the length of cable used.

The case of a telegraph line insulated in the air on poles is different from that of a cable; for here the capacity is small, and it is necessary to take into account the inductance. If in the general equation of telegraphy we write

we obtain the equation

as the capacity is small, we may replace the quantity under the radical by its second term: and thus we see that a typical elementary solution of the equation is

this shows that any harmonic disturbance, and therefore any disturbance whatever, is propagated along the wire with velocity (CL)- .

The difference between propagation in an aerial wire and propagation in an oceanic cable is, as Thomson remarked, similar to the difference between the propagation of an impulsive pressure through a long column of fluid in a tube when the tube is rigid (case of the aerial wire) and when it is elastic, so as to be capable of local distension (case of the cable, the distension corresponding to the effect of capacity): in the former case, as is well known, the impulse is propagated with a definite velocity, namely, the velocity of sound in the fluid.

The work of Thomson on signalling along cables was followed in 1857 by a celebrated investigation[68] of Kirchhoff’s, on the propagation of electric disturbance along an aerial wire of circular cross-section.

Kirchhoff assumed that the electric charge is practically all resident on the surface of the wire, and that the current is uniformly distributed over its cross-section; his idea of the current was the same as that of Fechner and Weber, namely, that it consists of equal streams of vitreous and resinous electricity flowing in opposite directions. Denoting the electric potential by V, the charge per unit length of wire by e, the length of the wire by l, and the radius of its cross-section by α, he showed that V is determined approximately by the equation[69]

The next factor to be considered is the mutual induction of the current-elements in different parts of the wire. Assuming with Weber that the electromotive force induced in an clement ds due to another element ds′ carrying a current i′ is derivable from a vector-potential

Kirchhoff found for the vector-potential due to the entire wire the approximate value

where i denotes the strength of the current;[70] the vector-potential being directed parallel to the wire. Ohm’s law then gives the equation where k denotes the specific conductivity of the material of which the wire is composed; and finally the principle of conservation of electricity gives the equation

Denoting log (l/α) by γ, and eliminating e, i, w from these four equations, we have

which is, as might have been expected, the equation of telegraphy. When the term in a ∂V/∂t is ignored, as we have seen is in certain cases permissible, the equation becomes

which shows that the electric disturbance is propagated along the wire with the velocity c.[71] Kirchhoff’s procedure has, in fact, involved the calculation of the capacity and self-induction of the wire, and is thus able to supply the definite values of the quantities which were left undetermined in the general equation of telegraphy.

The velocity c, whose importance was thus demonstrated, has already been noticed in connexion with Weber’s law of force; it is a factor of proportionality, which must be introduced when electrodynamic phenomena are described in terms of units which have been defined electrostatically,[72] or conversely when units which have been defined electrodynamically[73] are used in the description of electrostatic phenomena.

The factor which is introduced on such occasions must be of the dimensions (length/time), may be easily seen: for the electrostatic repulsion between electric charges is a quantity of the same kind as the electrodynamic repulsion between two definite lengths of wire, carrying currents which may be specified by the amount of charge which travels past any point in unit time.

Shortly before the publication of Kirchhoff’s memoir, the value of c had been determined by Weber and Kohlrausch;[74] their determination rested on a comparison of the measures of the charge of a Leyden jar, as obtained by a method depending on electrostatic attraction, and by a method depending on the magnetic effects of the current produced by discharging the jar. The resulting value was nearly

c = 3·1 × 1010 cm./sec.;

which was the same, within the limits of the errors of measurement, as the speed with which light travels in interplanetary space. The coincidence was noticed by Kirchhoff, who was thus the first to discover the important fact that the velocity with which an electric disturbance is propagated along a perfectlyconducting aerial wire is equal to the velocity of light.

In a second memoir published in the same year, Kirchhoff[75] extended the equations of propagation of electric disturbance to the case of three-dimensional conductors.

As in his earlier investigation, he divided the electromotive force at any point into two parts, of which one is the gradient of the electrostatic potential φ, and the other is the derivate with respect to the time (with sign reversed) of a vectorpotential a; so that if i denote the current and k the specific conductivity, Ohm’s law is expressed by the equation

Kirchhoff calculated the value of a by aid of Weber’s formula for the inductive action of one current element on another; the result is

where r denotes the vector from the point (x, y, z), at which a is measured, to any other point (x′, y′, z′) of the conductor, at which the current is i′; and the integration is extended over the whole volume of the conductor. The remaining general equations are the ordinary equation of the electrostatic potential

(where ρ denotes the density of electric charge), and the equation of conservation of electricity

It will be seen that Kirchhoff’s electrical researches were greatly influenced by those of Weber.

The latter investigations, however, did not enjoy unquestioned authority; for there was still a question as to whether the expressions given by Weber for the mutual energy of two current elements, and for the mutual energy of two electrons, were to be preferred to the rival formulae of Neumann and Riemann. The matter was examined in 1870 by Helmholtz, in a series of memoirs[76] to which reference has already been made.[77] Helmholtz remarked that, for two elements ds, ds′, carrying currents i, i′, the electrodynamic energy is

according to Neumann, and

according to Weber; and that these expressions differ from each other only by the quantity

since this vanishes when integrated round either circuit, the two formulae give the same result when applied to entire currents. A general formula including both that of Neumann and that of Weber is evidently

where k denotes an arbitrary constant.[78]

Helmholtz’s result suggested to Clausius[79] a new form for the law of force between electrons; namely, that which is ​obtained by supposing that two electrons of charges e, e′, and velocities v, v′, possess electrokinetic energy of amount

Subtracting from this the mutual electrostatic potential energy, which is ee′c2/r, we may write the mutual kinetic potential of the two electrons in the form

where (x, y, z) denote the coordinates of e, and (x′, y′, z′) those of e′.

The unknown constant k has clearly no influence so long as closed circuits only are considered: if k be replaced by zero, the expression for the kinetic potential becomes

which, as will appear later, closely resembles the corresponding expression in the modern theory of electrons.

Clausius’ formula has the great advantage over Weber’s, that it does not compel us to assume equal and opposite velocities for the vitreous and resinous charges in an electric current; on the other hand, Clausius’ expression involves the absolute velocities of the electrons, while Weber’s depends only on their relative motion; and therefore Clausius’ theory requires the assumption of a fixed aether in space, to which the velocities v and v′ may be referred.

When the behaviour of finite electrical systems is predicted from the formulae of Weber, Riemann, and Clausius, the three laws do not always lead to concordant results.

For instance, if a circular current be rotated with constant angular velocity round its axis, according to Weber’s law there would be a development of free electricity on a stationary conductor in the neighbourhood; whereas, according to Clausius’ formula there would be no induction on a stationary body, but electrification would appear on a body turning with the circuit as if rigidly connected with it.

Let a magnet be suspended within a hollow metallic body, and let the hollow body be suddenly charged or discharged; then, according to Clausius’ theory, the magnet is unaffected; but according to Weber’s and Riemann’s theories it experiences an impulsive couple.

If an electrified disk be rotated in its own plane, under certain circumstances a steady current will be induced in a neighbouring circuit according to Weber’s law, but not according to the other formulae.

An interesting objection to Clausius’ theory was brought forward in 1879 by Fröhlich. When a charge of free electricity and a constant electric current are at rest relatively to each other, but partake together of the translatory motion of the earth in space, a force should act between them if Clausius’ law were true.

Budde showed that the circuit itself acquires an electrostatic charge, partly as a result of the same action which causes the force on the external conductor, and partly as a result of electrostatic induction by the charge on the external conductor; and that the total force between the circuit and external conductor is thus reduced to zero.[83]