Gustav Kirchhoff

While Thomson was investigating the energy stored in connexion with electric currents, the equations of flow of the currents were being generalized by Gustav Kirchhoff (b. 1824, d. 1887).

In 1848 Kirchhoff[52] extended Ohm’s theory of linear conduction to the case of conduction in three dimensions; this could be done without much difficulty by making use of the analogy with the flow of heat, which had proved so useful to Ohm. In Kirchhoff’s memoir a system is supposed to be formed of three-dimensional conductors, through which steady currents are flowing.

At any point let V denote the “tension” or “electroscopic force”—a quantity the significance of which in electrostatics was not yet correctly known. Then, within the substance of any homogeneous conductor, the function V must satisfy Laplace’s equation ∇2V = 0; while at the air-surface of each conductor, the derivate of V taken along the normal must vanish.

At the interface between two conductors formed of different materials, the function V has a discontinuity, which is measured by the value of Volta’s contact force for the two conductors; and, moreover, the condition that the current shall be continuous across such an interface requires that k∂V/∂N shall be continuous, where k denotes the ohmic specific conductivity of the conductor, and ∂/∂N denotes differentiation along the normal to the interface. The equations which have now been mentioned suffice to determine the flow of electricity in the system.

Kirchhoff also showed that the currents distribute themselves in the conductors in such a way as to generate the least possible amount of Joulian heat; as is easily seen, since the quantity of Joulian heat generated in unit time is

where k, as before, denotes the specific conductivity, and this integral has a stationary value when V satisfies the equation

Kirchhoff next applied himself to establish harmony between electrostatical conceptions and the theory of Ohm.

That theory had now been before the world for twenty years, and had been verified by numerous experimental researches; in particular, a careful investigation was made at this time (1848) by Rudolph Kohlrausch (b. 1809, d. 1858), who showed[53] that the difference of the electric “tensions” at the extremities of a voltaic cell, measured electrostatically with the circuit open, was for different cells proportional to the electromotive force measured by the electrodynamic effects of the cell with the circuit closed; and, further,[54] that when the circuit was closed, the difference of the tensions, measured electrostatically, at any two points of the outer circuit was proportional to the ohmic resistance existing between them.

But it was still uncertain how “tension” or “electroscopic force” or “electromotive force” should be interpreted in the language of theoretical electrostatics.

Ohm himself perpetuated a confusion which had originated with Volta. He had:

• identified electroscopic force with density of electric charge, and
• assumed that the electricity in a conductor is at rest when it is distributed uniformly throughout the substance of the conductor.

The uncertainty was finally removed in 1849 by Kirchhoff[55]. He identified Ohm’s electroscopic force with the electrostatic potential.

This is seen by comparing the different expressions which have been obtained for electric energy.

Helmholtz’s expression[56] shows that the energy of a unit charge at any place is proportional to the value of the electrostatic potential at that place.

Joule’s result[57] shows that the energy liberated by a unit charge in passing from one place in a circuit to another is proportional to the difference of the electric tensions at the two places.

It follows that tension and potential are the same thing.

The work of Kirchhoff was followed by several other investigations which belong to the borderland between electrostatics and electrodynamics. One of the first of these was the study of the Leyden jar discharge.

Early in the century Wollaston, in the course of his experiments on the decomposition of water, had observed that when the decomposition is effected by a discharge of static electricity, the hydrogen and oxygen do not appear at separate electrodes; but that at each electrode there is evolved a mixture of the ​gases, as if the current had passed through the water in both directions. After this F. Savary[58] had noticed that the discharge of a Leyden jar magnetizes needles in alternating layers, and had conjectured that “the electric motion during the discharge consists of a series of oscillations.” A similar remark was made in connexion with a similar observation by Joseph Henry (b. 1799, d. 1878), of Washington, in 1842.[59]

“The phenomena,” he wrote, “require us to admit the existence of a principal discharge in one direction, and then several reflex actions backward and forward, each more feeble than the preceding, until equilibrium is restored.” Helmholtz had repeated the same suggestion in his essay on the conservation of energy: and in 1853 W. Thomson[60] verified it, by investigating the mathematical theory of the discharge, as follows:—

Let C denote the capacity of the jar, i.e., the measure of the charge when there is unit difference of potential between the coatings; let R denote the ohmic resistance of the discharging circuit, and L its coefficient of self-induction. Then if at any instant t the charge of the condenser be Q, and the current in the wire be i, we have i = dQ/dt; while Ohm’s law, modified by taking self-induction into account, gives the equation

Eliminating i, we have

an equation which shows that when R2C < 4L, the subsidence of Q to zero is effected by oscillations of period

This simple result may be regarded as the beginning of the theory of electric oscillations. Thomson was at this time much engaged in the problems of submarine telegraphy; and thus lie was led to examine the vexed question of the “velocity of electricity” over long insulated wires and cables. Various workers had made experiments on this subject at different times, but with hopelessly discordant results.

Their attempts had generally taken the form of measuring the interval of time between the appearance of sparks at two spark-gaps in the same circuit, between which a great length of wire intervened, but which were brought near each other in order that the discharges might be seen together, In one series of experiments, performed by Watson at Shooter’s Hill in 1747-8,[61] the circuit was four miles in length, two miles through wire and two miles through the ground; but the discharges appeared to be perfectly simultaneous; whence Watson concluded that the velocity of propagation of electric effects is too great to be measurable.

In 1834 Charles Wheatstone,[62] Professor of Experimental Philosophy in King’s College, London, by examining in a revolving mirror sparks formed at the extremities of a circuit, found the velocity of electricity in a copper wire to be about one and a half times the velocity of light.

In 1850 H. Fizeau and E. Gounelle,[63] I experimenting with the telegraph lines from Paris to Rouen and to Amiens, obtained a velocity about one-third that of light for the propagation of electricity in an iron wire, and nearly two-thirds that of light for the propagation in a copper wire.

The first step towards explaining these discrepancies was made by Faraday, who[64] early in 1854 showed experimentally that a submarine cable, formed of copper wire covered with gutta-percha, “may be assimilated exactly to an immense Leyden battery; the glass of the jars represents the gutta-percha; the internal coating is the surface of the copper wire,” while the outer coating corresponds to the sea-water. It follows that in all calculations relating to the propagation of electric disturbances along submarine cables, the electrostatic capacity of the cable must be taken into account.