Vortex Atoms

THE THEORY OF THE ELECTRIC TELEGRAPH.

[From the Proc. Royal Soc., May, 1855.]

THE following investigation was commenced in consequence of a letter received by the author from Prof. Stokes, dated Oct. 16, 1854. It is now communicated to the Royal Society, although only in an incomplete form, as it may serve to indicate some important practical applications of the theory, especially in esti- mating the dimensions of telegraph wires and cables required for long distances; and the author reserves a more complete develop- ment and illustration of the mathematical parts of the investigation for a paper on the conduction of Electricity and Heat through solids, which he intends to lay before the Royal Society on another occasion.

Extract from a letter to Prof. Stokes, dated Largs, Oct. 28, 1854.

“Let e be the electro-statical capacity per unit of length of the wire; that is, let e be such that cle is the quantity of electricity required to charge a length of the wire up to potential v. In a note communicated as an addition to a paper in the last June Number of the Philosophical Magazine, [Electrostatics and Mag- netism, Art. III.] and I believe at present in the Editor’s hands for publication, I proved that the value of c is

I R 2 log R if I

denote the specific inductive capacity of the gutta-percha, and R, R’ the radii of its inner and outer cylindrical surfaces.

Let k denote the galvanic resistance of the wire in absolute electro-statical measure [per unit of the wire’s length] (see a paper ‘On the application of the Principle of Mechanical Effect to the Measurement of Electro-motive Forces and Galvanic Resistances,” Phil. Mag. Dec. 1851 [Art. LIV. Vol. I. above]).

“Let denote the strength at the time t, of the current (also in electro-statical measure) at a point P of the wire at a distance a from one end which may be called O. Letv denote the potential at the same point P, at the time t

“The potential at the outside of the gutta-percha may be taken as at each instant rigorously zero (the resistance of the water, if the wire be extended as in a submarine telegraph, being certainly incapable of preventing the inductive action from being completed instantaneously round each point of the wire. If the wire be closely coiled, the resistance of the water may possibly produce sensible effects).

“Hence, at the time t, the quantity of electricity on a length da of the wire at P will be veda.

“The quantity that leaves it in the time dt will be dt do dry doc " Hence we must have dv -cd dtdt dy dx.. dt .(1).

“But the electromotive force, in electro-static units, at the point P, is dv da’ and therefore at each instant dv kry=- dx

“Eliminating y from (1) by means of this, we have dv d’e ck di da (2). .(3), which is the equation of electrical excitation in a submarine tele- graph-wire, perfectly insulated by its gutta-percha covering.

“This equation agrees with the well-known equation of the linear motion of heat in a solid conductor; and various forms of solution which Fourier has given are perfectly adapted for answer- ing practical questions regarding the use of the telegraph-wire. Thus first, suppose the wire infinitely long and communicating with the earth at its infinitely distant end: let the end 0 be suddenly raised to the potential V (by being put in communication with the positive pole of a galvanic battery, of which the negative pole is in communication with the ground, the resistance of the battery being small, say not more than a few yards of the wire); let it be kept at that potential for a time T’; and lastly, let it be put in communication with the ground (i.e. suddenly reduced to, and ever afterwards kept at, the zero of potential). An elementary expression for the solution of the equation in this case is VI -=[” dnes sin [2nt — zn13] — sin [(t − T) 2n − zn3] where for brevity, z=x√ko….. n .(4), .(5).”

That this expresses truly the solution with the stated conditions is proved by observing,-1st, that the second member of the equa- tion, (4), is convergent for all positive values of z and vanishes when is infinitely great; 2ndly, that it fulfils the differential equation (3); and 3rdly, that when 20 it vanishes except for values of t between 0 and 7, and for these it is equal to V. It is curious to remark, that we may conclude, by considering the physical circumstances of the problem, that the value of the definite integral in the second member of (4) is zero for all negative values of t, and positive values of z.