Page:Motion of Electrification through a Dielectric.djvu/9

of displacement externally upon a coaxial cylindrical surface; we then produce a real electromagnetic plane wave with electrification, and of finite energy. We have supposed the electrification to be carried through the dielectric at speed ν, to keep up with the wave, which would of course break up if the charge were stopped. But if perfectly conducting surfaces be given on which to terminate the displacement, the natural motion of the wave will itself carry the electrification along them. In fact we now have the rudimentary telegraph-circuit, with no allowance made for absorption of energy in the wires, and the consequent distortion. If the conductors be not coaxial, we only alter the distribution of the displacement and induction, without affecting the propagation without distortion.^[1]

If we now make the medium conduct electrically, and likewise magnetically, with equal rates of subsidence, we shall have the same solutions, with a time-factor ε^-ρt producing ultimate subsidence to zero; and, with only the real electric conductivity in the medium the wave is running through, it will approximately cancel the distortion produced by the resistance of the wires the wave is passing over when this resistance has a certain value.^[2] We should notice, however, that it could not do so perfectly, even if the magnetic retardation in the wires due to diffusion were zero; because in the case of the unreal magnetic conductivity its correcting influence is where it is wanted to be, in the body of the wave; whereas in the case of the wires, their resistance, correcting the distortion due to the external conductivity, is outside the wave; so that we virtually assume instantaneous propagation laterally from the wires of their correcting influence in the elementary theory of propagation along a telegraph-circuit which is symbolized by the equations

$\scriptstyle {-dV/dz=(R+Lp)C,\qquad -dC/dz=(K+Sp)V,\quad \cdots \cdots \cdots \cdots {\text{(32)}}}$

where R, L, K, and S are the resistance, inductance, leakage-conductance, and permittance per unit length of circuit, C the current, and V what I, for convenience, term the potential-difference, but which I have expressly disclaimed^[3] to represent the electrostatic difference of potential, and have shown to represent the transverse voltage or line-integral of the electric force across the circuit from wire to wire, including the electric force of inertia. Now in case of great distortion, as in a long submarine cable, this V approximates towards the electrostatic potential-difference, which it is in Sir W. Thomson's diffusion theory; but in case of little distortion, as in telephony through circuits of low resistance and large inductance, there may be a wide difference between my V and that of the electrostatic force. Consider, for instance, the extreme case of an isolated plane-wave disturbance with no spreading-out of the tubes of displacement. At the boundaries of the

1. The Electrician, Jan. 10, 1885. [p. 440, vol. I.]. Also "Self-Induction of Wires." Part iv. Phil. Mag., Nov. 1886. [p. 221, vol. II.].
2. "Electromagnetic Waves," § 6, Phil. Mag. Feb. 1888 [p. 379, vol. II.]. The Electrician, June 1887 [p. 123, vol. II.].
3. "Self-Induction of Wires," part ii. Phil. Mag. Sept 1886 [vol. II. p. 189].