Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/271

XVIII.

VELOCITY OF PROPAGATION OF ELECTROSTATIC FORCE.

[Nature, vol. liii. p. 509, April 2, 1896.]

As we may have to wait some time for the experimental solution of Lord Kelvin's very instructive and suggestive problem concerning two pairs of spheres charged with electricity (see Nature of February 6, p. 316), it may be interesting to see what the solution would be from the standpoint of existing electrical theories.

In applying Maxwell's theory to the problem it will be convenient to suppose the dimensions of both pairs of spheres very small in comparison with the unit of length, and the distance between the two pairs very great in comparison with the same unit. These conditions, which greatly simplify the equations which represent the phenomena, will hardly be regarded as affecting the essential nature of the question proposed.

Let us first consider what would happen on the discharge of (${\text{A, B}}$), if the system ($c,d$) were absent.

Let $m_{0}$ be the initial value of the moment of the charge of the system (${\text{A, B}}$), (this term being used in a sense analogous to that in which we speak of the moment of a magnet), and $m$ the value of the moment at any instant. If we set

$m=F(t),$

(1)

and suppose the discharge to commence when $t=0$, and to be completed when $t=h$, we shall have

$F(t)=m_{0}$when$t<0,$

(2)

and$F(t)=0$when$t>h,$

(3)

Let us set the origin of coordinates at the centre of the system (${\text{A, B}}$), and the axis of $x$ in the direction of the centre of the positively charged sphere. A unit vector in this direction we shall call $i$, and the vector from the origin to the point considered $\rho$. At any point outside of a sphere of unit radius about the origin, the electrical displacement (${\mathfrak {D}}$) is given by the vector equation

${\begin{aligned}4\pi {\mathfrak {D}}&=[3r^{-5}{\text{F}}(t-cr)+c^{2}r{-3}{\text{F}}''(t-cr)]x\rho \\&-[r^{-3}{\text{F}}(c-r)+cr^{-2}{\text{F}}'(t-cr)+c^{2}r^{-1}{\text{F}}''(t-cr)]i,\end{aligned}}$

(4)

where ${\text{F}}$ denotes the function determined by equation (1), ${\text{F}}''$ and ${\text{F}}''$ its derivatives, and $c$ the ratio of electrostatic and electromagnetic