# Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XVIII

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) |
$t=0$
$t=h$
| $F(t)=m_{0}$when$t<0,$ | (2) |
| and$F(t)=0$when$t>h,$ | (3) |
${\text{A, B}}$
$x$
$i$
$\rho$
${\mathfrak {D}}$
| ${\begin{aligned}4\pi {\mathfrak {D}}&=[3r^{-5}{\text{F}}(t-cr)+c^{2}r{-3}{\text{F}}''(t-cr)]x\rho \\&-[r^{-3}{\text{F}}(c-r)+cr^{-2}{\text{F}}'(t-cr)+c^{2}r^{-1}{\text{F}}''(t-cr)]i,\end{aligned}}$ | (4) |
${\text{F}}$
${\text{F}}''$
${\text{F}}''$
$c$
| $-\nabla ^{2}{\mathfrak {D}}=c^{2}d^{2}{\mathfrak {D}}/dt^{2},$ | (5) |
$t<0$
* $4\pi {\mathfrak {D}}=m_{0}(3r^{-5}x\rho -r^{-3i})$
* $4\pi {\mathfrak {D}}=m(3x\rho -i),$
$c$
$r.$
| $u=r^{-1}{\text{F}}(t-cr),$ | (6) |
| $-\nabla ^{2}u=c^{2}d^{2}u/dt^{2},$ | (7) |
${\mathfrak {D}}$
| $4\pi f$ | $=-d^{2}u/dy^{2}-d^{2}u/dz^{2}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (8) |
| $4\pi g$ | $=d^{2}u/dxdy$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (8) |
| $4\pi h$ | $=d^{2}u/dxdz$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (8) |
Equation (4) shows that the changes of the electrical displacement are represented by three systems of spherical waves, of forms determined by the rapidity of the discharge of the system (A, B), which expand with the velocity of light with amplitudes diminishing as $r^{-3},r^{-2},$ and $r^{-1},$ respectively. Outside of these waves, the electrical displacement is unchanged, inside of them it is zero.
If we write (with Maxwell) $-d{\mathfrak {A}}/dt$ for the force of electrodynamic induction at any point, and suppose its rectangular components calculated from those of $-d^{2}{\mathfrak {D}}/dt^{2}$ by the formula used in calculating the potential of a mass from its density, we shall have by Poisson's theorem
* $\nabla ^{2}(d{\mathfrak {A}}/dt)=4\pi d^{2}{\mathfrak {D}}/dt^{2},$
* $\nabla ^{2}(d{\mathfrak {A}}/dt)=-4\pi c^{-2}\nabla ^{2}{\mathfrak {D}},$
| $d{\mathfrak {A}}/dt=-4\pi c^{-2}{\mathfrak {D}}.$ | (9) |
* $d{\mathfrak {A}}/dt+4\pi c^{-2}{\mathfrak {D}}+\nabla {\text{V}}=0,$
$-\nabla {\text{V}}$
$-d{\mathfrak {A}}/dt.$
​If we write Curl for the differentiating vector operator which Maxwell calls by that name, equations (8) may be put in the form
* $4\pi {\mathfrak {D}}={\text{Curl Curl }}(iu),$
* $d{\mathfrak {D}}/dt=(4\pi )^{-1}{\text{ Curl Curl }}(idu/dt).$
$d{\mathfrak {D}}/dt$
${\mathfrak {B}}$
$(4\pi )^{-1}{\text{ Curl.}}$
* ${\mathfrak {B}}={\text{Curl }}(i\,du/dt),$
* ${\mathfrak {B}}=[r^{-3}{\text{F'}}(t-cr)+cr^{-2}{\text{F''}}(t-cr)](yk-zj).$
Equations (4) and (9) give the value of $d{\mathfrak {A}}/dt$ as function of $t$ and $r.$ By integration, we may find the value of ${\mathfrak {A}}$ Maxwell's "vector potential." This will be of the form of the second member of (4) multiplied by $-c^{-2}$ if we should give each ${\text{F}}$ one accent less, and for an unaccented ${\text{F}}$ should write ${\text{F}}_{1},$ to denote the primitive of ${\text{F}}$ which vanishes for the argument $\infty .$
That which seems most worthy of notice is that although simultaneously with the discharge of the system (A, B) the values of what we call the electric potential, the electrodynamic force of induction, and the "vector potential," are changed throughout all space, this does not appear connected with any physical change outside of the waves, which advance with the velocity of light.
If we now suppose that there is a second pair of charged spheres ($c,d$), as in the original problem, the discharge of this pair will evidently occur when the relaxation of electrical displacement reaches it. The time between the discharges is, therefore, by Maxwell's theory, the time required for light to pass from one pair to the other.
It may also be interesting to observe that in the axis of $x,$ on both sides of the origin, $x\rho =r^{2}i,$ and equation (4) reduces to
* $4\pi {\mathfrak {D}}=[2r^{-3}{\text{F}}(t-cr)+2cr^{-2}{\text{F'}}(t-cr)]i.$
${\text{F}}$
${\text{F'}}$
${\text{F''}},$
$x$
$x,$
J. Willard Gibbs.
New Haven, Conn., March 12 [1896].