Page:MortonCharge.djvu/4

These results have been obtained by Prof. Thomson and Mr. Heaviside. The particular case of a point charge, e, is got by putting

$\phi ={\frac {e}{4\pi {\sqrt {k^{2}(x^{2}+y^{2})+z^{2}}}}}$.

Evidently in the general case φ must vanish at infinity.

5. Mr. Heaviside points out that φ=constant is the condition holding at a surface of equilibrium. The matter may be stated thus : — If we suppose the field to terminate at the surface of a conductor, inside which the vectors vanish, we must see that the "curl" relations of the field are not violated for circuits which lie partly inside the empty space enclosed by the conductor. In particular, if there is a vector whose line integral round every circuit in the field vanishes, the lines of this vector must meet the surface at right angles. Otherwise we should have a finite value for the integral round a circuit drawn close to the surface outside and completed inside. In other words, if a vector is derived from a potential function, this function must be constant over the surface. In the ordinary static case it is the electric force (X, Y, Z) which is so derived; but in the case of a steadily moving field it is the vector (X, Y, ${\frac {Z}{k^{2}}}$) which meets the surface at right angles.

6. Let F(x y z)=C be the equation of the charged surface. Then φ(x y z) has to be constant over this surface and satisfy

${\frac {d^{2}\phi }{dx^{2}}}+{\frac {d^{2}\phi }{dy^{2}}}+k^{2}{\frac {d^{2}\phi }{dz^{2}}}=0$.

Put z=kζ, then φ is a function of x, y, ζ, which is constant when F(x, y, kζ)=C, and which satisfies

${\frac {d^{2}\phi }{dx^{2}}}+{\frac {d^{2}\phi }{dy^{2}}}+k^{2}{\frac {d^{2}\phi }{d\zeta ^{2}}}=0$.

Therefore if we regard (x y ζ) as Cartesian coordinates of a point, φ is the potential at external points of an electrostatic free distribution on the surface F(x, y, kζ)=C. The components of electric force due to this distribution, at a point