Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/120

We shall denote, as at Art. 26, the quantity

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}$ by $-\nabla ^{2}V$,

and we may express Poisson's equation in words by saying that the electric density multiplied by $4\pi$ is the concentration of the potential. Where there is no electrification, the potential . has no concentration, and this is the interpretation of Laplace's equation.

If we suppose that in the superficial and linear distributions of electricity the volume-density $\rho$ remains finite, and that the electricity exists in the form of a thin stratum or narrow fibre, then, by increasing $\rho$ and diminishing the depth of the stratum or the section of the fibre, we may approach the limit of true superficial or linear distribution, and the equation being true throughout the process will remain true at the limit, if interpreted in accordance with the actual circumstances.

On the Conditions to be fulfilled at an Electrified Surface.

78.] We shall consider the electrified surface as the limit to which an electrified stratum of density $\rho$ and thickness $v$ approaches when $\rho$ is increased and $v$ diminished without limit, the product $\rho \,v$ being always finite and equal to $\sigma$ the surface-density.

Let the stratum be that included between the surfaces

$F(x,\,y,\,z)=\,F\,=\,a$

(1)

and

and $F=a\,+\,h$

(2)

If we put

$R^{2}={\left.{\frac {\overline {dV}}{dy}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dy}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dz}}\right|}^{2}$,

(3)

and if $l,m,n$ are the direction-cosines of the normal to the surface,

$Rl={\frac {dF}{dx}},{\color {White}xxxxx}Rm={\frac {dF}{dy}},{\color {White}xxxxx}Rn={\frac {dF}{dz}}$

(4)

Now let $V_{1}$ be the value of the potential on the negative side of the surface $F=a,V'$ its value between the surfaces $F=a$ and $F=a+h$, and $V_{2}$ its value on the positive side of $F=a+h$.

Also, let $\rho _{1},\rho '$, and $\rho _{2}$ be the values of the density in these three portions of space. Then, since the density is everywhere finite, the second derivatives of $V$ are everywhere finite, and the first derivatives, and also the function itself, are everywhere continuous and finite.

At any point of the surface $F=a$ let a normal be drawn of