Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/161

On the Mechanical Action and Reaction of the Systems $E_{1}$ and $E_{2}$.

104.] If we now suppose the equipotential surface $V=C$ to become rigid and capable of sustaining the action of forces, we may prove the following theorem.

If on every element $dS$ of an equipotential surface a force ${\frac {1}{8\pi }}R^{2}dS$ be made to act in the direction of the normal reckoned outwards, where $R$ is the ‚electrical resultant force‘ along the normal, then the total statical effect of these forces on the surface considered as a rigid shell will be the same as the total statical effect of the electrical action of the electrified system $E_{1}$ outside the shell on the electrified system $E_{2}$ inside the shell, the parts of the interior system $E_{2}$ being supposed rigidly connected together.

We have seen that the action of the electrified surface in the last theorem on any external point was equal to that of the internal system $E_{2}$, and, since action and reaction are equal and opposite, the action of any external electrified body on the electrified surface, considered as a rigid system, is equal to that on the internal system $E_{2}$. Hence the statical action of the external system $E_{1}$ on the electrified surface is equal in all respects to the action of $E_{1}$ on the internal system $E_{2}$.

But at any point just outside the electrified surface the resultant force is $R$ in a direction normal to the surface, and reckoned positive when it acts outwards. The resultant inside the surface is zero, therefore, by Art. 79, the resultant force acting on the element $dS$ of the electrified surface is ${\frac {1}{2}}R\sigma dS$, where $\sigma$ is the surface- density.

Substituting the value of $\sigma$ in terms of $R$ from equation (2), and denoting by p dS the resultant force on the electricity spread over the element $dS$, we find

$p\ dS={\frac {1}{8\pi }}R^{2}dS$

This force always acts along the normal and outwards, whether $R$ be positive or negative, and may be considered as equal to a pressure $p={\frac {1}{8\pi }}R^{2}$ acting on the surface from within, or to a tension of the same numerical value acting from without.^[1]

1. See Sir W. Thomson ‚On the Attractions of Conducting and Non-conducting Electrified Bodies‘, Cambridge Mathematical Journal, May 1843, and Reprint, Art. VII, § 147.