Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/175

CHAPTER VI.

ON POINTS AND LINES OF EQUILIBRIUM.

112.] If at any point of the electric field the resultant force is zero, the point is called a Point of equilibrium.

If every point on a certain line is a point of equilibrium, the line is called a Line of equilibrium.

The conditions that a point shall be a point of equilibrium are that at that point

${\frac {dV}{dx}}=0,\ {\frac {dV}{dy}}=0,\ {\frac {dV}{dz}}=0$

At such a point, therefore, the value of $V$ is a maximum, or a minimum, or is stationary, with respect to variations of the coordinates. The potential, however, can have a maximum or a minimum value only at a point charged with positive or with negative electricity, or throughout a finite space bounded by a surface electrified positively or negatively. If, therefore, a point of equilibrium occurs in an unelectrified part of the field it must be a stationary point, and not a maximum or a minimum.

In fact, the first condition of a maximum or minimum is that

${\frac {d^{2}V}{dx^{2}}},\ {\frac {d^{2}V}{dy^{2}}},$ and ${\frac {d^{2}V}{dz^{2}}}$

must be all negative or all positive, if they have finite values.

Now, by Laplace’s equation, at a point where there is no electrification, the sum of these three quantities is zero, and therefore this condition cannot be fulfilled.

Instead of investigating the analytical conditions for the cases in which the components of the force simultaneously vanish, we shall give a general proof by means of the equipotential surfaces.

If at any point, $P$, there is a true maximum value of $V$, then, at all other points in the immediate neighbourhood of $P$, the value of $V$ is less than at $P$. Hence $P$ will be surrounded by a series of