Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/228

From Wikisource
Jump to navigation Jump to search
This page has been proofread, but needs to be validated.

Limiting Forms.

(1) When the surface is the part of the plane of between the two branches of the hyperbola whose equation is written above, (24).

(2) When ) the surface is the part of the plane of xy which is on the outside of the focal ellipse whose equation is

(25)


The Ellipsoids.

For any given ellipsoid is constant. If two ellipsoids, and , be maintained at potentials and then, for any point in the space between them, we have

(26)

The surface-density at any point is

(27)

where is the perpendicular from the centre on the tangent plane, and is the product of the semi-axes.

The whole charge of electricity on either surface is

(28)

a finite quantity.

When the surface of the ellipsoid is at an infinite distance in all directions.

If we make and , we find for the quantity of electricity on an ellipsoid maintained at potential in an infinitely extended field,

(29)

The limiting form of the ellipsoids occurs when , in which case the surface is the part of the plane of within the focal ellipse, whose equation is written above, (25).

The surface-density on the elliptic plate whose equation is (25), and whose eccentricity is , is

(30)

and its charge is

(31)