Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/85

46.] Let $$e$$ be the charge of the body, and $$F$$ the force acting on the body in a certain direction, then when $$e$$ is very small $$F$$ is proportional to $$e$$, or

where $$R$$ is a quantity depending on the other bodies in the field. If the charge $$e$$ could be made equal to unity without disturbing the electrification of other bodies we should have $$F = R$$.

We shall call $$R$$ the Resultant electric force at the given point of the field.

Electric Potential.

45.] If the small body carrying the small charge $$e$$ be moved from the given point to an indefinite distance from the electrified bodies, it will experience at each point of its course a force $$Re$$, where $$R$$ varies from point to point of the course. Let the whole work done on the body by these electrical forces be $$Ve$$, then $$V$$ is the potential at the point of the field from which the body started. If the charge$$ e$$ could be made equal to unity without disturbing the electrification of other bodies, we might define the potential at any point as the work done on a body charged with unit of electricity in moving from that point to an infinite distance.

A body electrified positively tends to move from places of greater positive potential to places of smaller positive, or of negative potential, and a body negatively electrified tends to move in the opposite direction.

In a conductor the electrification is distributed exactly as if it were free to move in the conductor according to the same law. If therefore two parts of a conductor have different potentials, positive electricity will move from the part having greater potential to the part having less potential as long as that difference continues. A conductor therefore cannot be in electrical equilibrium unless every point in it has the same potential. This potential is called the Potential of the Conductor.

Equipotential Surfaces.

46.] If a surface described or supposed to be described in the electric field is such that the electric potential is the same at every point of the surface it is called an Equipotential surface.

An electrified point constrained to rest upon such a surface will have no tendency to move from one part of the surface to another, because the potential is the same at every point. An equipotential surface is therefore a surface of equilibrium or a level surface.