Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/110

70 body of any form, but on an indefinitely small body, charged with an indefinitely small amount of electricity, and placed at any point of the space to which the electrical action extends. By making the charge of this body indefinitely small we render insensible its disturbing action on the charge of the first body.

Let $$e$$ be the charge of this body, and let the force acting on it when placed at the point $$(x, y, z)$$ be $$Re$$, and let the direction-cosines of the force be $$l, m, n$$, then we may call $$R$$ the resultant force at the point $$(x, y, z)$$.

In speaking of the resultant electrical force at a point, we do not necessarily imply that any force is actually exerted there, but only that if an electrified body were placed there it would be acted on by a force $$R e$$, where $$e$$ is the charge of the body.

Definition. The Resultant electrical force at any point is the force which would be exerted on a small body charged with the unit of positive electricity, if it were placed there without disturbing the actual distribution of electricity.

This force not only tends to move an electrified body, but to move the electricity within the body, so that the positive electricity tends to move in the direction of $$R$$ and the negative electricity in the opposite direction. Hence the force $$R$$ is also called the Electromotive Force at the point $$(x,\, y,\,z)$$.

When we wish to express the fact that the resultant force is a vector, we shall denote it by the German letter $$\mathfrak {C}$$. If the body is a dielectric, then, according to the theory adopted in this treatise, the electricity is displaced within it, so that the quantity of electricity which is forced in the direction of $$\mathfrak {C}$$ across unit of area fixed perpendicular to $$\mathfrak {C}$$ is

where $$\mathfrak {D}$$ is the displacement, $$\mathfrak {C}$$ the resultant force, and $$K$$ the specific inductive capacity of the dielectric. For air, $$K = 1$$.

If the body is a conductor, the state of constraint is continually giving way, so that a current of conduction is produced and maintained as long as the force $$\mathfrak {C}$$ acts on the medium.

Components of the Resultant Force.

If $$X,\, Y,\, Z$$ denote the components of R, then

where $$l, m, n$$ are the direction-cosines of $$R$$.