Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/125

82.] If lines of force be drawn from every point of a line they will form a surface such that the force at any point is parallel to the tangent plane at that point. The surface-integral of the force with respect to this surface or any part of it will therefore be zero.

If lines of force are drawn from every point of a closed curve $$L_l$$ they will form a tubular surface $$S_0$$. Let the surface $$S_1$$ bounded by the closed curve $$L_1$$ be a section of this tube, and let $$S_2$$ be any other section of the tube. Let $$Q_0, Q_1, Q_2$$ be the surface-integrals over $$S_0, S_1, S_2$$, then, since the three surfaces completely enclose a space in which there is no attracting matter, we have

But $$Q_0= 0$$, therefore $$Q_2 =-Q_1$$, or the surface-integral over the second section is equal and opposite to that over the first: but since the directions of the normal are opposite in the two cases, we may say that the surface-integrals of the two sections are equal, the direction of the line of force being supposed positive in both.

Such a tube is called a Solenoid, and such a distribution of force is called a Solenoidal distribution. The velocities of an in compressible fluid are distributed in this manner.

If we suppose any surface divided into elementary portions such that the surface-integral of each element is unity, and if solenoids are drawn through the field of force having these elements for their bases, then the surface-integral for any other surface will be re presented by the number of solenoids which it cuts. It is in this sense that Faraday uses his conception of lines of force to indicate not only the direction but the amount of the force at any place in the field.

We have used the phrase Lines of Force because it has been used by Faraday and others. In strictness, however, these lines should be called Lines of Electric Induction.

In the ordinary cases the lines of induction indicate the direction and magnitude of the resultant electromotive force at every point, because the force and the induction are in the same direction and in a constant ratio. There are other cases, however, in which it is important to remember that these lines indicate the induction, and that the force is indicated by the equipotential surfaces, being normal to these surfaces and inversely proportional to the distances of consecutive surfaces.