Page:Elementary Text-book of Physics (Anthony, 1897).djvu/77

 in the field will only do no work on the test unit when it is moved at right angles to their directions, it follows that the forces at the different points in an equipotential surface are normal to that surface. Draw the normals $$AP, BQ,$$ etc., of such lengths that the work done in moving the test unit over them is the same. The surface drawn through their end points is again an equipotential surface. By repeating this process, the whole field of force may be mapped out by equipotential surfaces. In the limit, as the lengths of the normals thus drawn become infinitesimal, the successive normals will form continuous curves, everywhere normal to the equipotential surfaces which they cut. These curves, which represent the direction of the force at the points through which they pass, are called lines of force. If a small area be described on an equipotential surface, the lines of force which pass through its boundary will form a tubular surface, which will cut out corresponding areas on the other equipotential surfaces. This tubular surface, with the region enclosed by it, is called a tube of force.

55. An Expression for Difference of Potential.—In what follows we will for convenience assume that the test unit is a unit mass, and that the field of force is due to the presence of particles which attract the unit mass with forces that are proportional to their masses and vary inversely with the square of the distance. By the proper choice of units the force due to any one particle may be set equal to $$\frac{m}{r^2},$$ where $$m$$ is a constant proportional to the mass of the particle and $$r$$ the distance between it and the test unit.

Let the point $$O$$ (Fig. 23) be the point at which a particle $$m$$ is placed, and let a unit mass traverse the path $$PRX$$ under the action of the force $$\frac{m}{r^2}$$ directed toward $$O.$$ When the particle is at