Page:Elementary Text-book of Physics (Anthony, 1897).djvu/76

 any path to the point $$B.$$ It is clear that if it be moved back over the same path from $$B$$ to $$A$$ the amount of work required to effect this motion will be equal and opposite to that done during the motion from $$A$$ to $$B.$$ This equality will also hold if the test unit be moved from $$B$$ to $$A$$ by any other path, provided the field of force is one which is nowhere interrupted by a region in which the force is not a function only of the position of the test unit, or is, as it is called, a singly connected region. The fields of force due to all forces known in Nature, except those caused by electrical currents, are singly connected regions. When the forces which act on the test unit at different points in its path are parallel, as in the case of gravity, this equality of the work done in carrying the test unit from one place to another over any path is obvious. If we assume the principle of the conservation of energy as a general principle, this equality may also be shown for fields in which the forces are not parallel; for, if the work done in moving the test unit over one path between $$A$$ and $$B$$ were not equal to that done in moving it over any other path between the same points, an endless supply of work could be obtained by repeatedly moving the unit over a path in which the work done by the forces of the field is greater, and returning it to its starting-point by motion over a path in which the work done is less. As this result is inconsistent with the principle of the conservation of energy, we conclude that the hypothesis from which it is deduced is untrue, and that the same amount of work will be done in moving the unit from the one point to the other, by whatever path the motion is effected. The difference of potential between the two points is therefore a function of their positions only.

54. Equipotential Surfaces and Lines of Force.—Let the test unit be moved from $$O$$ along the different paths $$OA, OB,$$ etc. (Fig. 32), so that the same amount of work is done upon it in each of these paths. The surface drawn through the end points of these paths is called an equipotential surface; as may be seen from the proposition just proved, it is a surface in which the test unit may be moved without doing any work upon it. Since the forces