Page:Elementary Text-book of Physics (Anthony, 1897).djvu/75

§ 53] they affect the motion of bodies and they may be considered, for purposes of mathematical representation, as acting between the particles of the bodies. A field of force is a region in which a particle, constituting a part of a mutually interacting system, will be acted on by a force, and will move, if free to do so, in the direction of the force. The strength of field or the intensity in the field at a point is measured by the force which acts upon a unit quantity or test unit of that agent to which the force is due when placed at that point. The test unit is supposed not to affect the forces of the field.

When the force acting on a particle depends only on the position of the particle the study of its effects is often very much facilitated by the use of a concept called the potential. To explain this concept and its relation to force, we begin with a definition of the difference of potential. The diilerence of potential between two points in a field of force is equal to the work done by the forces of the field in moving a test unit from the one point to the other.

If $$V_{P} - V_{Q}$$ represent the difference of potential between the points $$P$$ and $$Q,$$ and if $$F$$ represent the average force between those points and $$s$$ the distance between them, then the amount of work done by that force in moving a test unit from $$P$$ to $$Q,$$ and hence the difference of potential between $$P$$ and $$Q,$$ is represented by From this relation we have  If $$s$$ become indefinitely small, in the limit $$F$$ represents the force at the point $$P,$$ and $$- \frac{V_{Q} - V_{p}}{s} = - \frac{dV}{ds}$$ becomes the rate of change of potential at that point with respect to space, taken with the opposite sign. Hence we obtain a definition of potential. It is a function, the rate of change of which at any point, with respect to space, taken with the opposite sign, measures the force at that point.

Let the test unit be situated at the point $$A$$ and be moved over