ELECTRICAL WORK AND ENERGY

23.] Work in general is the act of producing a change of configuration in a material system in opposition to a force which resists this change. Energy is the capacity of doing work.

When the nature of a material system is such that if after the system has undergone any series of changes it is brought back in any manner to its orig- inal state, the whole work done by external agents on the system is equal to the whole work done by the system in overcoming external forces, the system is called a Conservative system.

The progress of physical science has led to the investigation of different forms of energy, and to the establishment of the doctrine, that all material sys- tems may be regarded as conservative systems provided that all the different forms of energy are taken into account.

This doctrine, of course, considered as a deduction from experiment, can assert no more than that no instance of a non-conservative system has hitherto been discovered, but as a scientific or science-producing doctrine it is always acquiring additional credibility from the constantly increasing number of deductions which have been drawn from it, which are found in all cases to be verified.

In fact, this doctrine is the one generalised statement which is found to be consistent with fact, not in one physical science only, but in all. When once apprehended it furnishes to the physical enquirer a principle on which he may hang every known law relating to physical actions, and by which he may be put in the way to discover the relations of such actions in new branches of science. For such reasons the doctrine is commonly called the Principle of the Conservation of Energy. General Statement of the Conservation of Energy.

24.] The total energy of any system of bodies is a quantity which can neither be increased nor diminished by any mutual action of those bodies, though it may be transformed into any of the forms of which energy is susceptible.

If, by the action of some external agent, the configuration of the system is changed, then, if the forces of the system are such as to resist this change of configuration, the external agent is said to do work on the system.

In this case the energy of the system is increased. If, on the contrary, the forces of the system tend to produce the change of configuration, so that the external agent has only to allow it to take place, the system is said to do work on the external agent, and in this case the energy of the system is diminished. Thus when a fish has swallowed the angler’s hook and swims off, the angler following him for fear his line should break, the fish is doing work against the angler, but when the fish becomes tired and the angler draws him to shore, the angler is doing work against the fish.

Work is always measured by the product of the change of configuration into the force which resists that change. Thus, when a man lifts a heavy body, the change of configuration is measured by the increase of distance between the body and the earth, and the force which resists it is the weight of the body.

The product of these measures the work done by the man. If the man, instead of lifting the heavy body vertically upwards, rolls it up an inclined plane to the same height above the ground, the work done against gravity is precisely the same; for though the heavy body is moved a greater distance, it is only the vertical component of that distance which coincides in direction with the force of gravity acting on the body.

25.] If a body having a positive charge of electricity is carried by a man from a place of low to a place of high potential, the motion is opposed by the electric force, and the man does work on the electric system, thereby increasing its energy. The amount of work is measured by the product of the number of units of electricity into the increase of potential in moving from the one place to the other.

We thus obtain the dynamical definition of electric potential. The electric potential at a given point of the field is measured by the amount of work which must be done by an external agent in carrying one unit of positive electricity from a place where the potential is zero to the given point. This definition is consistent with the imperfect definition given at Art. 6, for the work done in carrying a unit of electricity from one place to another will be positive or negative according as the displacement is from lower to higher or from higher to lower potential. In the latter case the motion, if not prevented, will take place, without any interference from without, in obedience to the electric forces of the system. Hence the flow of electricity along conductors is always from places of high to places of low potential.

26.] We have already defined the electromotive force from one place to another along a given path as the work done by the electric forces of the system on a unit of electricity carried along that path. It is therefore measured by the excess of the potential at the beginning over that at the end of the path.

The electromotive force at a point is the force with which the electrified system would act on a small body electrified with a unit of positive electricity, and placed at that point.

If the electrified body is moved in such a way as to remain on the same equipotential surface, no work is done by the electric forces or against them. Hence the direction of the electric force acting on the small body is such that any displacement of the body along any line drawn on the equipotential surface is at right angles to the force. The direction of the electromotive force, therefore, is at right angles to the equipotential surface.

The magnitude of this force, multiplied by the distance between two neighbouring equipotential surfaces, gives the work done in passing from the one equipotential surface to the other, that is to say, the difference of their potentials.

Hence the magnitude of the electric force may be found by dividing the difference of the potentials of two neighbouring equipotential surfaces by the distance between them, the distance being, of course, very small, and mea- sured perpendicularly to either surface. The direction of the force is that of the normal to the equipotential surface through the given point, and is reck- oned in the direction in which the potential diminishes.

Indicator Diagram of Electric Work.

27.] The indicator diagram, employed by Watt for measuring the work done by a steam engine∗ , may be made use of in investigating the work done during the charging of a conductor with electricity.

Maxwell’s ‘Theory of Heat,’ 4th ed., p. 102.DIAGRAM OF WORK. 26 Fig. 12.

Let the charge of the conductor at any instant be represented by a hori- zontal line OC, drawn from the point O, which is called the origin of the diagram, and let the potential of the conductor at the same instant be represented by a vertical line CA, drawn from the extremity of the first line, then the position of the extremity of the second line will indicate the electric state of the conductor, both with respect to its charge, and also with respect to its potential.

If during any electrical operation this point moves along the line AF GHB, we know not only that the charge has been increased from the value OC to the value OD, and that the potential has been increased from CA to DB, but that the charge and the potential at any instant, corresponding, say, to the point F of the curve, are represented respectively by Ox and xF. 28.] Theorem.

The work expended by an external agent in bringing the increment of charge from the walls of the room to the conductor is represented by the area enclosed by the base line CD, the two vertical lines CA and DB, and the curve AF GHB. For let CD, the increment of the charge, be divided into any number of equal parts at the points, x, y, z.