WORK DONE IN CHARGING A CONDUCTOR

The value of the potential just before the application of the charge Cx is represented by AC. Hence if the potential were to remain constant during the application of the charge Cx, the work expended in charging the conductor would be represented by the product of this potential and the charge, or by the area ACxQ.

As soon as the charge Cx has been applied the potential is xF. If this had been the value of the potential during the whole process, the work expended would have been represented by KCxF. But we know that the potential rises gradually during the application of the charge, and that during the whole process it is never less than CA or greater than xF. Hence the work expended in charging the conductor is not less than ACxQ, nor greater than KCxF.

In the same way we may determine the lower and higher limits of the work done during the application of any other portion of the entire charge. We conclude, therefore, that the work expended in increasing the charge from OC to OD is not less than the area of the figure CAQF RGSHT D, nor greater than CKF LGMHNBD. The difference between these two values is the sum of the parallelograms KQ, LR, MS, NT, the breadths of which are equal, and their united height is BV. Their united area is therefore equal to that of the parallelogram NvV B.

By increasing without limit the number of equal parts into which the charge is divided, the breadth of the parallelograms will be diminished without limit. In the limit, therefore, the difference of the two values of the work vanishes, and either value becomes ultimately equal to the area CAF GHBD, bounded by the curve, the extreme ordinates, and the base line. This method of proof is applicable to any case in which the potential is always increasing or always diminishing as the charge increases. When this is not the case, the process of charging may be divided into a number of parts, in each of which the potential is either always increasing or always diminishing, and the proof applied separately to each of these parts.

Superposition of Electric Effects.

29.] It appears from Experiment VII that several electrified bodies placed in a hollow vessel produce each its own effect on the electrification of theSUPERPOSITION OF ELECTRIC EFFECTS. vessel, in whatever positions they are placed. From this it follows that one electric phenomenon at least, that called electrification by induction, is such that the effect of the whole electrification is the sum of the effects due to the different parts of the electrification. The different electrical phenomena, however, are so intimately connected with each other that we are led to infer that all other electrical phenomena may be regarded as composed of parts, each part being due to a corresponding part of the electrification. Thus if a body A, electrified in a definite manner, would produce a given potential, P, at a given point of the field, and if a body, B, also electrified in a definite manner, would produce a potential, Q, at the same point of the field, then when both bodies, still electrified as before, are introduced simul- taneously into their former places in the field, the potential at the given point will be P + Q. This statement may be verified by direct experiment, but its most satisfactory verification is founded on a comparison of its consequences with actual phenomena.

As a particular case, let the electrification of every part of the system be multiplied n fold. The potential at every point of the system will also be multiplied by n.

30.] Let us now suppose that the electrical system consists of a number of conductors (which we shall call A1 , A2 , &c.) insulated from each other, and capable of being charged with electricity. Let the charges of these conductors be denoted by E1 , E2 , &c., and their potentials by P1 , P2 , &c.

If at first the conductors are all without charge, and at potential zero, and if at a certain instant each conductor begins to be charged with electricity, so that the charge increases uniformly with the time, and if this process is continued till the charges simultaneously become E1 for the first conductor, E2 for the second, and so on, then since the increment of the charge of any conductor is the same for every equal interval of time during the process, the increment of the potential of each conductor will also be the same for every equal increment of time, so that the line which represents, on the indicator diagram, the succession of states of a given conductor with respect to charge and potential will be described with a velocity, the horizontal and vertical components of which remain constant during the process.

This line on the diagram is therefore a straight line, drawn from the origin, which represents the initial state of the system when devoid of charge and at po- tential zero, to the point A1 which indicates the final state of the conductor when its charge is E1 , and its potential P1 , and will represent the process of charging the conductor A1 . The work expended in charging this conductor is represented by the area OCA, or half the product of the final charge E1 and the final potential P1 .

Fig. 13.

Energy of a System of Electrified Bodies

31.] When the relative positions of the conductors are fixed, the work done in charging them is entirely transformed into electrical energy. If they are charged in the manner just described, the work expended in charging any one of them is 12 EP, where E represents its final charge and P its final potential. Hence the work expended in charging the whole system may be written

E P + 12 E2 P2 + &c., 2 1 1 there being as many products as there are conductors in the system. It is convenient to write the sum of such a series of terms in the form 1 (EP ), 2∑

where the symbol ∑ (sigma) denotes that all the products of the form EP are to be summed together, there being one such product for each of the con- ductors of which the system consists.

Since an electrified system is subject to the law of Conservation of Energy, the work expended in charging it is entirely stored up in the system in the form of electrical energy. The value of this energy is therefore equal to that of the work which produced it, or 12 ∑(EP ). But the electrical energy of the system depends altogether on its actual state, and not on its previous history.

Theorem 1

We shall denote the electric energy of the system by the symbol Q, where Q = 12 ∑(EP ). (1)

Work done in altering the charges of the system.

32.] We shall next suppose that the conductors of the system, instead of being originally without charge and at potential zero, are originally charged with quantities E1 , E2 , &c. of electricity, and are at potentials P1 , P2 , &c. respectively.

When in this state let the charges of the conductors be changed, each at a uniform rate, the rate being, in general, different for each conductor, and let this process go on uniformly, till the charges have become E1 ′ , E2 ′ , &c., and the potentials P1 ′ , P2 ′ , &c. respectively.

By the principle of the superposition of electrical effects the incre- ment of the potential will vary as the increment of the charge, and the Fig. 14. potential of each conductor will increase or diminish at a uniform rate from P to P ′ , while its charge varies at a uniform rate from E to E ′

Hence the line AA′ , which represents the varying state of the conductor during the process, is the straight line which joins A, the point which indicates its original state, with A′ , which repre- sents its final state. The work spent in producing this increment of charge in the conductor is represented by the area ACC ′ A′ , or 12 CC ′ (CA + C ′ A′ ), or (E ′ − E) 12 (P + P ′ ), or, in words, it is the product of the increase of charge and the half sum of the potentials at the beginning and end of the operation, and this will be true for every conductor of the system.

As, during this process, the electric energy of the system changes from Q, its original, to Q′ , its final value, we may write Q′ = Q + 12 ∑{(E ′ − E)(P ′ + P )}, (2) hence,

Theorem 2

33. If all the charges but one are maintained constant (by the insulation of the conductors) the equation (2) is reduced to

Q′ − Q = (E ′ − E) 12 (P ′ + P ), or Q′ − Q 1 ′ = (P + P ). E′ − E 2 (3)

If the increment of the charge is taken always smaller and smaller till it ul- timately vanishes, P ′ becomes equal to P and the equation may be interpreted thus:—

The rate of increase of the electrical energy due to the increase of the charge of one of the conductors at a rate unity is numerically equal to the potential of that conductor.