the wings emerge from the shelter of the larger metallic sectors or inductors a a, of which one is connected with the acid, and the other with the earth. Suppose the acid to have a positive charge.

Then, at the instant of contact, an inductive movement of electricity takes place, producing an accumulation of negative electricity in the carrier which is next the positive inductor, and an accumulation of positive in the other. The next contacts are effected when the carrier which has thus acquired a positive charge is well under cover of the positive inductor, to which accordingly it gives up its electricity; for, being in great part surrounded by this inductor, and being connected with it by the spring, the carrier may be regarded as forming a portion of the interior of a concave conductor, and the electricity accordingly passes from it to the external surface, that is to the inductor, and to the acid connected with it, which forms the lining of the jar. The negative electricity on the other carrier is, in like manner, given off to the other inductor, and so to the earth.

Fig. 404.-Replenisher.

The jar thus receives an addition to its charge once in every halfrevolution of the replenisher; and, as these increments are very small, it is easy to regulate the charge so that the gauge shall indicate exactly the normal potential. If the charge is too strong, it can be diminished by turning the replenisher in the reverse direction.

NOTE ON THE ENERGY OF A SYSTEM OF CHARGED CONDUCTORS, WITH APPLICATION TO THOMSON'S QUADRANT ELECTROMETER.

(1.) By the energy of a system of charged conductors is meant the work which must have been spent in charging them, or, what is the same thing, the energy which will run down when the conductors are connected with the earth. We shall investigate its amount in terms of the charges Q1 Q &c., of the conductors, and their potentials V, V, &c., these latter being supposed to depend only on the charges of the system itself.

1 2

Let the conductors be charged gradually all at the same time, and let their charges at any time be Q1, xQ2 &c., the value of x being the same for all the conductors. The potentials at the same time will be xV1, xV2

NOTE ON CHARGED CONDUCTORS.

637* &c. By § 606, the work required to bring the small quantity of electricity Q1 dx from the earth to the conductor of potential жV1 is xV1 Q1 dĩ; thus, when x receives the small increase dx, the whole addition of energy to the system is

(V1Q1+V2 Q2+ &c.) x dx.

1

If this operation is repeated time after time, beginning with x = 0, and ending with x = 1, the conductors will begin with being uncharged, and will end by having the given charges. Since the integral of x dx between these limits is, the whole energy acquired by the system is § (V1Q1 + V2 Q2 + &c.), which may be written VQ.

(2.) Hence when any small changes dQ1, dQ, &c., are made in the charges, and any small changes dV1, dV, &c., in the potentials, either with or without displacement of the conductors, the increase of energy is Z(VdQ+QdV).

(3.) If the conductors are stationary, another simple, expression can be found for the increase of energy; for the work required to bring the electricity dQ1 to the conductor of potential V1 is V1 dQ1; thus the whole increase of energy is ZVdQ; and by comparing this with the expression for the same thing in (2.) we see that a third expression for the increase of energy will be Qd V.

(4.) If the conductors are insulated, so that their charges remain constant, the increase of energy when they are displaced will be the difference between the initial energy 2QV and the final energy QV', that is, will be Q(V'-V), where V' denotes the final potential of the conductor whose initial potential is If the charges are small the increase of energy will be Qd V. This, it will be noticed, is exactly half the increase in (3.), the changes of potential being supposed the same in both cases.

V.

When insulated charged conductors are allowed to move under the influence of their own mutual forces, these forces will do positive work, and the system will lose electrical energy of the same amount. On the other hand, if external forces move the conductors in opposition to their mutual forces, there will be a gain of electrical energy equal to the work done by external forces against the forces of the system. These consequences follow immediately from the principle of the conservation of

energy.

(5.) If the conductors while displaced are kept at constant potentials, their charges must change, and we cannot make the same direct application of the principle of conservation of energy which we have made above, unless we include in our reasonings the external sources from which electricity comes or to which it goes in making these alterations in the charges. We can, however, arrive at the relation between the change of energy in the system and the work done in the following way.

Divide the whole displacement into a series of small steps. In each step let the conductors be insulated, so that the potentials will change slightly, and then let the potentials be restored to their original values before the next step. The forces between the conductors will thus be sensibly the same as if the potentials were absolutely constant, and the work done by these forces will be the same.

In any one of the steps, the increase of energy, by (4.), is ΣQd V, and in the restoration of the potentials to what they were at the beginning of this step the increase of energy, by (3.), is – ΣQd V, the minus sign being rendered necessary by the fact that the change from V+dV back to V is - dV.

In the two operations combined the whole increase of energy is Qd V, and the mechanical work done by the forces of the system is, by (4.), equal to the loss of energy in the displacement which constitutes the first operation, that is to Qd V also. Hence in each step combined with its following restoration of potential, the change of energy is the same both in amount and in sign as the work which the forces of the system do in the movements. As this equality holds through all the steps, it holds for the complete result; that is, the gain of energy in a system of conductors which are displaced at constant potentials is equal to the work which the forces of the system do in the displacement. In any system cut off from external supplies of energy the work done is equal to the energy lost, but here it is equal to the energy gained. Hence the external sources which supply the electricity for keeping the conductors at constant potentials furnish an amount of energy double of the work done by the forces of the system. On the other hand, if the conductors are moved in opposition to the forces of the system, the external sources will draw energy from the system to double the amount of the work done against the forces of the system.

1

2

(6.) In Thomson's quadrant electrometer, let V denote the potential of the needle and sulphuric acid, V1 the potential of one pair of quadrants which we will call the first pair, and V2 the potential of the other pair. The needle and quadrants form two condensers, the inner coatings of both being at the same potential V, and the outer coatings at the respective potentials V1 and V2. When the needle turns through an angle ✪ from the first pair of quadrants towards the second, the capacity of the first condenser is diminished by a quantity proportional to 6, say co, and the capacity of the second condenser is increased by the same amount. Hence, supposing V to be higher than V1, and V1 than V, the charge of the needle is in one part diminished by cơ (V - V1), and in another part increased by co(V - V2), making a total increase of cơ (V1 − V2).

The charge of the first pair of quadrants, being opposite in sign to that of the included portion of the needle, is increased algebraically by cơ (V − V1), and that of the second pair is diminished by cơ (V − V2).

NOTE ON CHARGED CONDUCTORS.

639

The total increase in the expression (VQ+ V1Q1+V2Q) for the energy of the system composed of the needle and quadrants is therefore ¿co{V(V1− V2) +V1(V − V1) + V2(V2 − V) }

This is the increase of energy produced by the displacement of the needle through the angle while the three potentials remain unchanged, and is equal, by (5.), to the work done by the electrical forces against the mechanical forces of the bifilar suspension. Dividing the work by the angle, we get the average working couple. This quotient

is independent of 9, and of the sensibility of the bifilar suspension. It is therefore the value of the working couple itself. It is balanced by the couple due to the suspension, which is proportional to 0.

Hence the

(7.) In the ordinary use of the instrument, V is very large compared with V, and V. Hence is sensibly proportional to V1 - V2. 1

2.

1

2

1

2*

1

(8.) If the needle is connected with the first pair of quadrants, V, may be substituted for V, and the deflection is proportional to (V1 − V2)2. The direction of the deflection will be from the first pair towards the second, whether V1 - V2 be positive or negative. Joubert has taken advantage of this circumstance to use the instrument for measuring the difference of potential between the two terminals of an alternate-current dynamo. He connects these terminals with the electrodes of the quadrant electrometer, having first discharged the needle and sulphuric acid and connected them with one pair of quadrants. The difference of potential is reversed in sign, as well as changed in amount, some hundreds of times per second, and the needle gives a steady deflection which is proportional to the mean square of the difference of potential.

CHAPTER L.

ATMOSPHERIC ELECTRICITY.

646. Resemblance of Lightning to the Electric Spark.-The resemblance of the effects of lightning to those of the electric spark struck the minds of many of the early electricians. Lightning, in fact, ruptures and scatters non-conducting substances, inflaming those which are combustible; heats, reddens, melts, and volatilizes metals; and gives shocks, more or less severe, and frequently fatal, to men and animals; all of these being precisely the effects of the electric spark with merely a difference of intensity. We may add that lightning leaves behind it a characteristic odour precisely similar to that which is observed near an electrical machine when it is working, and which we now know to be due to the presence of ozone. Moreover, the form of the spark, its brilliancy, and the detonation which attends it, all remind one forcibly of lightning.

To Franklin, however, belongs the credit of putting the identity of the two phenomena beyond all question, and proving experimentally that the clouds in a thunder-storm are charged with electricity. This he did by sending up a kite, armed with an iron point with which the hempen string of the kite was connected. To the lower end of the string a key was fastened, and to this again was attached a silk ribbon intended to insulate the kite and string from the hand of the person holding it. Having sent up the kite on the approach of a storm, he waited in vain for some time even after a heavy cloud had passed directly over the kite. At length the fibres of the string began to bristle, and he was able to draw a strong spark by presenting his knuckle to the key. A shower now fell, and, by wetting the string, improved its conducting power, the silk ribbon being still kept dry by standing under a shed. Sparks in rapid succession were drawn from the key, a Leyden jar was charged by it, and a shock given.