The usual 'lamp and scale' arrangement gives us, in the reflected spot of light, a very sensitive means of noticing and measuring the deflexions of the needle.

Complete form of the quadrant electrometer.-In the complete form of Sir W. Thomson's quadrant electrometer there are many details of construction that we shall not discuss here. But we must mention two of the most important of these.

(i.) The gauge.--Connected with the inside of the Leyden jar (and so with the needle) is an attracted-disc electrometer. If the weights or spring be so arranged that the hair (see § 32) is in the proper line of sight when the jar, needie, and disc ƒ therewith connected, are at some fixed potential V, then any alteration in this potential will be at once detected by the movement of the hair. This is one detail.

(ii.) The replenisher.—The other detail is a replenisher (see Chapter VII. § 5), in which one armature is to earth while the other is connected with the Leyden jar.

We can thus remedy any error in V by turning the replenisher the one way or the other, so as to raise or lower the potential of the jar.

Formula for quadrant electrometer.-It can be shown that the following formula holds for the absolute quadrant electrometer.

where L is the electrostatic couple acting on the needle, balanced by and equal to the mechanical restoring couple; V the potential of the needle; V, and V, the potentials of the two pairs of quadrants respectively, and ha constant depending on the construction of the instrument.

2

If the needle be charged to a relatively very high potential, then we may neglect V1 + V2 in comparison with V ; and we have

1

2

We see then that we could measure the difference of potential (V1 - V) by observation of the deflexion; provided that we knew

the values of h, V, and L. In practice, however, these quantities are not calculated; but the instrument is graduated by means of a 'standard voltaic cell,' or by comparison with an absolute electrometer of the more simple 'attracted-disc' form.

In instruments devoid of the gauge and the replenisher, as well as in the above form, we may write

so long as the potential V of the needle remains constant. And, for small deflexions, we have

Here is the deflexion; and the constant b may be determined on each occasion of use by means of a 'standard voltaic cell.'

$ 34. Uses of the Quadrant Electrometer.

(i.) General use. The quadrant electrometer is mainly an instrument for the measurement, or comparison, of potentials and potential-differences. We have indicated in the above how it is possible to use simpler forms also, as well as the 'standard' patterns of the instrument, for such purposes of measurement.

It may be added that usually one pair of quadrants is put to earth, so as to be at zero potential; the potential of the other pair of quadrants being then measured with respect to the earth

as zero.

(ii.) Use as an electroscope. --The instrument can also be used for all purposes in which we desire merely a delicate electroscope, and not an electrometer. For such purposes any delicate form will serve, even though it be so faulty in construction that the law of its action (see § 33, end) is unknown. An important example of such a use will be given in Chapter XIV. § 10.

(iii.) Use in investigating the distribution on a conductor (see Chapter IV. § 16).—In the investigation of distribution upon conductors, of which we said something in Chapter IV. § 16, &c., it would at first sight appear impossible to use the electrometer. For, it would be argued, where we are dealing with a conductor whose potential is constant in every part, how can we employ an instrument that indicates only potential-differences?

A few words of explanation will enable the student to perceive how the instrument may be used in such investigation.

When we lay the proof-plane against any part of the conductor whose potential is V, it will also be at the potential V ; but the charge that it acquires will depend upon the density p that exists at the part touched. The fact is that the capacity of the proofplane changes as we change its position on the conductor; and so, though its charge is proportional to p, its potential is always V.

But when we remove the proof-plane and place it in an isolated position, its capacity assumes some constant value; and its potential will therefore be proportional to its charge, and so will be proportional to p.

When the proof-plane is connected with the one pair of quadrants of the electrometer, the combined conductors will have a potential that is directly proportional to the charge, and inversely proportional to their combined capacity. Since the latter remains constant, it follows that the quadrants will assume a potential that is directly proportional to p.

$ 35. Examples in Energy of Discharge, &c.—(a) In all cases where it is required to find the energy of discharge between two electrical systems A and B, the following method will be found to be a good one. First, find the energies of discharge of A to earth, and of B to earth, separately; and add these quantities Secondly, find what will be the energy of discharge to earth of the combined system A and B after these have been connected. It will then follow, by conservation of energy, that the difference between these two results will give the required energy of discharge between the two systems when they are connected with one another. Let us take an example.

A is first charged.

'There are 3 Leyden jars A, B, and C, equal in capacity, having their outer coatings connected with earth. Its knob is then connected with the knob of B. It is then disconnected from B, and connected with C. Finally, the knob of B is connected with the knobs of A and C. Find the energies of the several discharges.'

(i.) The initial discharge of A to earth would give energy to the amount of QV; and the final discharge of A and B together to

I

2

I

earth would give energy measured by Q., orQV. Since energy

2

4

cannot have been lost, it follows that the difference between these two quantities must measure the energy w of discharge between A and B when these two were connected. Hence

(ii.) The discharge of A to earth would now give

I QV. And the subsequent discharge of A and C together to earth

8

would give.

Q V

2 4

I

or QV. Hence, energy w, of discharge 16

between A and C is given by the difference, or

(iii.) The discharge of B to earth, and of A and C to earth, would

the energy w, of discharge when B is connected with A and C is given by the difference, or

It is to be noticed that all the discharges added together give the energyQV, which was that of the original discharge of A to earth. If this result had not followed there must have been some error in the above work; since 'Conservation of energy' demands that no energy be lost.

(3) A sphere A of 9 cms. diameter is connected by a thin wire (i.e. one of negligible capacity) with another sphere B of the same diameter. Round this latter, and concentric with it so as to form a spherical condenser, is a larger sphere of 10 cms. internal diameter, connected with earth. A charge of 33 units is given to A. Find (i.) in what proportion this charge is distributed between the two systems; and (ii) the energy of discharge of the two systems separately.'

Let VA be the potential of A, and therefore also of B.

CA be the capacity of A, and C1 that of B.

QA be the charge on A, and QB that on E.

Ws

be the energy of discharge of A, and w ̧ that of B.

§ 36. General Consideration of Electrostatic Fields of Force. It is probable that among all the important conceptions that are due in the first place to the genius and insight of Faraday, none has done more to place the physical theory of electrical and of magnetic phenomena upon a sound basis than his recognition of the part played by the medium across which the forces act. The former view of 'action at a distance,' and the purely geometric conception of lines of force, were essentially mathematical ideas; to the physicist they were both unreal and unsuggestive.

Taking our present case of electrostatic fields, we may explain the modern view, based upon Faraday's conception, somewhat as follows.

and

There is no such thing as a + charge' or 'a charge' by itself; but on the contrary, wherever electrostatic phenomena occur there is an electrostatic field, on the two sides of which occur equal charges respectively. The whole system. consists of these equal and opposite charges separated by a dielectric which is the seat of the field of force. The lines of force have a real physical meaning. Whether they are lines along which the dielectric undergoes a kind of tension, or whether they are lines along which the molecules of the dielectrics are 'polarised' by a separation of and charges in them, is not yet known. But it seems certain that the electrostatic potential energy, implied by the existence of an electrostatic field, resides in the dielectric that separates the two sides of the field; somewhat as mechanical potential energy resides in a bent spring, a stretched piece of elastic, or a strained elastic solid.

Thus, when we speak of an isolated body charged with + Q

M