Electrometers for the Measurement of Potentials

203. In all electrometers the moveable part is a body charged with electricity, and its potential is different from that of certain of the fixed parts round it.

When, as in Coulomb’s method, an insulated body having a certain charge is used, it is the charge which is the direct object of measurement. We may, however, connect the balls of Coulomb’s electrometer, by means of fine wires, with different conductors.

The charges of the balls will then depend on the values of the potentials of these conductors and on the potential of the case of the instrument. The charge on each ball will be approximately equal to its radius multiplied by the excess of its potential over that of the case of the instrument, provided the radii of the balls are small compared with their distances from each other and from the sides or opening of the case.

Coulomb’s form of apparatus, however, is not well adapted for measurements of this kind, owing to the smallness of the force between spheres at the proper distances when the difference of potentials is small.

A more convenient form is that of the Attracted Disk Electrometer. The first electrometers on this principle were constructed by Sir W. Snow Harris∗ . They have since been brought to great perfection, both in theory and construction, by Sir W. Thomson†

When two disks at different potentials are brought face to face with a small interval between them there will be a nearly uniform electrification on the op- posite faces and very little electrification on the backs of the disks, provided there are no other conductors or electrified bodies in the neighbourhood.

The charge on the positive disk will be approximately proportional to its area, and to the difference of potentials of the disks, and inversely as the distance between them. Hence, by making the areas of the disks large and the dis- tance between them small, a small difference of potential may give rise to a measurable force of attraction.

204. The addition of the guard-ring to the attracted disk is one of the chief improvements which Sir W. Thomson has made on the apparatus.

Instead of suspending the whole of one of the disks and determining the force acting upon it, a central portion of the disk is separated from the rest to form the attracted disk, and the outer ring forming the remainder of the disk is fixed.

In this way the force is measured only on that part of the disk where it is most regular, and the want of uniformity of the electrification near the edge is of no importance, as it occurs on the guard-ring and not on the suspended part of the disk.

Fig. 43.

Besides this, by connecting the guard-ring with a metal case surrounding the back of the attracted disk and all its suspending apparatus, the electrifica- tion of the back of the disk is rendered impossible, for it is part of the inner surface of a closed hollow conductor all at the same potential. Thomson’s Absolute Electrometer therefore consists essentially of two par- allel plates at different potentials, one of which is made so that a certain area, no part of which is near the edge of the plate, is moveable under the action of electric force. To fix our ideas we may suppose the attracted disk and guardring uppermost. The fixed disk is horizontal, and is mounted on an insulating stem which has a measurable vertical motion given to it by means of a mi- crometer screw. The guard-ring is at least as large as the fixed disk; its lower surface is truly plane and parallel to the fixed disk. A delicate balance is erected on the guard-ring to which is suspended a light moveable disk which almost fills the circular aperture in the guard-ring without rubbing against its sides. The lower surface of the suspended disk must be truly plane, and we must have the means of knowing when its plane coincides with that of the lower surface of the guard-ring, so as to form a single plane interrupted only by the narrow interval between the disk and its guard-ring. For this purpose the lower disk is screwed up till it is in contact with the guard-ring, and the suspended disk is allowed to rest upon the lower disk, so that its lower surface is in the same plane as that of the guard-ring. Its posi- tion with respect to the guard-ring is then ascertained by means of a system of fiducial marks. Sir W. Thomson generally uses for this purpose a black hair attached to the moveable part. This hair moves up or down just in front of two black dots on a white enamelled ground and is viewed along with these dots by means of a plano convex lens with the plane side next the eye. If the hair as seen through the lens appears straight and bisects the interval be- tween the black dots it is said to be in its sighted position, and indicates that the suspended disk with which it moves is in its proper position as regards height.

The horizontality of the suspended disk may be tested by compar- ing the reflexion of part of any object from its upper surface with that of the remainder of the same object from the upper surface of the guard-ring. The balance is then arranged so that when a known weight is placed on the centre of the suspended disk it is in equilibrium in its sighted position, the whole apparatus being freed from electrification by putting every part in metallic communication. A metal case is placed over the guard-ring so as to enclose the balance and suspended disk, sufficient apertures being left to see the fiducial marks.

The guard-ring, case, and suspended disk are all in metallic communication with each other, but are insulated from the other parts of the apparatus. Now let it be required to measure the difference of potentials of two con- ductors. The conductors are put in communication with the upper and lower disks respectively by means of wires, the weight is taken off the suspended disk, and the lower disk is moved up by means of the micrometer screw till the electrical attraction brings the suspended disk down to its sighted po- sition. We then know that the attraction between the disks is equal to the weight which brought the disk to its sighted position.

If W be the numerical value of the weight, and g the force of gravity, the force is W g, and if A is the area of the suspended disk, D the distance between the disks, and V the difference of the potentials of the disks,

If the suspended disk is circular, of radius R, and if the radius of the aper- ture of the guard-ring is R′ then

205. Since there is always some uncertainty in determining the micrometer reading corresponding to D = 0, and since any error in the position of the

Let us denote the radius of the suspended disk by R, and that of the aperture of the guard-ring by R′ , then the breadth of the annular interval between the disk and the ring will be B = R′ −R. If the distance between the suspended disk and the large fixed disk is D, and the difference of potentials between these disks is V, then (see Electricity and Magnetism, Art. 201) the quantity of electricity on the suspended disk will be

If the surface of the guard-ring is not exactly in the plane of the surface of the suspended disk, let us suppose that the distance between the fixed disk and the guard-ring is not D but D + z = D′ , then (see Electricity and Magnetism, Art. 225) there will be an additional charge of electricity near the edge of the disk on account of its height z above the general surface of the guard-ring. The whole charge in this case is therefore

and in the expression for the attraction we must substitute for A, the area of the disk, the cor- rected quantity

suspended disk is most important when D is small, Sir W. Thomson prefers to make all his measurements depend on differences of the electromotive force V. Thus, if V and V ′ are two potentials, and D and D′ the corresponding distances,

For instance, in order to measure the electromotive force of a galvanic bat- tery, two electrometers are used.

By means of a condenser, kept charged if necessary by a replenisher, the lower disk of the principal electrometer is maintained at a constant potential. This is tested by connecting the lower disk of the principal electrometer with the lower disk of a secondary electrometer, the suspended disk of which is connected with the earth. The distance between the disks of the secondary electrometer and the force required to bring the suspended disk to its sighted position being constant, if we raise the potential of the condenser till the secondary electrometer is in its sighted position, we know that the potential of the lower disk of the principal electrometer exceeds that of the earth by a constant quantity which we may call V.

If we now connect the positive electrode of the battery to earth, and con- nect the suspended disk of the principal electrometer to the negative electrode, the difference of potentials between the disks will be V + v, if v is the elec- tromotive force of the battery. Let D be the reading of the micrometer in this case, and let D′ be the reading when the suspended disk is connected with where

R = radius of suspended disk, R′ = radius of aperture in the guard-ring, D = distance between fixed and suspended disks, D′ = distance between fixed disk and guard-ring, α = 0.220635(R′ − R).

When α is small compared with D we may neglect the second term, and when D′ − D is small we may neglect the last term.

… earth, then 8πgW ..

In this way a small electromotive force v may be measured by the electrometer with the disks at conveniently measurable distances. When the distance is too small a small change of absolute distance makes a great change in the force, since the force varies inversely as the square of the distance, so that any error in the absolute distance introduces a large error in the result un- less the distance is large compared with the limits of error of the micrometer screw.

The effects of small irregularities of form in the surface of the disks and of the interval between them diminish according to the inverse cube and higher inverse powers of the distance, and whatever be the form of a corrugated sur- face, the eminences of which just reach a plane surface, the electrical effect at any distance which is considerable compared to the breadth of the corru- gations, is the same as that of a plane at a certain small distance behind the plane of the tops of the eminences.

By means of the auxiliary electrification, tested by the auxiliary electrom- eter, a proper interval between the disks is secured. The auxiliary electrometer may be of a simpler construction, in which there is no provision for the determination of the force of attraction in absolute measure, since all that is wanted is to secure a constant electrification. Such an electrometer may be called a gauge electrometer.

This method of using an auxiliary electrification besides the electrification to be measured is called the Heterostatic method of electrometry, in oppo- sition to the Idiostatic method in which the whole effect is produced by the electrification to be measured. In several forms of the attracted disk electrometer, the attracted disk is placed at one end of an arm which is supported by being attached to a plat- inum wire passing through its centre of gravity and kept stretched by means of a spring. The other end of the arm carries the hair which is brought to a sighted position by altering the distance between the disks, and so adjusting the force of the electric attraction to a constant value. In these electrometers

this force is not in general determined in absolute measure, but is known to be constant, provided the torsional elasticity of the platinum wire does not change.

The whole apparatus is placed in a Leyden jar, of which the inner surface is charged and connected with the attracted disk and guard-ring. The other disk is worked by a micrometer screw and is connected first with the earth and then with the conductor whose potential is to be measured. The difference of readings multiplied by a constant to be determined for each electrometer gives the potential required.