charge is moved from Q1

values it has at Q1 and Q2.

to Q, has a value intermediate between the Such an intermediate value will be obtained

by taking the geometrical mean, QKdd, of the forces at Q and QË Thus the work done while carrying the charge from Q1 to Q2 will be

In the same way, the work done while carrying the unit charge from Q2 to Q3 will be

Adding together the work done over all the elements of the path we shall obtain the whole work, that is, the potential, V, of the point Q Thus

Now it will be seen that in this expression the distance of each of the points Q, Q3, &c., occurs twice, once positively and once negatively, so that when we add together all the terms these positive and negative values will cancel, and we are left with the first term only, for the value of the last term 1 is zero. Thus

V=Q|Kd.

Hence the potential at a point at a distance d from a uniformly charged sphere is numerically equal to the charge on the sphere divided by K ́ times the distance of the point from the centre of the sphere. If the medium surrounding the sphere is air, the potential is obtained by putting

K=1.

If the point Q, is taken close to the surface of the sphere, the work which has to be done to carry the unit charge from Q, to infinity is the measure of the potential of the sphere. Thus the potential of a sphere, when at a great distance from all other conductors and charged with Q units, is QRK, or, if the medium is air, is QR. Now the capacity of a conductor is the ratio of the charge to the potential to which the conductor is raised by that charge. Thus the capacity of the sphere is KR, or, if surrounded by air, is R, that is, the capacity of a sphere in air is numerically equal to the radius.

465*. Capacity of a Spherical Condenser. The problem of calculating the capacity of a system of conductors of given form is in general very difficult to solve; the case however of a condenser, such as

that shown in Fig. 443, where the two coatings are concentric spheres, can be readily obtained. Let R be the radius of the outside sphere which is connected to earth, and the radius of the inside sphere. Let the charge on the inside sphere be Q, and the difference of potential between the two spheres be V. Then Q lines of force leave the inside sphere, and, since each of these tubes of force terminates on the inner surface of the outside sphere, there must be a charge of Q units, but of opposite sign to the charge on the inside sphere, induced on the outside sphere. If the charge on the outside sphere alone were present, the potential within this sphere would be everywhere constant, and equal to the value it has at the surface of the sphere, for, as we have shown in § 449, the potential inside a charged conductor is everywhere equal to the potential at the surface of the conductor. Hence, owing to the charge on the outside sphere, the potential everywhere inside is

Q

R'

for this is

the potential to which a charge Q will raise a sphere of radius R, the capacity of such a sphere being numerically equal to the radius. If the charge on the inside sphere were alone present, the potential at its surface would be Qr. Since the potential at any point due to the simultaneous action of two charges is the sum of the potentials which each would produce if it acted alone, the potential, V, of the inside sphere, when the outer sphere is present, is given by

V=Qr=QR = Q( − })⋅

But the capacity, C, of the condenser is equal to Q/V. Hence

rR
C=
R-r

If the dielectric separating the two spheres, instead of being air, has a specific inductive capacity K, the capacity will be

KrR
R-r

If the thickness, R-r, of the dielectric is small, the radii R and r will
be very nearly equal, so that, if d is the thickness of the dielectric,
KR2
If S is the surface of the inside sphere, we have S=4′′R2, or

C = d
R2=S/4. Hence under these circumstances the capacity, C, can be

written C=

SK 4d'

Although this formula only strictly applies to the case of a spherical condenser, yet it holds approximately in the case of the ordinary form of Leyden jar, in which the outside coating does not completely surround

the inside coating, and it is sometimes of use for calculating the approximate capacity of jars.

The expression for the capacity of a spherical condenser can be written in the form C=If in this expression we make ✯ infinite, R I-rR

we get C=r, which corresponds to the case of a sphere removed from all other conductors. Hence this case may be regarded as a condenser in which the outer coating has been removed to an infinite distance. This corresponds to what was said in § 444, as to the fact that the lines of force which leave a charged body must terminate on some body, and that where they terminate will be found a charge equal in magnitude, but opposite in sign to the charge on the electrified body.

CHAPTER VI

ELECTROMETERS AND ELECTRICAL MACHINES

466. The Attracted Disc Electrometer. Suppose that two conducting planes, AB and CED (Fig. 450), are placed parallel to one another, and at a distance d apart, so small compared to their size that the disturbing effect of their edges produces no effect at the central portions,

E

FIG. 450.

so that the field of force between the plates is in these parts uniform. We require to find the attraction exerted on a portion, E, of the one plane of area S, when the two planes are charged to a difference of potential, I, the dielectric being air. Suppose that the surface density of the charge on AB is + σ, and that on CD is σ, then a tubes of force will terminate on each square centimetre of the plane CD, or at any rate on each square centimetre .of the central portion, E. Hence, since these tubes are all normal to the surface of E, and each exerts a mechanical force F2 (§ 460), the total attraction exerted on E by the charged plate AB is FSo/2.

Now the cross-section of each tube of force being 1o, the electrical force, F, acting at any point between the plates is 4σ. Hence the attraction, f, acting on E is given by

f=2πSo.

Since the electrical force, F, acting on the unit charge anywhere between the plates is 4′′σ, the work that must be done to carry the unit charge from one plane to the other is 4od, and therefore l'=47σd. Hence σ = 4d, and substituting this value for σ in the expression for the attraction exerted on E, we get

Hence by measuring the force exerted on a portion of area S of the plate CD, when the distance between the plates is d, and they are charged to a difference of potential, we can calculate the value of this

difference of potential. This then gives a method of obtaining the value of a given difference of potential in terms of the units of mass, time (involved in the value of the force), and length, that is, of determining a difference of potential in absolute units (§ 8).

The portion of the plate CD, which surrounds the part E on which the attractive force is measured, is called by Lord Kelvin, to whom the arrangement is due, the guard ring. The functions of the guard ring are simply to insure that the electrical field at the part of the plates where the attracted part E is placed shall be uniform.

In the instrument depending on this principle invented by Lord Kelvin, and called the attracted disc electrometer, or the absolute electrometer, the part E on which the force is measured consists of a metal disc supported by three springs, so that it lies concentrically within a circular hole in the guard ring, to which it is electrically connected. The springs are so arranged that when the attracted disc is attracted with a certain force by the opposite plate, AB, it lies exactly in the plane of the guard ring, as indicated by means of two sights which are attached. The plate AB can be moved in a direction parallel to its normal by means of a micrometer screw. When using the instrument the guard ring and attracted disc are connected with earth, so that their potential is zero, and the other plate is connected with the body of which the potential is to be measured. The distance between the two plates is then altered till the disc comes into its sighted position. The force necessary to bring the disc into its sighted position is determined once for all by placing weights on it, and hence, knowing this quantity (f in the formula), and also knowing the distance between the plates from the reading of the micrometer screw, the potential can be obtained.

467. The Quadrant Electrometer. The absolute electrometer, although it permits of our measuring a given potential in absolute measure, is not very sensitive, and is not at all suited for comparing two small differences of potential, or for detecting the existence of a small difference of potential. Hence another form of