the opposite kind of electrification to that of the inducing body. In the case when the inducing charge is removed before the conductor has been put to earth, the reason why, on the removal of the inducing charge, the conductor was unelectrified was that the two kinds of electrification, produced in equal quantities by the induction, neutralise each other. We shall return to the consideration of the subject of induction after we have dealt with the quantitative measurement of electrification.

443. Coulomb's Law. By means of the torsion balance, Coulomb was able to show that the force exerted on one another by two charged conductors is directly proportional to the product of their charges, and inversely proportional to the square of the distance between the bodies.

Hence, as in the case of the unit magnetic pole, we may define the unit electrification or charge as such that if two small bodies, each charged with a unit, are placed at one centimetre apart in air, the force they will exert on one another will be one dyne. The reason the medium air is specified is that, as we shall see later, the force exerted between two charged bodies depends on the nature of the medium which fills the space between them, while the reason the bodies on which the charges are supposed to exist are taken as small is that if the bodies were of appreciable magnitude the distribution of the electrification would be altered by the action of the one charge on the other.

Suppose then we had two points, charged with e and e' units of electricity respectively, placed at a distance r apart in air, the force, F, which they would exert one on the other, due to their electrification, will be given by the equation—

The force will be an attraction if the charges e and e' are of opposite sign, and a repulsion if they are of the same sign. In the case of the unit of electrification, as we shall see later, we meet with a case where there are two separate relations commonly employed to connect the quantity to be measured with the fundamental units (§ 8). Hence, in order to distinguish the unit as defined above, which depends for its definition on the force exerted between two charged bodies, and another unit which we shall consider later, and which depends for its definition on another physical property of a charged body, the unit above defined is called the electro-static unit of quantity of electricity or charge.

CHAPTER IV

THE ELECTRICAL FIELD

444. Electrical Lines of Force.-If a small body, charged with the unit positive charge, is brought into the neighbourhood of a charged body, this unit charge will be acted upon by an electrical force, which at every point of the space surrounding the charged body will have a definite magnitude and direction. As in the case of magnetism, a line, such that its direction at every point is the same as the direction of the force acting on the unit charge when placed at the point, is called a line of force. The direction in which a line of force is supposed to run is the direction in which a small positively electrified body would tend to move. Hence a line of force will always start from a body which is positively electrified and end on a body which is negatively electrified.

If we take an area on the surface of a positively electrified body, such that this portion of the surface contains a unit of electricity, and, starting from all points on the curve bounding this area, draw the lines of force, these lines of force will enclose a tube-shaped space which is called a tube of force. Since each of the lines of force must terminate on a negatively electrified body, we see that every tube of force must end on a negatively electrified body, and it can be shown that the quantity of electricity on that portion of the surface enclosed by the tube of force will be a unit of negative electricity. By means, therefore, of tubes of force, we can indicate the distribution of the electrification on the surface of a charged body, for the greater the charge the smaller will be the cross-section of the tubes of force, and hence the larger the number of them which will leave each square centimetre of the surface of the charged body. Since it would be rather inconvenient to draw a series of tubes, it is usual to suppose that a single line of force is drawn along the axis of each unit tube of force, that is, that from the centre of each element of the surface of the positively electrified body on which the unit quantity of positive electricity exists we draw a line of force. Under these circumstances the number of these lines of force which leave the surface of the charged body will represent the charge on the surface. If a body is charged with e units of electricity, then e lines of force will leave the surface of the body, while if the body is charged with e units of negative electricity, e lines of force will terminate

on the surface of the body. It will thus be seen that if a body is charged with e units of positive electricity, so that e lines of force leave the body and must terminate on a negatively charged body, somewhere or other there must necessarily exist e units of negative electricity. This negative charge may, however, be so far removed from the spot where we are making our experiments that it does not in any way affect the results, and hence we are able to perform experiments in which we practically have only to deal with one kind of electrification. Here we have a marked difference between magnetism and electricity, for in the case of magnetism we are unable to obtain a body which has only one pole, and so cannot deal with a single pole.

The space in the neighbourhood of electrified bodies in which electrical phenomena, such as attraction, are exhibited is called an electrical field. A field in which the force acting on a small electrified body is everywhere the same both in magnitude and direction is called a uniform field. In a uniform field the lines of force must be everywhere parallel, and therefore the tubes of force must everywhere have the same crosssection.

The quantity of electrification on the unit of area of the surface of an electrified body is called the surface density of the electrification, and, as we have seen, the number of lines of force which leave or terminate on the unit of area of the surface is also equal to the charge on the unit area. Hence the surface density may also be defined as the number of lines or tubes of force which leave or terminate on the unit area of the surface of the electrified body. In the case where the electrification of the body is not uniform, the surface density at a given point is defined, as in the case of other variable quantities, as the quantity of electricity on a small element of surface surrounding the given point divided by this area.

The lines of force in the case of two small bodies, one of them positively and the other negatively electrified, and placed at a very great distance from all other conductors, so that all the lines of force which leave the positively electrified body terminate on the negatively electrified body, are shown in Fig. 432,1 while in Fig. 433 the lines of force in the case where the two bodies are electrified with the same kind of electrification are shown.

As in the corresponding case in magnetism, we may account for the attraction which takes place in the one case, and the repulsion in the other, if we suppose that there exists a tension along the lines of force, and that something of the nature of an hydrostatic pressure acts at right angles to the direction of the lines, so that they repel one another.

When the electrical charge on any system of conductors alters its distribution, we may consider that each unit of the charge, as it moves over the surface of the conductors, drags the end of its tube of force after

1 The lines of force are symmetrical about the line joining the charges, and so to save space only half are shown.

it, but that, on account of the tension acting along the tube, the tendency is for the tube to become as short as possible. When the two conductors,

ends of the line of force to tend to approach each other, and can only be kept apart by the interposition of a non-conductor, we shall be able to explain how it is that one of the kinds of electricity produced by induction remains on the body when the latter is put to earth, while the other kind of electrification escapes to earth.

In Fig. 434 let A represent the inducing body, which we may suppose charged with positive electricity, and B be an insulated conductor which is electrified by induction by A. Then some of the lines of force (shown by the full lines) which leave A will terminate on B, and B will therefore be negatively electrified at the part where these lines meet the surface. In addition a number of lines of force will leave B, and terminate on surrounding conductors, such as the walls of the room in which the two

bodies are placed. The part of B where these lines leave the surface will be positively electrified, the corresponding negative charge being on the walls. The lines of force which stretch from A to B, by their tension, cause the negative charge on B to accumulate on the side next A. The whole charge does not accumulate at the nearest point, however, because of the mutual repulsion which the lines of force exert on one another. It

is owing to this repulsion between the lines that the lines leaving the body B accumulate at the other end.

When the body B is put in conducting communication with the earth, i.e. with the bodies on which the lines of force which leave it terminate, owing to the action of the tension on the electrification itself, the latter will escape, but the negative electrification corresponding to the lines of force which leave A and terminate on B will not be able to reach A, since these two bodies are not in conducting communication. The distribution