CHAPTER III.

ELECTRIC INDUCTION.

21. Lines of Force.—We have already (§ 18) defined the intensity of electric force at a given point; it is the resultant of the forces exerted by every portion of electricity upon a unit of positive electricity imagined as being at that point.1

The intensity of electric force has a definite magnitude and direction at every point in an electric field. If a line is drawn in the field so that it is, at every point in its course, tangential to the direction of electric force, it is called a line of force. It represents the path which would be followed by an electrified particle without mass, entirely free and unacted on by any other force. It is reckoned as running in the direction in which a positively electrified particle could move. Lines of force, therefore, are to be thought of as traversing the whole electric field from the positive to the negative boundary.

One line and one only passes through every point of an electric field. If it were possible for two lines of force to pass through the same point, the resultant electric force at this point would have two directions at the same time, which is absurd.

As we have seen (§§ 14, 18), there is no electric force inside a conductor, and, just outside, the direction of force is everywhere normal to the surface. Consequently, lines of force terminate at conducting surfaces, and always meet them normally. A surface from which lines of force start is a positively electrified surface; one at which they end is negatively electrified.

If it were possible to draw all the lines of force of an electric field, the resulting diagram, being completely full of lines, would be unintelligible; but by drawing representative lines selected according to some easily recognisable system, the general course

1 As it is only in imagination that a unit of electricity is put at the point in question, there is no need to be concerned about possible disturbances of equilibrium which it might cause if put there in reality.

of the whole set may be indicated, and so the character of all parts of the electric field, as to direction and intensity, can be represented. The rule generally followed is to make the number of lines such that the number traversing the unit area of a surface, supposed perpendicular to their direction at any part of the field, shall represent the electric intensity at that part. Hence, if the electric intensity diminishes as we follow a line of force, this is represented by divergence of the lines, and increase of intensity by convergence. A uniform field, that is, one in which the intensity and direction are everywhere the same, is represented by equidistant parallel straight lines, and reciprocally.

In many cases it facilitates thinking and reasoning about the properties of the electric field to refer the stresses of which it is the seat to the lines of force, and to think of these as having not merely a geometrical significance, but as possessing certain physical properties, namely, a tendency for each line to shorten, and a tendency for separate lines to repel each other.

22. Electrification by Induction.—There being no electric force within a conductor, there can be no lines of force within it. Suppose, then, an unelectrified piece of metal placed in an electric field the space outside the surface of the metal is to be thought of as full of lines of force, and the space inside the surface as free from such lines. The conductor thus represents a gap in the field, or an interruption of the lines of force. The lines which previously traversed the space occupied by it are each cut into two branches, one branch ending on one part of the surface, and the other, originally the continuation of the same line, starting from another part. But (§ 21) a surface at which lines of force end is negatively electrified, and one from which such lines start is positively electrified. Hence we should expect an insulated unelectrified conductor immersed in an electric field to acquire equal opposite electrifications at different parts of its surface, that part becoming positive which is presented towards the negative boundary of the field, and vice versa.

Such electrification is found experimentally to occur, and is known as electrification by induction.

Fig. 19 represents the distribution of lines of force for the case of a metal ball placed in an originally uniform electric field. It will be seen that in this case there is not only an interruption, but also a deflexion of the lines by the ball. This is a necessary consequence of the fact that the lines meet a conducting sur

face at right angles. Again, a deflexion of the lines that actually encounter the conductor results, on account of the transverse stress in the field or the mutual repulsion of the lines of force, in an appreciable deflexion of some of those which do not encounter the surface.

FIG. 19.

23. Laws of Electrostatic Induction.-In order to give a more complete account of electrostatic induction we will consider a special typical case of a kind that frequently occurs in actual experiments.

a. Consider an insulated metal ball or other conductor that has been positively electrified by an ordinary electrical machine, the rubber of which is in electrical connection with the floor of the room. We have, then, an electric field extending from the ball on all sides to the floor, walls, and ceiling, which together constitute the second boundary. To assist the mind in picturing the distribution of force in the field, we may think of the lines of force as if they were elastic threads extending between the two boundaries of the field, each one repelling every other, and their extremities capable of moving freely about on the conducting surfaces to which they are attached, but not able to leave these surfaces, except when, under extreme conditions, by the crowding together of lines, the force in any part of the field becomes so great that the medium breaks down and disruptive discharge takes place.

Mechanical consequences of the suppositions made as to the properties of the lines of force are that they would all meet the

boundaries of the field at right angles, and that there would be the greatest concentration of lines at the parts of the field where the opposite boundaries are nearest together, and where, therefore, the lines are shortest. These results correspond to the facts already pointed out (§§ 18, 20), that the direction of electric force close to the surface of a conductor is normal to the surface, and that the electric force is relatively great at any part of an electric field where the opposite boundaries come near together. Again, since the beginning of a line of force corresponds to a positive charge, and the termination of a line to a negative charge, and since the number of beginnings and endings are necessarily equal, the conception of lines of force includes the essential equality of the opposite charges on the two boundaries of an electric field. It was, moreover, pointed out in § 19 that the electric force, just outside a uniformly electrified sphere, is equal to the surface-density of the charge multiplied by 4′′ and divided by the dielectric coefficient of the surrounding medium; and we shall see subsequently (§ 43) that this relation between electric force and surface density of electrification is quite general. Hence, a statement of the number of lines of force starting from or ending upon a given area, and a statement of the surface-density of the charge of the same area, are only different modes of expressing the same facts.

room.

b. Let the conductor, A, be completely surrounded by a hollow conductor, B, of any shape whatever. This conductor divides the electric field into two parts, that between A and the inner surface of B, and that between the outer surface of B and the inside of the All the lines of force issuing from A are cut by B, so that the number starting from a and ending on the inner surface of B is identical, and so that also the continuation of every line ending inside B is represented by a line issuing from its outer surface and ending on the surface of the room. In other words, the surface of the charged body A, the inner and outer surfaces of the surrounding conductor B, and the surface of the room, are all charged with equal quantities of electricity, the charges of the two external surfaces (that of A and the outer surface of B) being of one kind, and those of the two internal surfaces (the inner surface of B and that of the room) being of the opposite kind.

Since no force acts inside the conductor B, no force can be transmitted through it, and the electric fields inside it and outside it are entirely independent of each other so far as concerns the distribution of force in each, or (what comes to the same thing)

the density of charge at any part of the boundary of each. In fact, surface-density and distribution of lines of force in each field are entirely determined by the forces of that field.

c. The two fields-for convenience we may call them internal and external-are indeed so entirely independent of each other, that either of them may be destroyed without in any way affecting the other. For example, if the electrified body A be displaced so as to come into contact with the inside of B, the internal field ceases to exist, but no electrical effect of any kind is thereby produced outside B. Again, if the conductor B be put into contact with the inside of the room, either directly or by means of a wire, the external field is destroyed, but nothing whatever is thereby altered inside B. To say that an electric field anywhere ceases to exist is the same thing as to say that there all electric force ceases to act; hence, from the point of view which refers electric force to the action at a distance of electric charges, we may express the facts last stated by saying that the two internal charges mutually neutralise each other so far as regards all action outside the conductor B, and that the two external charges mutually neutralise each other in regard to action inside B.

d. If, after the external field has been destroyed, the conductor B is opened and the electrified body A is withdrawn, what was previously the internal field remains still unchanged as to the number of lines of force in it-that is, as to the quantity of electricity on its bounding surfaces; but the form of the field is altered by the changed relative positions of its boundaries. In accordance with their tendency to shorten and to separate from each other, the lines of force which previously all ended on the inner surface of B come, more and more of them, to end on the outer surface as A is removed to a greater distance. If A is put in contact with the inside of the room, it becomes electrically part of a conducting enclosure surrounding B, and the result is that we have now an electric field whose positive boundary is the surface of the room, and whose negative boundary is the outside of the conductor B. We have then, in a sense, reversed the original field extending between A and the room: the charges remain throughout of the same magnitude as at first (always assuming perfect insulation), but the charge of the room is positive instead of negative, and that of the enclosed conductor is negative instead of positive.

e. If an insulated conductor is placed in the electric field existing

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