FARADAY’S LAW OF LINES OF INDUCTION

55.] Faraday in his electrical researches employs the lines of force to indicate, not only the direction of the electric force at each point of the field, but also the quantity of electrification on any given portion of the electrified surface.

If we consider a portion of an electrified surface as cut off from the rest by the bounding line which surrounds it, and if from every point of this bounding line we draw a line of force, producing it till it meets the surface of some other body in a point which is said to correspond to the point of the body from which the line was drawn, these lines will form a tubular surface, and will cut off a certain portion from the surface of the other body corresponding to the portion of the surface of the first body, and the total electrifications of the two corresponding portions are equal in numerical magnitude but opposite in kind.

56.] A particular instance of Faraday’s law is that which we have already proved by experiment, namely, that the electrification of the inner surface of a closed conducting vessel is equal and opposite to that of an electrified body placed within it. Here we have a relation between the whole electrification of the inner surface and that of the opposed surface of the interior body.

Faraday’s law asserts that, by drawing lines of force from the one surface to the other, points corresponding to each other in the two surfaces may be found; that corresponding lines are such that any point of one has its corresponding point in the other; and that the electrifications of the two portions of the opposed surfaces bounded by such corresponding lines are equal and opposite. 57.] We have called these lines ‘lines of force’ because we began by defin- ing them as lines whose direction at every point coincides with that of the electric force.

Every line of force begins at a positively electrified surface and ends at a negatively electrified surface, and the points of these surfaces at which it begins and ends are called corresponding points.

A system of lines of force forming a tubular surface closed at the one end by a portion of the positively electrified surface and at the other by the corresponding portion of the negative surface, is called by Faraday a Tube of Induction, because electric induction, according to Faraday, is that condition of the dielectric by which the electrifications of the opposed surfaces are placed in that physical relation to one another, which we express by saying that their electrifications are equal and opposite.

Properties of a Tube of Induction.

58.] (1) The electrification of the portion of the positively electrified sur- face from which the tube of induction proceeds is equal in numerical value but opposite in sign to the negative electrification of the portion of the surface at which the tube of induction terminates.

By dividing the positive surface into portions, the electrification of each of which is unity, and drawing tubes corresponding to each portion, we obtain a system of unit tubes, which will be very convenient in describing electric phenomena. For in this case the electrification of any surface is measured by the number of tubes which abut on it. If they proceed from the surface, this number is to be taken as representing the positive electrification; if the tubes terminate at the surface, the electrification is negative. It is in this sense that Faraday so often speaks of the number of lines of force which fall on a given area.

If we suppose an imaginary surface drawn in the electric field, then the quantity of electrostatic induction through this surface is measured by the number of tubes of induction which pass through it, and is reckoned positive or negative accordingly as the tubes pass through it in the positive or negative direction.

Note. By an imaginary surface is meant a surface which has no physi- cal existence, but which may be imagined to exist in space without interfer- ing with the physical properties of the substance which occupies that space. Thus we may imagine a vertical plane dividing a man’s head longitudinally into two equal parts, and by means of this imaginary surface we may render our ideas of the form of his head more precise, though any attempt to con- vert this imaginary surface into a physical one would be criminal. Imaginary quantities, such as are mentioned in treatises on analytical geometry, have no place in physical science.

59.] In every part of the course of a line of electrostatic induction it is passing from places of higher to places of lower potential, and in a direction at right angles to the equipotential surfaces which it cuts. We have seen that the electric field is divided by the equipotential surfaces into a series of shells, like the coats of an onion, the thickness of each shell at any point being inversely as the electric force at that point. We have now divided the electric field into a system of unit tubes of induc- tion, the section of each tube at any point varying inversely as the intensity of the electric induction at that point.

Each of these tubes is cut by the equipotential surfaces into a number of segments which we may call unit cells.

60.] If we take the simplest case—that of a single positively electrified body placed within a closed conducting vessel, all the tubes of induction be- gin at the positively electrified body and end at the negatively electrified inner surface of the vessel. The number of these tubes, since they are unit-tubes, is equal to the number of electrical units in the charge of the electrified body. Each of them cuts all the equipotential surfaces which enclose the electrified body and are enclosed by the vessel. Each tube, therefore, is divided into a number of cells representing the difference of the potential of the electrified body from that of the vessel.

If e is the charge of the body and p its potential, E and P being the charge and potential of the vessel, the whole number of cells is e(p − P ), or, since E = −e, we may write this expression ep + EP.

This is double of the expression which we formerly obtained for the electrical energy of the system (see Art. 31). Hence in this simple case the number of cells is double the number of units of energy in the system. If there are several electrified bodies, A, B, C, &c., the tubes of induction proceeding from one of them, A, may abut either on the inner surface of the surrounding vessel or on one of the other electrified bodies.

ENERGY OF AN ELECTRIFIED SYSTEM.

Let E1 , E2 , E3 be the charges of A, B, C and P1 , P2 , P3 their potentials, the charge and potential of the vessel being E0 and P0 . Let EAB , EAC , EAO denote the number of tubes of induction which pass from A to the conductors B and C and the vessel O, respectively. Then the whole number of cells will be EAB (P1 − P2 ) + EAC (P1 − P3 ) + EAO (P1 − P0 ), +EBC (P2 − P3 ) + EBO (P2 − P0 ),

• ECO (P3 − P0 ). By arranging the terms according to the potentials involved in them, and remembering that since EAB denotes the number of tubes which pass from A to B, EBA must denote the number which pass from B to A and therefore EBA = −EAB , the expression may be written P1 (EAB + EAC + EAO ), +P2 (EBC + EBO + EBA ), +P3 (ECO + ECA + ECB ), +P0 (EOA + EOB + EOC ). Now EAB + EAC + EAO is the whole number of tubes issuing from A and this therefore is equal to E1 , the charge of A, and the coefficients of the other potentials are also the charges of the bodies to which they refer, so that the final expression is P 0 E0 + P 1 E1 + P 2 E2 + P 3 E3 , and this is double the energy of the system. Hence, whether there is one electrified body or several, the number of cells is twice the number of units of electrical energy in the system. 61.] This remarkable correspondence between the number of cells into which the tubes of induction are cut by the equipotential surfaces, and the electrical energy of the system, leads us to enquire whether the electrical energy may not have its true seat in the dielectric medium which is thus cut up into cells, each cell being a portion of the medium in which half a unit of energy is stored up.

ELECTRIC DISPLACEMENT

We have only to suppose that the electromotive force, when it acts on a dielectric, puts it into a certain state of constraint, from which it is always endeavouring to relieve itself. To make our conception of what takes place more precise, let us consider a single cell belonging to a tube of induction proceeding from a positively elec- trified body, the cell being bounded by two consecutive equipotential surfaces surrounding the body. We know that there is an electro- motive force acting outwards from the electrified body. This force, if it acted on a conducting medium, would pro- duce a current of electricity which would last as long as the force contin- ued to act.

The medium however is a non-conducting or dielectric medium, and the effect of the electromotive force is to produce what we may call electric displacement, that is to say, the electricity is forced outwards in the di- Fig. 17. rection of the electromotive force, but its condition when so displaced is such that, as soon as the electromotive force is removed, the electricity resumes the position which it had before dis- placement.

The amount of electric displacement is measured by the quantity of elec- tricity which crosses an imaginary fixed surface drawn parallel to the equipo- tential surfaces.

We know absolutely nothing with respect to the distance through which any particular portion of electricity is displaced from its original position. The only thing we know is the quantity which crosses a given surface. The greater the amount of electricity which we suppose to exist, say, in a cubic inch, the smaller the distance through which we must suppose it displaced in order that a given quantity of electricity may be displaced across a square inch of area

ELECTRIC TENSION.

fixed in the medium. It is probable that the actual motion of displacement is exceedingly small, in which case we must suppose the quantity of electricity in a cubic inch of the medium to be exceedingly great. If this is really the case the actual velocity of electricity in a telegraph wire may be very small, less, say, than the hundredth of an inch in an hour, though the signals which it transmits may be propagated with great velocity.

62.] The displacement across any section of a unit tube of induction is one unit of electricity and the direction of the displacement is that of the electro- motive force, namely, from places of higher to places of lower potential. Besides the electric displacement within the cell we have to consider the state of the two ends of the cell which are formed by the equipotential sur- faces. We must suppose that in every cell the end formed by the surface of higher potential is coated with one unit of positive electricity, the opposite end, that formed by the surface of lower potential, being coated with one unit of negative electricity. In the interior of the medium where the positive end of one cell is in contact with the negative end of the next, these two electrifications exactly neutralise each other, but where the dielectric medium is bounded by a conductor, the electrification is no longer neutralised, but constitutes the observed electrification at the surface of the conductor. According to this view of electrification, we must regard electrification as a property of the dielectric medium rather than of the conductor which is bounded by it.

63.] If we further admit that in every part of a dielectric medium through which electric induction is taking place there is a tension, like that of a rope, in the direction of the lines of force, and a pressure in all directions at right angles to the lines of force, we may account for all the mechanical actions which take place between electrified bodies. The tension, referred to unit of surface, is proportional to the square of the electromotive force at the point. The pressure has the same numerical value, but is, of course, opposite in sign.

In my larger treatise on electricity a proof is given of the fact that a system of stress such as is here described is consistent with the equilibrium of a fluid dielectric medium, and that this state of stress in the medium is mechanically equivalent to the attraction or repulsion which electrified bodies manifest.

ANALOGIES BETWEEN ELECTROSTATICS AND HEAT.

I have not, however, attempted, by any hypothesis as to the internal consti- tution of the dielectric medium, to explain in what way the electric displace- ment causes or is associated with this state of stress.

We have thus, by means of the tubes of induction and the equipotential surfaces, constructed a geometrical model of the field of electric force. Dia- grams of particular cases are given in the figures at the end of this book. The direction and magnitude of the electric force at any point may be indi- cated either by means of the equipotential surfaces or by means of the tubes of induction. Hence, when it is expressed in both ways, we may by the study of the relation between the equipotential surfaces and the tubes of induction deduce important theorems in the theory of electricity.