Experiment 17: Coulomb’s Proof Plane

68.] It appears, therefore, that the analogy between the conduction of heat and electrostatic phenomena has its limits, beyond which we must not attempt to push it. At the time when it was pointed out by Thomson, men of science ∗ Exp. Res. 1173.

were already acquainted with the great work of Fourier on the conduction of heat in solid bodies, and their minds were more familiar with the ideas there developed than with those belonging to current electricity, or to the theory of the displacements of a medium.

It is true that Ohm had, in 1827, applied the results obtained by Fourier for heat to the theory of the distribution of electric currents in conductors, but it was long before the practical value of Ohm’s work was understood, and till men became familiar with the idea of electric currents in solid conductors, any illustration of electrostatic phenomena drawn from such currents would have served rather to obscure than to enlighten their minds.

69.] When an electric current flows through a solid conductor, the direc- tion of the current at any point is from places of higher to places of lower potential, and its intensity is proportional to the rate at which the potential decreases from point to point of a line drawn in the direction of the current. We may suppose equipotential surfaces drawn in the conducting medium. The lines of flow of the current are everywhere at right angles to the equipo- tential surfaces, and the rate of flow is proportional to the number of equipo- tential surfaces which would be cut by a line of unit length drawn in the direction of the current.

It appears, therefore, that this case of a conducting medium through which an electric current is passing has certain points of analogy with that of a dielectric medium bounded by electrified bodies. In both the medium is divided into layers by a series of equipotential sur- faces. In both there is a system of lines which are everywhere perpendicular to these surfaces. In the one case these lines are called current lines or lines of flow; in the other they are called lines of electric force or electric induction.

An assemblage of such lines drawn from every point of a given line is called a surface of flow. Since the lines of which this surface is formed are ev- erywhere in the direction of the electric current, no part of the current passes through the surface of flow. Such a surface therefore may be regarded as impervious to the current without in any way altering the state of things. If the line from which the assemblage of lines of flow is drawn is one which returns into itself, which we shall call a closed curve, or, more briefly,CURRENT.

a ring, the surface of flow will have the form of a tube and is called a tube of flow. Any two sections of the same tube of flow correspond to each other in the sense defined in Art. 54, and the quantities of electricity which in the same time flow across these two sections are equal. Here then we have the analogue of Faraday’s law, that the quantities of electricity upon corresponding areas of opposed conducting surfaces are equal and opposite.

Faraday made great use of this analogy between electrostatic phenomena and those of the electric current, or, as he expressed it, between induction in dielectrics and conduction in conductors, and he proved that, in many cases, induction and conduction are associated phenomena. Exp. Res. 1320, 1326. We must remember, however, that the electric current cannot be maintained constant through a conductor which resists its passage except by a continual expenditure of energy, whereas induction in a perfectly insulating dielectric between oppositely electrified conductors may be maintained in it for an in- definitely long time without any expenditure of energy, except that which is required to produce the original electrification. The element of time enters into the question of conduction in a way in which it does not appear in that of induction.

70.] But we may arrive at a more perfect mental representation of induction by comparing it, not with the instantaneous state of a current, but with the small displacements of a medium of invariable density. Returning to the case of an electric current through a solid conductor, let us suppose that the current, after flowing for a very short time, ceases. If we consider a surface drawn within the solid, then if this surface intersects the tubes of flow, a certain quantity of electricity will have passed from one side of the surface to the other during the time when the current was flowing. This passage of electricity through the surface is called electric displacement, and the displacement through a given surface is the quantity of electricity which passes through it. In the case of a continuous current the displacement increases continuously as long as the current is kept up, but if the current lasts for a finite time, the displacement reaches its final value and then remains constant. The lines, surfaces, and tubes of flow of the transient current are also lines, surfaces, and tubes of displacement. The displacements across any 2 sections of the same tube of displacement are equal. At the beginning of each unit tube of displacement there is a unit of positive electricity, and at the end of the tube there is a unit of negative electricity. At every point of the medium there is a state of stress consisting of a ten- sion in the direction of the line of displacement through the point and a pres- sure in all directions at right angles to this line. The numerical value of the tension is equal to that of the pressure, namely, the square of the intensity of the electric force divided by 4π.

71.] By the consideration of the properties of the tubes of induction and the equipotential surfaces we may easily prove several important general the- orems in the theory of electricity, the demonstration of which by the older methods is long and difficult. The properties of a tube of induction have already been stated, but for the sake of what follows we may state them again:— (1) If a tube of induction is cut by an imaginary surface, the quantity of electricity displaced across a section of the tube is the same at whatever part of the tube the section be made. (2) In every part of the course of a line of electrostatic force it cuts the equipotential surfaces at right angles, and is proceeding from a place of higher to a place of lower potential. Note. This statement is true only when the distribution of electric force can be completely represented by means of a set of equipotential surfaces. This is always the case when the electricity is in equilibrium, but when there are electric currents, though in some parts of the field it may be possible to draw a set of equipotential surfaces, there are other parts of the field where the distribution of electric force cannot be represented by means of such surfaces. For an electric current is always of the nature of a circuit which returns into itself, and such a circuit cannot in every part of its course be proceeding from places of higher to places of lower potential. 72.] It may be observed that in (1) we have used the words ‘tube of induc- tion,’ and in (2) the words ‘line of electrostatic force.’ In a fluid dielectric, such as air, the line of electrostatic force is always in the same direction as the tube of induction, and it may seem pedantic to maintain a distinction be- tween them. There are other cases, however, in which it is very important toELECTRIFICATION AT ENDS OF INDUCTION TUBE. 63 remember that a tube of induction is defined with respect to the phenomenon which we have called electric displacement, while a line of force is defined with respect to the electric force. In fluids the electric displacement is always in the direction of the electric force, but there are solid bodies in which this is not the case∗ , and in which, therefore, the tubes of induction do not coincide in direction with the lines of force. 73.] It follows from (1) that every tube of induction begins at a place where there is a certain quantity of positive electricity and ends at a place where there is an equal quantity of negative electricity, and that, conversely, from any place where there is positive electricity a tube may be drawn, and that wherever there is negative electricity a tube must terminate. 74.] It follows from (2) that the potential at the beginning of a tube is higher than at the end of it. Hence, no tube can return into itself, for in that case the same point would have two different potentials, which is impossible. 75.] From this we may prove that if the potential at every point of a closed surface is the same, and if there is no electrified body within that surface, the potential at any point within the region enclosed within the closed surface is the same as that at the surface. For if there were any difference of potential between one point and another within this region, there would be lines of force from the places of higher towards the places of lower potential. These lines, as we have seen, cannot return into themselves. Hence they must have their extremities either within the region or without it. Neither extremity of a line of force can be within the region, for there must be positive electrification at the beginning and negative electrification at the end of a line of force, but by our hypothesis there is no electrification within the region. On the other hand, a line of force within the region cannot have its extremities without the region, for in that case it must enter the region at one point of the surface and leave it at another, and therefore by (2) the potential must be higher at the point of entry than at the point of issue, which is contrary to our hypothesis that the potential is the same at every point of the surface. Hence no line of force can exist within the region, and therefore the po- ∗ See the experiments of Boltzmann on crystals of sulphur. Vienna Sitzungsb. 9 Jan. 1873.NO ELECTRIFICATION WITHIN A HOLLOW CONDUCTOR. 64 tential at any point within the region is the same as that at the surface itself. 76.] It follows from this theorem, that if the closed surface is the internal surface of a hollow conducting vessel, and if no electrified body is within the surface, there is no electrification on the internal surface. For if there were, lines of force would proceed from the electrified parts of the surface into the region within, and we have already proved that there are no lines of force in that region.

We have already proved this by experiment (Art. 20), but we now see that it is a necessary consequence of the properties of lines of force.