PARTICULAR CASES OF ELECTRIFICATION

98.] These diagrams are constructed in the following manner:

First, take the case of a single centre of force, a small electrified body with

E E a charge E. The potential at a distance r is V = ; hence, if we make r = , r V

we shall find r, the radius of the sphere for which the potential is V. If we now give to V the values 1, 2, 3, &c., and draw the corresponding spheres, we shall obtain a series of equipotential surfaces, the potentials corresponding to which are measured by the natural numbers.

The sections of these spheres by a plane passing through their common centre will be circles, which we may mark with the number denoting the potential of each. These are indicated by the dotted circles on the right hand of Fig. 21.

If there be another centre of force, we may in the same way draw the equipotential surfaces belonging to it, and if we now wish to find the form of the equipotential surfaces due to both centres together, we must remember that if V1 be the potential due to one centre, and V2 that due to the other, the potential due to both will be V1 + V2 = V, Hence, since at every intersection of the equipotential surfaces belonging to the two series we know both V1 and V2 , we also know the value of V. If therefore we draw a surface which passes through all those intersections for which the value of V is the same, this surface will coincide with a true equipotential surface at all these inter- sections, and if the original systems of surfaces be drawn sufficiently close, the new surface may be drawn with any required degree of accuracy. The equipotential surfaces due to two points whose charges are equal and oppo- site are represented by the continuous lines on the right hand side of Fig. 21.

This method may be applied to the drawing of any system of equipotential surfaces when the potential is the sum of two potentials, for which we have already drawn the equipotential surfaces. The lines of force due to a single centre of force are straight lines radiating from that centre. If we wish to indicate by these lines the intensity as well as the direction of the force at any point, we must draw them so that they mark out on the equipotential surfaces portions over which the surface-integral of induction has definite values. The best way of doing this is to suppose our plane figure to be the section of a figure in space formed by the revolution of the plane figure about an axis passing through the centre of force. Any straight line radiating from the centre and making an angle θ with the axis will then trace out a cone, and the surface-integral of the induction through that part of any surface which is cut off by this cone on the side next the positive direction of the axis, is 2πE(1 − cosθ).

If we further suppose this surface to be bounded by its intersection with two planes passing through the axis, and inclined at the angle whose arc is equal to half the radius, then the induction through the surface so bounded is E(1 − cosθ) = 2Ψ, say; and Ψ θ = cos−1 (1 − 2 ) . E

If we now give to Ψ a series of values 1, 2, 3 … E, we shall find a corre- sponding series of values of θ, and if E be an integer, the number of corre- sponding lines of force, including the axis, will be equal to E. We have therefore a method of drawing lines of force so that the charge of any centre is indicated by the number of lines which converge to it, and the induction through any surface cut off in the way described is measured by the number of lines of force which pass through it. The dotted straight lines on the left hand side of Fig. 21 represent the lines of force due to each of two electrified points whose charges are 10 and −10 respectively.

If there are two centres of force on the axis of the figure we may draw the lines of force for each axis corresponding to values of Ψ1 and Ψ2 , and then, by drawing lines through the consecutive intersections of these lines, for which the value of Ψ1 + Ψ2 is the same, we may find the lines of force due to both centres, and in the same way we may combine any two systems of lines of force which are symmetrically situated about the same axis. The continuous curves on the left hand side of Fig. 21 represent the lines of force due to the electrified points acting at once.

After the equipotential surfaces and lines of force have been constructed by this method, the accuracy of the drawing may be tested by observing whetherFig. 21. Method of drawing Lines of Force and Equipotential Surfaces.EQUIPOTENTIAL SURFACES AND LINES OF INDUCTION. 89 the two systems of lines are everywhere orthogonal, and whether the distance between consecutive equipotential surfaces is to the distance between consec- utive lines of force as half the distance from the axis is to the assumed unit of length. In the case of any such system of finite dimensions the line of force whose index number is Ψ has an asymptote which passes through the centre of grav- ity of the system, and is inclined to the axis at an angle whose cosine is Ψ 1 − 2 , where E is the total electrification of the system, provided Ψ is less E than E. Lines of force whose index is greater than E are finite lines. The lines of force corresponding to a field of uniform force parallel to the axis are lines parallel to the axis, the distances from the axis being the square roots of an arithmetical series.