30 Electrostatics Field of Force [OH. n 



intensity is in the direction of V decreasing. Thus the lines of force run 

 from higher to lower values of F, and, as we have already seen, cut all 

 equipotentials at right angles. 



37. At a point which is occupied by conducting material, the electric 

 charges, as has already been said, must be in equilibrium under the action of 

 the forces from all the other charges in the field. The resultant force from 

 all these charges on any element of charge e is however .Re, so that we must 

 have R = 0. Hence X=Y=Z=0, so that 



9F_8F_aF_ 

 dx dy dz 



In other words, V must be constant throughout a conductor, for electro- 

 static equilibrium to be possible. And in particular the surface of a 

 conductor must be an equipotential surface, or part of one. The equi- 

 potential which consists of the surface of a conductor has the peculiarity 

 of being three-dimensional instead of two-dimensional, for it occupies the 

 whole interior as well as the surface of the conductor. 



In the same way, in considering the analogous arrangement of contour-lines and lines 

 of greatest slope on a map of the earth's surface, we find that the edge of a lake or sea 

 must be a contour-line, but that in strictness this particular contour must be regarded as 

 two-dimensional rather than one-dimensional, since it coincides with the whole surface of 

 the lake or sea. 



If V is not constant in any conductor, the intensity is in the direction of 

 V decreasing. Hence positive electricity tends to flow in the direction of V 

 decreasing, and negative electricity in the direction of V increasing. If two 

 conductors in which the potential has different values are joined by a third 

 conductor, the intensity in the third conductor will be in direction from 

 the conductor at higher potential to that at lower potential. Electricity will 

 flow through this conductor, and will continue to flow until the redistribution 

 of potential caused by the transfer of this electricity is such that the potential 

 is the same at all points of the conductors, which may now be regarded as 

 forming one single conductor. 



Thus although the potential has been defined only with reference to 

 single points, it is possible to speak of the potential of a whole conductor. 

 In fact, the mathematical expression of the condition that equilibrium shall 

 be possible for a given system of charges is simply that the potential shall 

 be constant throughout each conductor. And when electric contact is 

 established between two conductors, either by joining them by a wire or by 

 other means, the new condition for equilibrium which is made necessary by 

 the new physical condition introduced, is simply that the potentials of the 

 two conductors shall be equal. 



