33-36] The Potential 29 



35. DEFINITION. A surface in the electric field such that at every point 

 on it the potential has the same value, is called an Equipotential Surface. 



In discussing the phenomena of the electrostatic field, it is convenient to think of the 

 whole field as mapped out by systems of equipotential surfaces and lines of force, just as 

 in geography we think of the earth's surface as divided up by parallels of latitude and of 

 longitude. A more exact parallel is obtained if we think of the earth's surface as mapped 

 out by "contour-lines" of equal height above sea-level, and by lines of greatest slope. 

 These reproduce all the properties of equipotentials and lines of force, for in point of fact 

 they are actual equipotentials and lines of force for the gravitational field of force. 



THEOREM. Equipotential surfaces cut lines of force at right angles. 



Let P be any point in the electric field, and let Q be an adjacent point 

 on the same equipotential as P. Then, by definition, V P = VQ, so that by 

 equation (8) TF=0, W being the amount of work done in moving a charge e 

 from P to Q. If R is the intensity at Q, and the angle which its direction 

 makes with QP, the amount of this work must be Re cos x PQ, so that 



Re cos 6 = 0. 



Hence cos = 0, so that the line of force cuts the equipotential at right 

 angles. As in a former theorem, an exception has to be made in favour of 

 the case in which R = 0. 



36. Instead of P, Q being on the same equipotential, let them now be 

 on a line parallel to the axis of x, their coordinates being x, y, z and x 4- dx y 

 y, z respectively. In moving the charge e from P to Q the work done is 

 Xedx, and by equation (8) it is also (Vq V P )e. Hence 



- Xdx = F Q - V P . 



Since Q and P are adjacent, we have, from the definition of a differential 

 coefficient, 



ox 

 hence we have the relations 



, 3F 8F W 



-> -' Z= ~ 



results which are of course obvious on differentiating equation (6) with 

 respect to x, y and z respectively. 



Similarly, if we imagine P, Q to be two points on the same line of force 

 we obtain 



w 



H = ^-, 



ds 



where ^- denotes differentiation along a line of force. Since R is necessarily 

 positive, it follows that -~- is negative, i.e. V decreases as s increases, or the 



