198 ELEMENTARY LESSONS ON [CHAP. iv. 



at its centre. 1 We have seen that the force exerted by 

 such a charge falls off at a distance from the ball, the 

 force becoming less and less as the square of the 

 distance increases. But the force is the same in 

 amount at all points equally distant from the small charged 

 sphere. And the potential is the same at all points 

 that are equally distant from the charged sphere. If, in 

 Fig. 96, the point A represents the sphere charged with 

 q units of electricity, then the potential at P, which we 



will call V P , will be equal to ~. where r is the distance 

 from A to P. But if we take any other point at the 

 same distance from A its potential will also be i Now 



all the points that are the same distance from A as 

 P is, will be found to lie upon the surface of a sphere 

 whose centre is at A, and which is represented by the 

 circle drawn through P, in Fig. 97. All round this circle 

 the potential will have equal values ; hence this circle 

 represents an equipotential surface. The work to 

 be done in bringing up a + unit from an infinite distance 

 will be the same, no matter what point of this equi- 

 potential junace it is tirougnf to, ana to move it a6out 

 from one point to another in the equipotential surface 

 requires no further overcoming of the electrical forces, 

 and involves therefore no further expenditure of work. 

 At another distance, say at the point Q, the potential 

 will have another value, and through this point Q 

 another equipotential surface may be drawn. Suppose 

 we chose Q so far from P that to push up a unit of + 

 electricity against the repelling force of A required the 

 expenditure of just one erg of work (for the definition 



1 The student must be warned that this ceases to be true if other charges 

 are brought very near to the sphere, for then the electricity will no longer 

 be distributed uniformly over its surface. It is for this reason that we have 

 said, in describing the measurement of electrical forces with the torsion 

 balance, that " the balls must be very small in proportion to the distances 

 between them." 



