49-54] Maxima and Minima of Potential 43 



DEDUCTIONS FROM LAW OF INVERSE SQUARE. 



51. THEOREM. The potential cannot have a maximum or a minimum 

 value at any point in space which is not occupied by an electric charge. 



For if the potential is to be a maximum at any point 0, the potential at 

 every point on a sphere of small radius r surrounding must be less than 

 that at 0. Hence the average value of the potential on a small sphere 

 surrounding must be less than the value at 0, a result in opposition to 

 that of the last section. 



A similar proof shews that the value of V cannot be a minimum. 



52. A second proof of this theorem is obtained at once from Laplace's 

 equation. Regarding V simply as a function of x, y, z, a necessary condition 



92 y ffY ffY 



for V to have a maximum value at any point is that - -, ^ and ^ shall 



B&- 2 8 ( y 2 3^ 2 



each be negative at the point in question, a condition which is inconsistent 

 with Laplace's equation 



a 2 F a 2 F a 2 F 



lW + df + dz 2 ~ 



So also for V to be a minimum, the three differential coefficients would 

 have to be all positive, and this again would be inconsistent with Laplace's 

 equation. 



53. If V is a maximum at any point 0, which as we have just seen 



8F 

 must be occupied by an electric charge, then the value of ~~ must be 



negative as we cross a sphere of small radius r. Thus 1 1 ^ dS is negative 



where the integration is taken over a small sphere surrounding 0, and by 

 Gauss' Theorem the value of the surface integral is 4?re, where e is the 

 total charge inside the sphere. Thus e must be positive, and similarly if F 

 is a minimum, e must be negative. Thus : 



If V is a maximum at any point, the point must be occupied by a positive 

 charge, and if V is a minimum at any point, the point must be occupied by a 

 negative charge. 



54. We have seen ( 36) that in moving along a line of force we are 

 moving, at every point, from higher to lower potential, so that the potential 

 continually decreases as we move along a line of force. Hence a line of 

 force can end only at a point at which the potential is a minimum, and 

 similarly by tracing a line of force backwards, we see that it can begin only 

 at a point of which the potential is a maximum. Combining this result 

 with that of the previous theorem, it follows that : 



