206 ELEMENTARY LESSONS ON [CHAP. iv. 



Sphere at a point near to it. It was shown 

 above that the quantity of electricity Q upon a sphere 

 charged until its surface-density was /o, was 



The problem is to find the force exercised by this 

 charge upon a + unit of electricity, placed at a point 

 infinitely near the surface of the sphere. The charge on 

 the sphere acts as if at its centre. The distance between 

 the two quantities is therefore r. By Coulomb's law the 

 force/ =^i = 4-!P j,^ ' 



This important result may be stated in words as 

 follows : The force (in dynes) exerted by a charged 

 sphere upon a unit of electricity placed infinitely near to 

 its surface, is numerically equal to 4?r times the surface- 

 density of the charge. 



252. Electric Force exerted by a charged 

 plate of indefinite extent on a point near it. 

 Suppose a plate of indefinite extent to be charged so that 

 it has a surface-density p. This surface-density will be 

 uniform, for the edges of the plate are supposed to be 

 so far off as to exercise no influence. It can be shown 

 that the force exerted by such a plate upon a + unit any- 

 where near it, will be expressed (in dynes) numerically 

 as 27rp. This will be of opposite signs on opposite sides 

 of the plate, being + 2?r/o on one side, and 27r/o on the 

 other side, since in one case the force tends to move the 

 unit from right to left, in the other from left to right. 

 It is to be observed, therefore, that the force changes its 

 value by the amount of ^irp as the point passes through 

 the surface. The same was true of the charged sphere, 

 where the force outside was 4777), and inside was zero. 

 The same is true of all charged surfaces. These two 

 propositions are of the utmost importance in the theory 

 of Electrostatics. 



253. The elementary geometrical proof of the latter theorem 

 is as follows : 



