188 Methods for the Solution of Special Problems [CH. vin 



216. For instance if in fig. 62 we have a point charge e and the con- 

 ductor raised to potential V, we superpose on to the field of force already 

 found, the field which is obtained by raising the conductor to potential V 

 when the point charge is absent. The charge on the sphere in the second 

 field is aV, so that the total charge is 



ea 



o 



If the sphere is to be uncharged, we must have V= ,, so that a point 

 charge placed at a distance f from the centre of an uncharged sphere raises 



p 



it to potential - , a result which is also obvious from the theorem of 104. 



Sphere in a uniform field of force. 



217. A uniform field of force of which the lines are parallel to the axis 

 of x may be regarded as due to an infinite charge E at x = R, and a charge 

 E at x = R, when in the limit E and R both become infinite. The 

 intensity at any point is 



parallel to the axis of x, so that to produce a uniform field in which the 

 intensity is F parallel to the axis of x, we must suppose E and R to 

 become infinite in such a way that 



Since, in this case, F= ^, the potential of such a field will clearly 



ox 



be -Fac + C. 



Suppose that a sphere is placed in a uniform field of force of this kind, 

 its centre being at the origin. We can suppose the charge E at x = R to 



have an image of strength 



Ea a? 



'If at * = R> 

 while the other charge has an image 



Ea a* 



-R at X " -R' 



These two images may be regarded as a doublet (cf. 64) of strength 

 -p- x -p- , and of direction parallel to the negative axis of x. The strength 



