POTENTIAL AND CAPACITY 83 



plates. Take two circular areas of which AB, ATJ' are sections 

 with equal radii, and with centres in the common normal to the 

 plates through P. 



Let the radii of these areas be large compared with the distance 

 d of the plates from each other. The surface densities being 

 uniform and equal to <r, the intensity at P due to the circular area 

 on the positive plate will be STPT, and away from it ; and that due 

 to the circular area on the negative plate will be STTO-, and towards 

 it (Chapter III, p. 36), or TT<T in all. Outside these circular 

 areas the charges can be divided into equal positive and negative 

 elements, each pair lying close together and at practically equal 

 distances from P. These, acting practically in the same line, 



neutralise each other and do not contribute towards the intensity, 

 intensity between the plates is therefore 47r<r and normal 

 to each. 



\Vi thin either conductor, say, for instance, at P or P", the two 

 circular areas neutralise each other, and, as above, the outlying 

 areas also neutralise each other and the intensity is zero. The 

 want of uniformity round the edges may be neglected (if these are 

 sufficiently distant) in calculating the intensity at points well 

 within the boundary. 



int< -n>it v is 4frr(T and the distance apart is rf, the work 

 done in carrying unit charge from one plate to the other is l-To-d. 

 II 'he difference of potential V = 4t7r<rd. 



It A i> the area of either charged face, the total charge on the 

 positive plate is approximately ACT- The capacity then is 



Q AT _A_ 



' V 4W 47rcT 



Two long co-axial circular cylinders. The cylinders are 

 to l>e supposed of length great compared with the radius of either. 

 Tin- inner cylinder is charged to potential V above the outer. The 

 radii are a and h. 



The lines of force will evidently be radial at a distance from 

 the ends, and the tubes of force will be wedge-shaped as in Fig. 



, the apex of each wedge lying along the axis of the cylinders. 

 Th-n the area of the cross-section of a tube at distance r from the axis 



