119, 120] POLARIZED DISTRIBUTIONS. 227 



the other a space filled with negative matter, (Fig. 51). The 

 volumes of these two spaces must be equal, since the bodies A 

 and B originally coincided. The effect of the system is now that 

 of a body covered on part of its surface with a ^ 



positive, and on the remainder with a negative 

 surface distribution. If n, Fig. 52, is the normal 

 drawn inwards, and h represent the direction as 

 well as magnitude of the displacement, the amount 

 of matter contained in a right prism standing on the 



element of surface dS will be 



FIG. 52. 

 ph cos (hn) dS, 



but this is equal to crdS where a- is the surface density. Accord- 

 ingly the surface density of the equivalent distribution is 



( I ) a = ph cos (hn). 



If we now decrease h and increase p without limit, keeping their 

 product finite and equal to /, we obtain a body charged with 

 surface density 



(2) o- = I cos (hn), 



possessing the property of experiencing a couple, but no resultant 

 force when placed in a uniform field. 



To find the magnitude of the couple let us divide the body up 

 into prisms with their generators parallel to h and standing on the 

 elements dS. Such a cylinder of length I carries upon one end 

 the charge adS which experiences the force FadS, and upon the 

 other the charge - adS which experiences the force Fad 8 in the 

 opposite direction. The moment of the couple thus produced is 



I sin (hF) Fa-dS. 



For the whole moment we must take the integral of this over the 

 positively charged surface, 



(3) III sin (hF) FvdS = -F sin (hF)jjlI cos (hn) dS. 



Now cos (hn) dS is the area of a right section of the prism on dS, 



rl 

 so that I cos (hn) dS is its volume I dr, and the total moment 



becomes 



(4) F sin (hF) l&r = fl sin (hF) . r. 



152 



