412 Note 22: capacity of a square plate 



NOTE 22, ART. 283. 

 Electric Capacity of a Square. 



I am not aware of any method by which the capacity of a square can be 

 found exactly. I have therefore endeavoured to find an approximate value 

 by dividing the square into 36 equal squares and calculating the charge of 

 each so as to make the potential at the middle of each square equal to unity. 



The potential at the middle of a square whose side is i and whose charge 

 is i, distributed with uniform density, is 



4 log (i + A/2) = 3-52549- 



In calculating the potential at the middle of any of the small squares which 

 do not touch the sides of the great square I have used this formula, but for those 

 which touch a side I have supposed the value to be 3-1583, and for a corner 

 square 2-9247. 



A B C C B A 



B D E E D B 

 If the 30 squares are arranged as in the margin, and __,_, 



if the charges of the corner squares be taken for unity, __,_,_,, 

 the charges will be as follows : R n F F D R 



A B C C B A 

 A B C D E F 



I-OOO -599 -562 -265 -210 -2OI 



and the capacity of a square whose side is i will be 0-3607. 



The ratio of the capacity of a square to that of a globe whose diameter is 

 equal to a side of the square is therefore 0-7214. 



In Art. 654 Cavendish deduces this ratio from the measures in Art. 478 

 and finds it 0-73, which is very near to our result. If, however, we take the 

 numbers given in Art. 478, we find the ratio 0-79. From Art. 281 we obtain 

 the ratio 0-747. 



The ratio of the charge of a square to that of a circle whose diameter is 

 equal to a side of the square is by our calculation 1-133. 



In Art. 648 Cavendish says that the ratio is that of 9 to 8 or 1-128, which 

 is very close to our result, but in Arts. 283 and 682 he makes it 1-153. 



The numbers in Art. 281 from which Cavendish deduces this would make 

 it 1-1514. 



The numbers given in Art. 478 would make it 1-176. 



Cavendish supposes that the capacity of a rectangle is the same as that of 

 a square of equal area, and he deduces this from a comparison of the square 

 15-5 with the rectangle 17-9 x 13-4. 



It is not easy to calculate the capacity of a rectangle in terms of its sides, 

 but we may be certain that it is greater than that of a square of equal area. 



