14 ELECTRICAL MEASUREMENTS 



ridsdydx _y 



and for the whole filament is 



Therefore, if c is the mean radius of the coil 



+ a+\/(c + a) 2 '+ b 



G = 27rn 

 but 



c+ 



-. 



.'. O = 47rn'6 log. 



8 



Cot 2 



cot 



Theoretically, there is a best ratio of total breadth to radius 

 for the coil. If the coil has a fixed volume and it be assumed that 

 the diameter of the core is zero, G will be a maximum when = 

 30.8. If the area of the cross-section be fixed, a given number 

 of turns of wire of a definite diameter will render G a maximum 

 when |8 = 16.66. 



Best Form of Cross-section. Theoretically, the rectangular 

 form of cross-section is not the best. With a definite length of 

 wire of a certain size, the volume of the coil and its resistance are 

 fixed. The question is, how shall this wire be arranged so that 

 it will produce the maximum effect at the needle? 



The effect at the needle of a unit length of wire, bent into an 



arc of a circle and carrying unit current, is F = 5 where r 



and a are the polar coordinates of the traces of the wire on a plane 

 including the axis. The origin of coordinates being the axis of 



the coil and the coil center, call % the efficacy of a unit length 

 of wire, denoted by e v . Then every unit length of wire whose 



traces are on the curve r 2 = sin a, where e v has a definite value, 



e v 



will produce the same effect at the center of the coil. Suppose 



