Magnetic Effect to Thickness of Wire with insulation. Ill 



used in winding the galvanometer, which, if the radius be p, will 

 equal in resistance the given external resistance, which will be 



equal consequently to — • 



The magnetic effect will be 



a(r + s)' 



c\ Im _ acXiri^ (r + s) 2 4- Impair 



mp^irr 2. 



ackr* (r + s) 2 + Imp 12 



Neglecting the constant multiplier mp^ir and differentiating, 



2ackr*{r + s) 2 + 2rlmp* -2ac\r 3 (r + s) 2 - 2ac\r 4 (r + s) = ; 



therefore 



acXr 4 + ac\sr 3 — Imp 2 = 0, 

 and 



** + »*- ^£=0, (1) 



an equation having only two real roots, one positive and the 

 other negative. 



Ifs = (that is, neglecting the silk covering), the equation 

 would become 



7 ? <*> 



-y 



In this case also r 4 = — ^ ; multiplying both sides by it and 

 transposing, we get 



therefore 



ar 4 7r 



p^TT 



Im 



c\ 



Im 



c\ 



ar 4 7r 



<2 ' 



(3) 



The first member of equation (3) is identical with the expres- 

 sion given for the resistance of the coil if 5 = 0, and the second 

 member is the expression for the external resistance. The value 

 of r in equation (2) is that of the radius of the wire, which, if we 

 neglect the insulating covering, will give the maximum magnetic 

 effect ; and equation (3) shows that when wire of this size is used, 

 the coil resistance will be equal to the external resistance. 



Equation (1) is a biquadratic, and consequently rather trouble- 



