of Unipolar Induction. 



405 



If we let the jacket enter into rotation it will produce ex- 

 ternal mechanical work, and the development of heat will 

 consequently be less than before. This is brought about by 

 the production, in the jacket in rotation, of an electromotive 

 force sending a current in the opposite direction to that of the 

 battery. If we denote this current by i, the production of 

 heat in this case will be 



A{{I-i) 2 m-E(I-i)\=A{I 2 m-EI-2Iim + i 2 m + m\. 



In other terms [if we pay attention to equation (1)], there is 

 produced in the closed circuit a loss of heat equal to 



Aim(I-i) (2) 



The value of the external work done will then be equal to 

 this loss. 



In conformity with the law of Biot-Savart, a magnetic pole 

 acts upon an element of the current with a force equal to the 

 intensity of the pole divided by the square of the distance to 

 the element, and multiplied also by the intensity of the cur- 

 rent and the length of the element, and by the sine of the angle 

 a formed by the element and the line of junction between the 

 pole and the element : the direction of this force is normal to 

 the plane passing through the pole and the element. If the 

 direction of motion of the element makes an angle <£ with the 

 normal in question, we obtain of course the component of the 

 force along the line of motion by multiplying the expression 

 further by cos </>. If we denote the velocity of the element 

 by h, the intensity of the pole by M, and the intensity of the 

 current by (I — i), and, lastly, the distance between the pole 

 and the element by p, we have, as the expression of the me- 

 chanical work due to the action of the pole upon an element 

 of the current dz, 



M(I-0 



sin a. cos (f> . hdz. . 

 P 



In fig. 2, let sn represent a magnet, and ab a 

 jacket through which a current (I — i) passes from 

 a to b. Let the distance between the two poles 

 be 21, and the length of a b half this distance, viz. 

 /; and let the radius of the jacket be r. Let us 

 consider at first an element dz of the jacket situated 

 at the point at the distance z from a. The dis- 

 tance p from the south pole to dz is then equal to 



Fig. 2. 



(3) 



The 



vV+ (/-*)», and sina = 7 =_. 



normal to the plane passing through the south 

 pole and the element coinciding with the direction 



