32 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE 



Hence the currents in the disk would be proportional to 



coB 



(T 



where cr is the specific resistance of the disk. The rate of production of 

 heat would then be proportional to 



co 2 B 2 

 cr 

 and the heat per revolution proportional to 



(oB 2 



Hence if these most elementary assumptions were sufficient, as might 

 easily be supposed to be the case, the heat per revolution would be a linear 

 function of the speed instead of reaching a maximum and then decreasing. 



In the actual case the disk is of finite thickness and the current paths 

 possess a self-inductance depending on the form of the paths and on the 

 magnetic constants of the iron pole-pieces. Hence the current density 

 is not uniform through the thickness of the disk, and the magnetic field 

 is distorted and modified by the reaction of the eddy currents and by the 

 changes in permeability, to an extent that it is not easy to predict. 



Nevertheless, by making the following three fundamental assumptions, 

 it is possible to establish relations between speed, resultant magnetic 

 induction, and rate of production of heat, which are capable of experi- 

 mental verification. These three assumptions are obviously valid only 

 at low speeds; still, our observations were not extensive enough to fur- 

 nish more than a rough test for the theory. 



Owing to self-induction the currents are not symmetrically situated 

 with respect to the magnet poles, but are advanced in the direction of 

 rotation of the disk through a certain angle which we will call 6. In 

 accordance with Hertz's results we may assume that for moderate speeds 



= k\(O (1) 



where (o is the angular velocity of the disk and k x a constant independent 

 of the strength of the magnetic field. This condition is shown in fig. 16, 

 page 40, in which the disk is supposed to rotate counter-clockwise, and 

 the magnetic induction to be directed from the observer into the disk. 

 It is evident that the system of current loops whose magnetic field is 

 opposed to the field of the electro-magnet is now brought more nearty be- 

 tween the magnet poles. If there were no such displacement of the loops, 

 the magnetic induction would be weakened on the side where the current 

 paths enter the air-gap, and strengthened on the other side. This would 

 result in a crowding of the lines away from the "leading" edge of the pole, 

 toward the "trailing" edge. The resulting decrease in permeability on 

 the trailing half would be expected of itself to diminish the total magnetic 

 flux somewhat. But owing to the self-inductance, the field must be 

 weakened on one side much more than it is strengthened on the other. 



