1902.] The Dissipation of Energy by Electric Currents. 363 



It may be mentioned that as the above graphical methods rest upon 

 calculation, the paths of the respective curves were determined by 

 many more points than were possible in connection with the experi- 

 mental results.' 



Theoretical Considerations. 



Let B = average induction per square centimetre in C.G.S. units. 

 / = frequency in complete periods per second. 

 p = specific resistance of the material of the cylinder in ohms. 

 t = 21 = length of cylinder in centimetres along its longi- 

 tudinal axis. 

 2nl = diameter of cylinder in centimetres. 



R = maximum radius of cylinder in centimetres. 

 x. y, z be axes as shown in figs. 1 and 2. 



In any plane section such as a, fig. 2, one can consider three distri- 

 butions of the currents induced in the cylinder by rotation in a mag- 

 netic field, the lines of force being parallel to the planes a, f3, y, . . . 

 and at right angles to the longitudinal axis of the cylinder. 



(1.) There is the distribution assumed by Baily* in which the 

 electric currents flow in rectangular paths similar to the boundary of 

 the plane of section ccf3y .... On the assumption that the rate of 

 cutting lines of force is proportional to the distance from the centre of 

 a plate, and that the distribution of magnetism is such that any point 

 in the plate is cutting lines of force during two-thirds of a revolution, 

 and that for the remaining one-third revolution the point is travelling 

 along the lines of force, Baily has shown that, when R is great com- 

 pared with t, the average rate of dissipation of energy per cubic centi- 



metre of the plate is given in watts by the formula 3 . Assume 



that the cylinder rotates about its longitudinal axis in a truly uniform 



magnetic field. Assume also with Baily that the length of the path of 



the induced current 



, , , . , a fi> J(nH 2 -x 2 )\ 

 = iz + £nz sin d 2 = 42<l+7&-^ ^ 1 cms., 



and that the electric resistance of the path 



ax dz in nt J 



The electromotive force in volts = 8z 2 7rfnB10~ 8 . Then the total rate 

 of dissipation of energy in watts 



nl I 



= lWBJtnH f Ix ' 



10 16 /> J l + n J(riW~x 2 ) 

 o 



* See 'Phil. Trans.,' A, vol. 187 (1896), pp. 715—746. 

 VOL. LXX. 2 C 



