1902.] The Dissipation of Energy by Electric Currents. 369 



Theoretical Appendix. By J. B. Dale, M.A. 



I. The rotation of a conducting sphere in a magnetic field is dis- 

 cussed by J. J. Thomson (' Researches in Electricity,' p. 546). 



When the sphere is rotating about the axis of z, in a uniform magnetic 

 field of force B parallel to x, the values of the current components 

 when ^/(/aw/o-) a is very small are, 



r ~(p + 2)<r 3 (/A + 2)<r 



(o being the angular velocity and a the radius. In the case of an iron 

 sphere we may take ft = 10 3 , o- = 10 4 , and the current components are 

 approximately 



-|-zB, 0, t-zB. 



* or ■ a- 



The currents therefore flow in circles in planes parallel to yz, having 

 their centres on the axis of y. 



Integrating round a circle the value of the E.M.F. is found to be 

 37iy> 2 wB. This is likewise the value of the E.M.F. found on the 

 assumption that the circuit is insulated and rotating in a field of 

 magnetic induction equal to that in the sphere at rest. 



The rate of heat production 



= a- ^(p 2 + r 2 ) dx dy dz, 



taken throughout the volume of the sphere. 



Transforming to polar co-ordinates this becomes 



f 2 *- , , ' 7ra 5 o) 2 B 2 



" Jo Jo Jo 



Writing for to, 2irf where / is the frequency, we find for the rate of heat 

 production per cubic centimetre, 



°" 



where 3B is the induction when the sphere is stationary. 



2. In the case of a cylinder rotating about its axis of figure in a 

 uniform magnetic field at right angles to the axis, we may assume, in 

 analogy with the result for a sphere, that the induced currents circu- 

 late in planes parallel to the axis and the magnetic force, and that the 

 E.M.F. in any circuit is the same as that which would be induced in 

 the circuit supposed insulated and rotating in the magnetic field which 

 exists in the cylinder at rest. 



The form of the stream lines in any plane section is unknown. It 

 is reasonable to assume that they are approximately rectangular. 



