370 



Prof. E. Wilson. 



[May 28, 



Different assumptions may be made regarding the law of formation of 

 the rectangular circuits. We shall assume, (1) that in any section 

 the currents flow in rectangles similar to the boundary of the section ; 

 (2) that the currents flow on the surfaces of cylinders similar to the 

 cylinder ; (3) that the current density in any circuit is constant 

 throughout that circuit • and compare the results obtained for the 

 value of the rate of heat production according as one or other of these 

 hypotheses is adopted. 



3. Take axes through the centre of the cylinder, O z the axis of 

 rotation, 0^ parallel to the magnetic induction in the cylinder. 



y 



-vj 



3 





% 









N 



00 











y 



p 





\ 





a 

















L 



/ 





q! 



\ 





p' 



Let a cm. be the radius, and 2ma cm. the length of the cylinder, 

 2rrf the angular velocity. In a section PNP' distant x cm. from the 

 axis, assume that the currents flow in circuits similar to the boundary 

 of the section. 



Let NL = z cm., then NQ 



/ 2 -a 2 ) 



V 



ma 



cm. 



If B is the magnetic induction, and p the specific resistance of the 

 material in ohms, the E.M.F. in the circuit QLQ' 



= 2tt/B . 4=z 



and the resistance 



, / J(a 2 -x 2 ) - ■ 



4/) z -^S i + z 



ma 



ma 



. 10- 8 VOlt, 



ma 



J{a 2 -x2)J! 



/fa 



Sx Sz. 



Hence the rate of dissipation of energy in watts per cubic centi- 

 metre 



a ma 



J J [(l+m 2 )a*-x*] 



This may be integrated by substituting a sin 6 for x, and the 

 result is 



/ 2 B 2 10- 



2m 2 



lot 



a 



