THE THEORY OF THE DYNAMO 225 



whole number of lines of force which pass through the magnet. We 

 may write B, proportional to N, the number of turns of the wire around 

 the magnet, and C, the current; and inversely proportional to the re- 

 sistance to these lines of force in going around the circuit. The resist- 

 ance to the lines of force is proportional to L, the length of the iron of 

 the system, divided by S, the cross-section of the magnet, supposing it 

 to be uniform, into //, the magnetic permeability of the iron (or the 

 conductivity of the iron for the lines of force). This quantity ft varies 

 with the current, and can readily be obtained. Some years ago I gave 

 a formula for it. It can be expressed simply as dependent upon the 

 magnetization of the iron and a constant depending upon the iron 

 alone. We have something more to add: 



Let I be twice the width of the opening between armature and pole 

 piece, and A the area across which the lines of force flow; then we 



have to add -i and another quantity, which we can call p, which depends 



^L 



upon the resistance of these lines of force which escape in all direc- 

 tions and represents the loss due to that escapement. Thus we have 

 the final value for the number of lines of force (or rather induction) 

 in the magnet 



NC 



T> 



ti A + p 



This gives us an equation which may be solved with respect to fi. 

 The curve for the magnetic permeability is of this nature (Fig. 7). It 

 will be of a more or less flat form, according to the value of I and p. 

 Therefore, in increasing the magnetic force upon the magnet, it becomes 

 easier and easier to magnetize it until a certain point is reached, and 

 after that it becomes harder and harder. In practice the core should 

 have sufficient cross-section to produce a very strong magnetic field, 

 but not so great as to require too much wire to wind it. The two must 

 be balanced, which can only be done by calculation or, better, by experi- 

 ments on the machine. By examining the force of the magnet at each 

 point, and in that way getting an idea of how these lines of force go, 

 we can see whether the cross-section of the core is large enough to 

 produce all the lines of force necessary for our purpose or not. Of 

 course, in order to have sufficient magneto-motive force to send lines of 

 force across the opening in sufficient quantity, we must have sufficient 

 wire. As the thickness of the coil is increased, we have to use more 

 wire in proportion for a certain diameter of core, which is a disadvan- 

 15 



