ELECTROMAGNETS. MAGNETISM OF IRON. 3/7 



Proof of equation (14). In order to avoid the complications which arise on 

 account of the perceptible demagnetizing action of the ends of a short iron rod, let us 

 consider a very long slim rod, / centimeters in length, and s square centimeters in 

 sectional area. Suppose this rod to be placed in a long coil having z turns of wire 

 per centimeter of length, or Iz total turns. When the coil of wire is first connected 

 to the battery or other source of current, the current in the coil, beginning at zero, 

 rises in value during the time that the rod is being magnetized, and during this time 

 the magnetic flux through the rod is growing in value. Let dbjdt be the rate at 

 which the flux is increasing at a given instant, and let i be the value of the current at 

 this instant. Then h X d$\dt is the induced electromotive force in the coil which at 

 the given instant is opposing the current z, so that Iz X d$\dt X i is the rate, dlV\dt, 

 at which work is being done at the given instant in magnetizing the rod. That is : 



dW _ . . d$ 

 ~di - dt 



so that 



dW=lzi.d<b (a) 



in which dW\s the amount of work done during the time that the flux is increasing 

 by the amount d$ and while the current has the mean value i. 



Now $ = c&s or d$ = s d&> from equation (7), and zi=.d{\qK from equation 

 (9), Chapter I., so that the equation (a) becomes : 



dW=z W- d> 



47T 



or, since Is = V, we have 



W-d& 



4 7T 



In magnetizing a short iron rod, more work is done than is 

 accounted for by equation (14). The additional work goes to 

 establish the magnetic field in the neighborhood of the magnetic 

 poles of the rod. Equation (14) expresses the work which is 

 spent within the iron. 



21. Graphical representation of work done in magnetizing iron. 

 Let the curve opp' , Fig. 16, be drawn so that the coordinates 

 represent corresponding values of cB and <M for a given sample of 

 iron. The branch op represents the values of cB and c^when 

 the iron is magnetized for the first time, df beginning at the value 

 zero and increasing until it reaches the value represented by the 

 abscissa ap ; and the branch pp' represents the values of c& 



/ 



