246 MAGNETISM 



distance r from the axis through the centre of the circle, is given by 



where n is the number of turns of the solenoid and C is the current 

 in it. 



Then 



ZnC 

 xi = 



r 



If a wire of permeability /UL and of uniform cross-section forms 

 a closed rinj* of mean radius r within the solenoid, the iron is 

 magnetised; but there are no poles formed, for the tubes of 

 induction go round through the iron and do not pass into or out 

 of it anywhere. The intensity in the iron must be the same as 

 that in the air close to it and is therefore given by the above 

 equation. The induction and the intensity of magnetisation are 



- l)nC 



Energy per cubic centimetre in a magnetised body 

 with constant permeability. We may use the last case to 

 obtain an expression for the distribution of energy in a magnetised 

 body when the permeability is constant as, for instance, in iron in 

 the first stage of magnetisation represented by O A Fig. 130. 



We shall suppose that a thin iron circular ring forms the core 

 of a circular solenoid as above and that the field intensity within 

 the solenoid, due to the current in the solenoid, is H. Let a be the 



cross-section of the iron. Let B = /xH 

 be the induction in it. Then the total 

 flux of induction through the iron is 

 aB. Now imagine a cut made through 

 the iron and a gap formed as in Fig. 

 190. If H due to the solenoid is in 

 the direction of the arrow, the poles 

 at the gap will be as indicated by N 

 and S. Let the gap formed be so 

 narrow that the magnetisation of the 

 iron remains the same as before it 

 was made. The induction in the gap, 

 then, is B,and this would be produced 

 by polarity of surface density B/4?r 

 or total polarity aB/4x. 

 It is to be noted that the intensity of magnetisation I in the 

 iron does not give us this polarity. From it we should get 

 <r = I = /cH, and the intensity of field in the gap due to <r would 



FIG. 190. 



