412 Induced Magnetism [OH. xn 



10. A sphere of soft iron of permeability p. is introduced into a field of force in which 

 the potential is a homogeneous polynomial of degree n in x, y, z. Shew that the potential 

 inside the sphere is reduced to 



of its original value. 



11. If a shell of radii a, b is introduced in place of the sphere in the last question, 

 shew that the force inside the cavity is altered in the ratio 



2n + l 



12. An infinitely long hollow iron cylinder of permeability /z, the cross-section being 

 concentric circles of radii a, b, is placed in a uniform field of magnetic force the direction 

 of which is perpendicular to the generators of the cylinder. Shew that the number of 

 lines of induction through the space occupied by the cylinder is changed by inserting the 

 cylinder in the field, in the ratio 



13. A cylinder of iron of permeability p has for cross-section the curve 



r =a(l+e cos 20), 



where e 2 may be neglected. Find the distribution of potential when the cylinder is placed 

 in a field of force of which the potential before the introduction of the cylinder was 



14. An infinite elliptic cylinder of soft iron is placed in a uniform field of potential 

 (Xx+ Yy\ the equation of the cylinder being -2 + p = l. Shew that the potential of 

 the induced magnetism at any internal point is 



15. A solid elliptic cylinder whose equation is =a given by 



#-f?# = c cosh ( + 177) 



is placed in a field of magnetic force whose potential is A(x*-y 2 }. Shew that in the 

 space external to the cylinder the potential of the induced magnetism is 



- \A c 2 cosech 2 (a + /3) sin 4a e 2 (a ~ ^ " * > cos 217, 

 where coth 2/3 is the permeability. 



16. A solid ellipsoid of soft iron, semi-axes a, 6, c and permeability /*, is placed in a 

 uniform field of force X parallel to the axis of x, which is the major axis. Verify that the 

 internal and external potentials of the induced magnetisation are 



where A 1 



and X is the parameter of the confocal through the point considered. 



