479] Examples 411 



EXAMPLES. 



1. A small magnet is placed at the centre of a spherical shell of radii a and b. 

 Determine the magnetic force at any point outside the shell. 



2. A system of permanent magnets is such that the distribution in all planes parallel 

 to a certain plane is the same. Prove that if a right circular solid cylinder be placed in 

 the field with its axis perpendicular to these planes, the strength of the field at any point 

 inside the cylinder is thereby altered in a constant ratio. 



3. A magnetic particle of moment m lies at a distance a in front of an infinite block 

 of soft iron bounded by a plane face, to which the axis of the particle is perpendicular. 

 Find the force acting on the magnet, and shew that the potential energy of the system is 



4. The whole of the space on the negative side of the yz plane is filled with soft iron, 

 and a magnetic particle of moment m at the point (a, 0, 0) points in the direction 

 (cos a, 0, sin a). Prove that the magnetic potential at the point #, y, z inside the iron is 



2m z sin a - (a - x] cos a 



5. A small magnet of moment M is held in the presence of a very large fixed mass of 

 soft iron of permeability p. with a very large plane face : the magnet is at a distance a 

 from the plane face and makes an angle B with the shortest distance from it to the plane. 

 Shew that a certain force, and a couple 



(p - 1) M 2 sin e cos 6/8 (p + 1 ) a 3 , 

 are required to keep the magnet in position. 



6. A small sphere of radius b is placed near a circuit which, when carrying unit 

 current, would produce a field of strength H at the point where the centre of the sphere is 

 placed. Shew that if < is the coefficient of magnetic induction for the sphere, the presence 

 of the sphere increases the self-induction of the wire by, approximately, 



(3 + 47TK) 2 



7. If the magnetic field within a body of permeability p. be uniform, shew that any 

 spherical portion can be removed and the cavity filled up with a concentric spherical 

 nucleus of permeability p>i and a concentric shell of permeability /n 2 without affecting the 

 external field, provided p. lies between ^ and p. 2 , and the ratio of the volume of the nucleus 

 to that of the shell is properly chosen. Prove also that the field inside the nucleus is 

 uniform, and that its intensity is greater or less than that outside according as p. is greater 

 or less than m . 



8. A sphere of radius a has at any point (#, y, z) components of permanent magneti- 

 sation (P#, Qy, 0), the origin of coordinates being at its centre. It is surrounded by a 

 spherical shell of uniform permeability /x, the bounding radii being a and b. Determine 

 the vector potential at an outside point. 



9. A sphere of soft iron of radius a is placed in a field of uniform magnetic force 

 parallel to the axis of z. Shew that the lines of force external to the sphere lie on surfaces 

 of revolution, the equation of which is of the form 



r being the distance from the centre of the sphere. 



