E.camplm 473 



EXAMPLEa 



1. Prove that the currents induced in A solid with an infinite plane face, owing to 

 magnetic changes near the face, circulate parallel to it, and may be regarded as due to 

 the diffusion into the solid of current-sheets induced at each instant on the surface so as 

 to screen off the magnetic changes from the interior. 



Shew that for periodic changes, the current penetrates to a depth proportional to the 

 square root of the period. Give a solution for the case in which the strength of a fixed 

 inducing magnet varies as cospt. 



2. A magnetic system is moving towards an infinite plane conducting sheet with 

 velocity w. Shew that the magnetic potential on the other side of the sheet is the same 

 as it would be if the sheet were away, and the strengths of all the elements of the magnetic 

 system were changed in the ratio R/(R+w\ where 2*R is the specific resistance of the 

 sheet per unit area. Shew that the result is unaltered if the system is moving away from 

 the sheet, and examine the case of ir= /?. 



If the system is a magnetic particle of mass Jf and moment m, with its axis perpen- 

 dicular to the sheet, prove that if the particle has been projected at right angles to the 

 sheet, then when it is at a distance z from the sheet, its velocity I is given by 



K= C- 



3. A small magnet horizontally magnetised is moving with a velocity w parallel to a 

 thin horizontal plate of metal Shew that the retarding force on the magnet due to the 

 currents induced in the plate is 



where m is the moment of the magnet, c its distance above the plate, 2irff the resistance 

 of a sq. cm. of the plate, and Q*=u*-{-R*. 



4. A slowly alternating current /cos/?/ is traversing a small circular coil whose 

 magnetic moment for a unit current is M. A thin spherical shell, of radius a and specific 

 resistance <r, has its centre on the axis of the coil at a distance / from the centre of the 

 coil. Shew that the currents in the shell form circles round the axis of the coil, and that 

 the strength of the current in any circle whose radius subtends an angle cos" 1 /* at the 

 centre is 



where (2+l).r 



4*pd 



5. An infinite iron plate is bounded by the parallel planes #=A, x= -h ; wire is 

 wound uniformly round the plate, the layers of wire being parallel to the axis of y. If an 

 alternating current is sent through the wire producing outside the plate a magnetic force 

 ffo cos pt parallel to 2, prove that H y the magnetic force in the plate at a distance x from 

 the centre, will be given by 



* 



= sinh m (h + 3?) sin TO (h - x) - sinh TO (h - JT) sin m (h +x) 

 cosh m (A + x) cos m (A - x) +cosh m (h - JT) cos TO (A +ar) ' 



where wi*=2ir/ip/<r. 



Discuss the special cases of (i) mh small, (ii) mh large, 



