457] Examples 393 



4. The axis of a small magnet makes an angle <f> with the normal to a plane. Prove 

 that the line from the magnet to the point in the plane where the number of lines of 

 force crossing it per unit area is a maximum makes an angle 6 with the axis of the 



magnet, such that 



2 tan 6 = 3 tan 2 (0-0). 



5. Two small magnets lie in the same plane, and make angles 0, 6' with the line 

 joining their centres. Shew that the line of action of the resultant force between them 

 divides the line of centres in the ratio 



tan0' + 2tan0 : tan0 + 2tan0'. 



6. Two small magnets have their centres at distance r apart, make angles 6, 6' with 

 the line joining them, and an angle e with each other. Shew that the force on the first 

 magnet in its own direction is 



| (5 cos 2 6 cos 6' cos 6' 2 cos e cos 0). 



Shew that the couple about the line joining them which the magnets exert on one 

 another, is 



mm' , . 

 j- a sin e, 



where d is the shortest distance between their axes produced. 



7. Two magnetic needles of moments M, M' are soldered together so that their 

 directions include an angle a. Shew that when they are suspended so as to swing freely 

 in a uniform horizontal magnetic field, their directions will make angles 0, B' with the 

 lines of force, given by 



sin 6 sin0' sin a 



8. Prove that if there are two magnetic molecules, of moments M and JT, with their 

 centres fixed at A and B, where AB = r, and one of the molecules swings freely, while the 

 other is acted on by a given couple, so that when the system, is in equilibrium this 

 molecule makes an angle 6 with AB, then the moment of the couple is 



f MM' sin 20/r 3 (3 cos 2 + 1)*, 

 where there is no external field. 



9. Two small equal magnets have their centres fixed, and can turn about them in a 

 magnetic field of uniform intensity If, whose direction is perpendicular to the line r 

 joining the centres. Shew that the position in which the magnets both point in the 

 direction of the lines of force of the uniform field is stable only if 



10. Two magnetic particles of equal moment are fixed with their axes parallel to the 

 axis of 2, and in the same direction, and with their centres at the points a, 0, 0. Shew 

 that if another magnetic molecule is free to turn about its centre, which is fixed at the 

 point (0, y, z\ its axis will rest in the plane #=0, and will make with the axis of z the 

 angle 



tan ~ 1 2g 2 -a 2 -y 2 ' 

 Examine which of the two positions of equilibrium is stable. 



