438 The Magnetic Field produced by Electric Currents [CH. xm 



15. A current i flows in a rectangular circuit whose sides are of lengths 2a, 26, and 

 the circuit is free to rotate about an axis through its centre parallel to the sides of length 

 2. Another current i' flows in a long straight wire parallel to the axis and at a distance 

 d from it. Prove that the couple required to keep the plane of the rectangle inclined at 

 an angle to the plane through its centre and the straight current is 



16. Two circular wires lie with their planes parallel on the same sphere, and carry 

 opposite currents inversely proportional to the areas of the circuits. A small magnet has 

 its centre fixed at the centre of the sphere, and moves freely about it. Shew that it will 

 be in equilibrium when its axis either is at right angles to the planes of the circuits, or 

 makes an angle tan ~ x ^ with them. 



17. An infinitely long straight wire conveys a current and lies in front of and parallel 

 to an infinite block of soft iron bounded by a plane face. Find the magnetic potential at 

 all points, and the force which tends to displace the wire. 



18. A small sphere of radius b is placed in the neighbourhood of a circuit, which 

 when carrying a current of unit strength would produce magnetic force // at the point 

 where the centre of the sphere is placed. Shew that, if * is the coefficient of induced 

 magnetization for the sphere, the presence of the sphere increases the coefficient of self- 

 induction of the wire by an amount approximately equal to 



(3-J-47TK) 2 



19. A circular wire of radius a is concentric with a spherical shell of soft iron of radii 

 b and c. If a steady unit current flow round the wire, shew that the presence of the iron 

 increases the number of lines of induction through the wire by 



approximately, where a is small compared with b and c. 



20. A right circular cylindrical cavity is made in an infinite mass of iron of perme- 

 ability p.. In this cavity a wire runs parallel to the axis of the cylinder carrying a steady 

 current of strength /. Prove that the wire is attracted towards the nearest part of the 

 surface of the cavity with a force per unit length equal to 



where d is the distance of the wire from its electrostatic image in the cylinder. 



21. A steady current C flows along one wire and back along another one, inside a 

 long cylindrical tube of soft iron of permeability /*, whose internal and external radii are 

 i and 2 , the wires being parallel to the axis of the cylinder and at equal distance a on 

 opposite sides of it. Shew that the magnetic potential outside the tube will be 



V=~sm6-\ sin 30 + - sin 50 + ..., 





Hence shew that a tube of soft iron, of 150 cm. radius and 5 cm. thickness, for which the 

 effective value of p is 1200 C.G.S., will reduce the magnetic field at a distance, due to the 

 current, to less than one-twentieth of its natural strength. 



