Examples 439 



22. A wire is wound in a spiral of angle a on the surface of an insulating cylinder of 

 radius a, so that it makes n complete turns on the cylinder. . A current i flows through 

 the wire. Prove that the resultant magnetic force at the centre of the cylinder is 



along the axis. 



23. A current of strength i flows along an infinitely long straight wire, and returns in 

 a parallel wire. These wires are insulated and touch along generators the surface of an 

 infinite uniform circular cylinder of material whose coefficient of induction is k. Prove, that 

 the cylinder becomes magnetised as a lamellar magnet whose strength is 2irt/(l+2ir)' 



24. A fine wire covered with insulating material is wound in the form of a circular 

 disc, the ends being at the centre and the circumference. A current is sent through the 

 wire such that I is the quantity of electricity that flows per unit time across unit length 

 of any radius of the disc. Shew that the magnetic force at any point on the axis of the 

 disc is 



27T/ (cosh " 1 (sec a) - sin a}, 



where a is the angle subtended at the point by any radius of the disc. 



25. Coils of wire in the form of circles of latitude are wound upon a sphere and 

 produce a magnetic potential Ar n P n at internal points when a current is sent through 

 them. Find the mode of winding and the potential at external points. 



26. A tangent galvanometer is to have five turns of copper wire, and is to be made so 

 that the tangent of the angle of deflection is to be equal to the number of amperes flowing 

 in the coil. If the earth's horizontal force is '18 dynes, shew that the radius of the coil 

 must be about 17*45 cms. 



27. A given current sent through a tangent galvanometer deflects the magnet through 

 an angle 6. The plane of the coil is slowly rotated round the vertical axis through the 

 centre of the magnet. Prove that if ^>^TT, the magnet will describe complete revolu- 

 tions, but if 6 < Jir, the magnet will oscillate through an angle sin " 1 (tan 0) on each side of 

 the meridian. 



28. Prove that, if a slight error is made in reading the angle of deflection of a tangent 

 galvanometer, the percentage error in the deduced value of the current is a minimum if the 

 angle of deflection is |TT. 



29. The circumference of a sine galvanometer is 1 metre : the earth's horizontal 

 magnetic force is '18 c.G.s. units. Shew that the greatest current which can be measured 

 by the galvanometer is 4*56 amperes approximately. 



30. The poles of a battery (of electromotive force 2*9 volts and internal resistance 

 4 ohms) are joined to those of a tangent galvanometer whose coil has 20 turns of wire and 

 is of mean radius 10 cms. : shew that the deflection of the galvanometer is approximately 

 45. The horizontal intensity of the earth's magnetic force is 1-8 and the resistance of 

 the galvanometer is 16 ohms. 



31. A tangent galvanometer is incorrectly fixed, so that equal and opposite currents 

 give angular readings a and /3 measured in the same sense. Shew that the plane of the 

 coil, supposed vertical, makes an angle e with its proper position such that 



2 tan e=tan a + tan /3. 



