458 Induction of Currents in Linear Circuits [CH. xiv 



EXAMPLES. 



1. A coil is rotated with constant angular velocity co about an axis in its plane in a 

 uniform field of force perpendicular to the axis of rotation. Find the current in the coil 

 at any time, and shew that it is greatest when the plane of the coil makes an angle 



tan~M-^-] with the lines of magnetic force. 



2. The resistance and self-induction of a coil are R and Z, and its ends A and B are 

 connected with the electrodes of a condenser of capacity C by wires of negligible resistance. 

 There is a current Icospt in a circuit connecting A and B, and the charge of the con- 

 denser is in the same phase as this current. Shew that the charge at any time is 



-^-COSJP, and that C(R?+p 2 L' 2 ) = L. Obtain also the current in the coil. 



3. The ends B, D of a wire (R, L] are connected with the plates of condenser of 

 capacity C. The wire rotates about ED which is vertical with angular velocity to, the 

 area between the wire and ED being A. If H is the horizontal component of the earth's 

 magnetism, shew that the average rate at which work must be done to maintain the 

 rotation is 



+ (l - (7Zc) 2 ]. 



4. A closed solenoid consists of a large number JV of circular coils of wire, each of 

 radius a, wound uniformly upon a circular cylinder of height 2/i. At the centre of the 

 cylinder is a small magnet whose axis coincides with that of the cylinder, and whose 

 moment is a periodic quantity psinpt. Shew that a current flows in the solenoid whose 

 intensity is approximately 



where R, L are the resistance and self-induction of the solenoid, and t&ua 



5. A circular coil of n turns, of radius a and resistance R, spins with angular velocity 

 o> round a vertical diameter in the earth's horizontal magnetic field H : shew that the 

 average electromagnetic damping couple which resists its motion is ^ZT 2 ^ 2 7r 2 a 4 o)/2. Given 

 /7=0'17, n = 50, R = l ohm, a = 10cm., and that the coil makes 20 turns per second, 

 express the couple in dyne-centimetres, and the mean square of the current in amperes. 



6. A condenser, capacity (7, is discharged through a circuit, resistance R, induction Z, 

 containing a periodic electromotive force Esin nt. Shew that the "forced" current in the 

 circuit is 



E sin (nt - 6} ^ + (nL - J] * , 

 where tan B = (ri'CL - l)/nCR. 



1. Two circuits, resistances R l and R 2 , coefficients of induction L, M, N lie near each 

 other, and an electromotive force E is switched into one of them. Shew that the total 

 quantity of electricity that traverses the other is 



