Examples 459 



8. A current is induced in a coil B by a current / smpt in a coil A. Shew that the 

 mean force tending to increase any coordinate of position 0, is 



t PjPLM 'bM 



where L, M, N are the coefficients of induction of the coils, and R is the resistance of B. 



9. A plane circuit, area S, rotates with uniform velocity o> about the axis of z, which 

 lies in its plane at a distance h from the centre of gravity of the area. A magnetic 

 molecule of strength p, is fixed in the axis of x at a great distance a from the origin, 

 pointing in the direction Ox. Prove that the current at time t is approximately 



where 77, e are determinate constants. 



10. Two points A, B are joined by a wire of resistance R without self-induction ; 

 B is joined to a third point C by two wires each of resistance R, of which one is without 

 self-induction, and the other has a coefficient of induction L. If the ends A, C are kept 

 at a potential difference JScospt, prove that the difference of potentials at B and C will 

 be E' cos (pt - y), where 



pLR 



11. A condenser, capacity (7, charge Q, is discharged through a circuit of resistance 

 7?, there being another circuit of resistance S in the field. If LN=M*, shew that there 

 will be initial currents - JYQ/C(RN+SL) and MQIC(RN+SL\ and find the currents at 

 any time. 



12. Two insulated wires A, of the same resistance have the same coefficient of 

 self-induction L, while that of mutual induction is slightly less than L. The ends of B 

 are connected by a wire of small resistance, and those of A by a battery of small resistance, 

 and at the end of a time t a current i is passing through A. Prove that except when t is 

 very small, 



-*(*+*) 



approximately, where IQ is the permanent current in A, and i' is the current in each after 

 a time t, when the ends of both are connected in multiple arc by the battery. 



13. The ends of a coil forming a long straight uniform solenoid of m turns per unit 

 length are connected with a short solenoidal coil of n turns and cross-section J, situated 

 inside the solenoid, so that the whole forms a single complete circuit. The latter coil can 

 rotate freely about an axis at right angles to the length of the solenoid. Shew that in free 

 motion without any external field, the current i arid the angle 6 between the cross-sections 

 of the coils are determined by the equations 



Ri= --r.(Lii+L&+&irmnAico8 0), 



d 2 d 

 1 -r-2 + 4:7rmnAi 2 sin 0=0, 



where 7 1} 7 2 , are the coefficients of self-induction of the two coils, / is the moment of 

 inertia of the rotating coil, R is the resistance of the whole circuit, and the effect of the 

 ends of the long solenoid is neglected. 



