237, 238] INDUCTION OF CURRENTS. 477 



The fact that the secondary jumps abruptly from zero to its 

 maximum value 7 at starting may be reached from considering 

 the preceding case with ^=00. The time between the secon- 

 dary's taking the value zero and attaining its maximum and the 

 time from then to the inflexion is (log Xa/^O/^i - ^-2), which is less 

 the greater R 1} vanishing for ^=00. 



The effects here described may be illustrated by means of any 

 of the mechanical models described in 71. For instance suppose 

 that the mass raj, Fig. 30, is revolving with a uniform angular 

 velocity, the centrifugal force, which represents the electromagnetic 

 force, being just balanced by an applied force so that the distance 

 of ra a from the axis remains constant. If ra 2 is at rest and we 

 suddenly apply a force to the upper bar so as to increase its 

 angular velocity, the lower bar will begin to turn in the reverse 

 direction, the velocity representing the secondary induced current. 

 If on the other hand the upper bar is suddenly retarded, the 

 lower begins to move forward in the direct sense. Similar effects 

 may be produced by suddenly changing the distance of either m 1 

 or m a from the axis, corresponding to a relative motion of the two 

 circuits, producing a change in the mutual inductance. We have 

 not in this section explicitly considered this case, but since if the 

 change is made suddenly, and the circuits then remain at rest, 

 the differential equations are the same as those we have used, and 

 the solution is obtained from those here given. 



238. Periodically-varying Electromotive-force. 



(1) SINGLE CIRCUIT. Suppose that in the circuit is included 

 a variable electromotive-force varying proportionately to the cosine 

 of a linear function of the time, as would be the case if a coil of 

 wire should rotate in a uniform magnetic field about an axis in 

 the plane of the coil, and perpendicular to the direction of the 

 field. Then the equation for the current is 



(i) L -j- + RI = EQ cos cot. 



A convenient way of treating such an equation is by replacing 

 the trigonometric term cos wt by the exponential e i<at , whose real 

 part is the trigonometric part in question. The value of I thus 

 obtained will be complex, and its real part will be the solution of 

 the differential equation with the cosine term on the right, while 

 its imaginary part will have as the coefficient of i the solution of 



