Space is divided by the line AD into two regions. Region I is field free and the Region II has a unifrom magnetic field B direction into the plane of the paper. ACD is a simicircular conducting loop of radius r with center at O, hte plane of the loop being in the plane of the paper. The loop is now made to rotate with a constant angular velocity `omega` about an axis passing through O and the perpendicular to the plane of the paper. The effective resistance of the loop is R.

(i) obtain an expression for hte magnitude of the induced cureent in the loop.
(ii) Show the direction of the current when the loop is entering into the Rigion II.
Plot a graph between the induced e.m.f and the time of roation for two periods or rotation.

Answer

A, B, C

Explanation

(i) Induced emf
`E=-(d phi)/(dt)=-(d)/(dt)(BxxA)`
`=-(d)/(dt)[B(1/2 r^(2)theta)]=-1/2B(r^2)(d theta)/(dt)=-1/2B(r^2)(omega)`
`:. I=E/R=-1/2 (b(r^2)omega)/(R) implies|I|=1/2 (B(r^2)omega)/(R)`

(ii)The loop is entering in the magnetic field and hance magnetic lines of force passing through the loop is increasing in the downward direction. Therefore, current will flow in the loop in such a direction which will oppose the change. The current will flow in the anticlockwise direction.
(iii) Graph between induced enf and period of rotation: For first half rotation, (t=T//2), when the loop enters the fields, the current is in anticlockwise direction.
Magnitude of current remains constant at
`I=B(omega)(r^2)//(2R)`
`(B omega (r^2))/(2R)`

For next half rotation, when the loop comes out of the field, current of the same magnitude is set up clockwise.Anticlock current is supposed to be positive, The I-t graph is shown in the figure for two periods of rotation.