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.

Explanation

(a) When the loop is rotated about an axis passing through center `O` and perpendicular to the plane of the paper, the angle between magnetic field vector `B` and area `A` is always `0^(@)`. When the loop ` = Bacos 0 = 0` the (Since `B = 0` in region I).
When the loop eneters the magnetic field in region II, the magnetic flux linked with it is given by `phi = BA` where
`A = (1)/(2)r^(2_)theta`. Therefore, emf induced
`e = -(dphi)/(dt) = -(d)/(dt)(BA) = -B(dA)/(dt) = (-Br^(2))/(2)(d theta)/(dt) = -(Br^(2))/(2)omega`


As resistance of the loop is `R`, the current induced is given by
`i = (e)/(R) = (1)/(2)(Br^(2)omega)/(R)`
This is the required expression for current induced in the loop.
(b) According to Lenz's law, the direction of current induced is to oppose the change in magnetic flux. So, when entering into region II the field produced by the current induced anticlockwise as shown in Fig.3.18(a).
( c) When the loop enters the magnetic flux linked with it increases and the emf `e = (1)/(2) Br^(2)omega` is induced in one direction. when the loop comes out of the field the flux decreases and emf is induced in opposite sence. The graph for representing the emf induced versus time for two taken aniclockwise direction as positive.