A long solenoid of radius R carries a time (t) - dependent current $I (t) = I_{0} t (1 - t)$ . A ring of radius 2R is placed coaxially near its middle. During the time interval $0 \leq t \leq 1$ , the inducted current $(I_{R})$ and the induced EMF ( $V_{R}$ ) in the ring changes as:

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Direction of $I_{R}$ remains unchanged and $V_{R}$ is maximum at t=0.5

At t=0.5 direction of $I_{R}$ reverses and $V_{R}$ is zero

Direction of $I_{R}$ remains unchanged and $V_{R}$ is zero at t=0.25

At t=0.25 direction of $I_{R}$ reverses and $V_{R}$ is maximum

answer is B.

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Detailed Solution

$\begin{array}{l} C u r r e n t, I = I_{0} t - I_{0} t^{2} \\ M a g n e t i c f l u x, ϕ = B A = μ_{0} n I A \\ I n d u c e d e m f, V_{R} = - \frac{d ϕ}{d t} = - μ_{0} n A I_{0} (1 - 2 t) \\ H e n c e, V_{R} = 0 a t t = \frac{1}{2} \\ A n d I_{R} = \frac{V_{R}}{Resistance of loop} \\ S o, 0 < t < 0.5, V_{R} a n d I_{R} b o t h n e g a t i v e \\ 0.5 < t, V_{R} a n d I_{R} b o t h p o s i t i v e \end{array}$