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(A) The magnetic field inside the solenoid is $B = \mu_{0} n I(t) = \mu_{0} n I_{0} (t - t^{2})$.The magnetic flux $\phi$ through the ring of radius $

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

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(A) The magnetic field inside the solenoid is $B = \mu_{0} n I(t) = \mu_{0} n I_{0} (t - t^{2})$.The magnetic flux $\phi$ through the ring of radius $2R$ is limited to the area of the solenoid $(R)$,so $\phi = B \cdot A = B \cdot \pi R^{2}$.$\phi = \pi R^{2} \mu_{0} n I_{0} (t - t^{2})$.The induced $EMF$ is $V_{R} = -\frac{d\phi}{dt} = -\pi R^{2} \mu_{0} n I_{0} (1 - 2t) = \pi R^{2} \mu_{0} n I_{0} (2t - 1)$.The induced current is $I_{R} = \frac{V_{R}}{R_{R}}$,where $R_{R}$ is the resistance of the ring.At $t = 0.5$,$V_{R} = \pi R^{2} \mu_{0} n I_{0} (2(0.5) - 1) = 0$.Since the expression $(2t - 1)$ changes sign at $t = 0.5$,the direction of the induced $EMF$ and the induced current reverses at $t = 0.5$.

$A$ 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 induced current $(I_{R})$ and the induced $EMF$ $(V_{R})$ in the ring change as

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