Question

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

• A. Direction of $I_R$ remains unchanged and $V_R$ is maximum at $t=0.5$
• B. Direction of $I_R$ remains unchanged and $V_R$ is zero at $t=0.25$
• C. At $t=0.5$ direction of $I_R$ reverses and $V_R$ is zero
• D. At $t=0.25$ direction of $I_R$ reverses and $V_R$ is maximum

Answer 1

The correct option is C At $t=0.5$ direction of $I_R$ reverses and $V_R$ is zero


Field due to solenoid near the middle $={\mu }_{o}NI$

$$\begin{array}{cc}& \text{Flux,}\varphi =BA\text{where}(A=\pi (R{)}^2)\\ & ={\mu }_{o}N{I}_{o}t(1-t)\pi R^2\\ & E=-\frac{d\varphi }{dt}\text{[By Lenz’s law]}\\ & E=-\frac{d}{dt}\left({\mu }_{o}N{I}_{o}t\right(1-t\left)\pi R^2\right)\\ & E=-{\mu }_{o}N{I}_o\pi R^2\frac{d}{dt}\left[t\right(1-t\left)\right]\\ & E=-\pi {\mu }_o{I}_{o}N{R}^2(1-2t)\end{array}$$
Current will change its direction when EMF will be zero.
$⟹(1-2t)=0$
$So,t=0.5sec$