question_answer

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 \[0\le t\le 1\], the induced current \[({{I}_{R}})\] and the induced \[EMF({{V}_{R}})\] in the ring change as: [JEE MAIN Held on 07-01-2020 Morning]

A) Direction of \[{{I}_{R}}\] remains unchanged and \[{{V}_{R}}\]is zero at t = 0.25

B) Direction of \[{{I}_{R}}\] remains unchanged and \[{{V}_{R}}\] is maximum at t = 0.5

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

Correct Answer: C

Solution :

[c]

\[I={{I}_{0}}t-{{I}_{0}}{{t}^{2}}\]

\[\phi =\left( {{\mu }_{0}}nI \right)\times \left( \pi {{R}^{2}} \right)\]

\[\therefore \varepsilon =\frac{-d\phi }{dt}\]

\[\varepsilon ={{\mu }_{0}}n\pi {{R}^{2}}\left( {{I}_{0}}-2{{I}_{0}}t \right)\]

\[\Rightarrow \varepsilon =0\] at \[t=\frac{1}{2}s\]