A long solenoid of radius R carries a time (t) - dependent current
I(t)=I0t(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 (IR) and the induced EMF(VR) in the ring change as :
Solution
I(t) = I<sub>0</sub>t(1 - t)
<br><br>We know, $\phi$ = BA
<br><br>$\Rightarrow$ $\phi$ = $\mu$<sub>0</sub>nIA
<br><br>$\Rightarrow$ $\phi$ = $\mu$<sub>0</sub>nAI<sub>0</sub>(t - t<sup>2</sup>)
<br><br>Also V<sub>R</sub> = $- {{d\phi } \over {dt}}$
<br><br>= - $\mu$<sub>0</sub>nAI<sub>0</sub>(1 - 2t)
<br><br>V<sub>R</sub> = 0 when 1 - 2t = 0
<br><br>$\Rightarrow$ t = 0.5
<br><br>Also we know, V<sub>R</sub> = I<sub>R</sub>r
<br><br>$\Rightarrow$ I<sub>R</sub> = ${{{\mu _0}nA{I_0}\left( {1 - 2t} \right)} \over r}$
<br><br>after t = 0.5, I<sub>R</sub> reverses its direction.
About this question
Subject: Physics · Chapter: Electromagnetic Induction · Topic: Faraday's Laws
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