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Physics · 2024 · 2 marks
CBSE 2024 · Region 2 · Set 1 · Q24
A 100 -turn coil of radius 1.6 cm and resistance 5.0 Ω is co-axial with a solenoid of 250 turns / cm and radius 1.8 cm. The solenoid current drops…
A $\displaystyle 100$ -turn coil of radius $\displaystyle 1.6$ cm and resistance $\displaystyle 5.0 \Omega$ is co-axial with a solenoid of $\displaystyle 250$ turns $\displaystyle / \mathrm{cm}$ and radius $\displaystyle 1.8$ cm . The solenoid current drops from $\displaystyle 1.5$ A to zero in $\displaystyle 25$ ms . Calculate the current induced in the coil in this duration. (Take $\displaystyle \pi^{2}=10$ )
Marking-scheme solution
Induced emf $\displaystyle (\varepsilon)=\frac{-N d \phi}{d t}$
$\displaystyle =\frac{-N A d B}{d t}$
$\displaystyle =-N A \frac{d}{d t}\left(\mu_{0} n I\right)$
$\displaystyle =-N \mu_{0} n\left(\pi r^{2}\right) \frac{d I}{d t}$
$\displaystyle \varepsilon=\frac{100 \times 4 \pi \times 10^{-7} \times 250 \times 10^{2} \times \pi \times\left(1.6 \times 10^{-2}\right)^{2} \times 1.5}{25 \times 10^{-3}}$
$\displaystyle =0.1536 \mathrm{~V}$
$\displaystyle I=\frac{\varepsilon}{R}$
$\displaystyle =0.03 \mathrm{~A}$
$\displaystyle \varepsilon=-M \frac{d I}{d t}$
$\displaystyle M=\mu_{0} n_{1} n_{2} \pi r_{1}^{2} l$
$\displaystyle =\mu_{0}\left(n_{1} l\right) n_{2} \pi r_{1}^{2}$
$\displaystyle =4 \pi \times 10^{-7} \times 100 \times 250 \times 10^{2} \times \pi \times\left(1.6 \times 10^{-2}\right)^{2}$
$\displaystyle =2.56 \times 10^{-3} \mathrm{H}$
$\displaystyle \varepsilon=-2.56 \times 10^{-3} \times \frac{(0-1.5)}{25 \times 10^{-3}}$
$\displaystyle =0.1536 \mathrm{~V}$
$\displaystyle I=\frac{\varepsilon}{R}=\frac{0.1536}{5}$
$\displaystyle =0.03 \mathrm{~A}$
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