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A long solenoid having 100 turns per $\mathrm{cm}$ carries a current of $\frac{4}{\pi}$ A. At the centre of it is placed a coil of 200 turns of cross-sectional area $25 \mathrm{~cm}^2$ having its axis parallel to the field produced by the solenoid. When the direction of the current in the solenoid is reversed with in $0.04 \mathrm{~s}$, the induced emf in the coil is
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Verified Answer
The correct answer is:
$0.4 \mathrm{~V}$
Due to solenoid a magnetic field exists around coil placed at centre
Magnetic field intensity produced by solenoid,
$B=\mu_0 n I$
Flux linked with coil is
$\phi_B=N . B \cdot A$
Where, $N=$ Number of turns in coil
$A=$ Area of coil
As current in solenoid is reversed, change in flux is
$\Delta \phi_B=2 N \cdot B \cdot A$
This change in flux produces an emf in coil given by
Induced emf $e=$ Rate of change of flux
$\Rightarrow \quad e=\frac{2 N B A}{\Delta t}=\frac{2 N\left(\mu_0 n I\right) A}{\Delta t}$
Here $N=200, A=25 \mathrm{~cm}^2=25 \times 10^{-4} \mathrm{~m}^2$
$I=\frac{4}{\pi} \mathrm{A}, n=100$ tuns $/ \mathrm{cm}=100 \times 100$ turns $/ \mathrm{m}$
$\mu_0=4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{mA}^{-1}$ and $\Delta t=0.04 \mathrm{~s}$
So induced emf,
$e=\frac{2 \times 200 \times 4 \pi \times 10^{-7} \times 10^4 \times \frac{4}{\pi} \times 25 \times 10^{-4}}{0.04}=0.4 \mathrm{~V}$
Induced emf in coil is $0.4 \mathrm{~V}$
Magnetic field intensity produced by solenoid,
$B=\mu_0 n I$
Flux linked with coil is
$\phi_B=N . B \cdot A$
Where, $N=$ Number of turns in coil
$A=$ Area of coil
As current in solenoid is reversed, change in flux is
$\Delta \phi_B=2 N \cdot B \cdot A$
This change in flux produces an emf in coil given by
Induced emf $e=$ Rate of change of flux
$\Rightarrow \quad e=\frac{2 N B A}{\Delta t}=\frac{2 N\left(\mu_0 n I\right) A}{\Delta t}$
Here $N=200, A=25 \mathrm{~cm}^2=25 \times 10^{-4} \mathrm{~m}^2$
$I=\frac{4}{\pi} \mathrm{A}, n=100$ tuns $/ \mathrm{cm}=100 \times 100$ turns $/ \mathrm{m}$
$\mu_0=4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{mA}^{-1}$ and $\Delta t=0.04 \mathrm{~s}$
So induced emf,
$e=\frac{2 \times 200 \times 4 \pi \times 10^{-7} \times 10^4 \times \frac{4}{\pi} \times 25 \times 10^{-4}}{0.04}=0.4 \mathrm{~V}$
Induced emf in coil is $0.4 \mathrm{~V}$
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