Question 30: Electromagnetic Induction

Question

A square loop of side 2 cm is placed in a time varying magnetic field with magnitude as $B=0.4 \sin (300 t)$ Tesla. The normal to the plane of loop makes an angle of $60^{\circ}$ with the field. The maximum induced emf produced in the loop is $\_\_\_\_$ mV .

Accepted Answer

Magnetic flux $\phi$ through a loop is $\phi=\overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{A}}=\mathrm{BA} \cos 	heta$.
According to Faraday's Law, the induced emf $\varepsilon$ is the negative rate of change of flux:
$$ \varepsilon=-\frac{\mathrm{d} \phi}{\mathrm{dt}}=-\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{BA} \cos 	heta) $$
Since area (A) and angle ( $	heta$ ) are constant.
$$ \varepsilon=-\mathrm{A} \cos 	heta \frac{\mathrm{~dB}}{\mathrm{dt}} $$
The length of side of square is $\mathrm{s}=2 \mathrm{~cm}=0.02 \mathrm{~m}$.
Then the area of loop is,
$$ A=s^2=(0.02)^2=4 	imes 10^{-4} \mathrm{~m}^2 $$
The angle made by magnetic field with the normal to the plane of loop is $	heta=60^{\circ}$
The time varying magnetic field is $B=0.4 \sin (300 t)$.
The rate of change of magnetic field is
$$ \frac{\mathrm{dB}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(0.4 \sin (300 \mathrm{t}))=0.4 	imes 300 \cos (300 \mathrm{t})=120 \cos (300 \mathrm{t}) $$
Substituting into the emf equation :
$$ \varepsilon=-\left(4 	imes 10^{-4}ight) \cos \left(60^{\circ}ight)[120 \cos (300 \mathrm{t})] $$
$$\Rightarrow $$ $$ \varepsilon=-\left(4 	imes 10^{-4}ight) 	imes 0.5 	imes 120 \cos (300 t)=-0.024 \cos (300 t) V $$
The maximum induced emf $\left(\varepsilon_{	ext {max }}ight)$ occurs when $\cos (300 t)=1$ :
$$ \varepsilon_{\max }=0.024 \mathrm{~V}=24 \mathrm{mV} $$
Therefore, the magnitude of maximum induced emf is 24 mV .
Thus, the correct option is (D).

Answer

12

Answer

18

Answer

21

Answer

24

Electromagnetic Induction

A square loop of side 2 cm is placed in a time varying magnetic field with magnitude as B=0.4 \sin (300 t) Tesla. The normal to the plane of loop makes an angle of 60^{\circ} with the field. The maximum induced emf produced in the loop is \_\_\_\_ mV .

Choose the correct answer: