A coil with 100 turns and cross-sectional area \( 0.02 \, \text{m}^2 \) is placed in a uniform magnetic field of \( 0.5 \, \text{T} \). The coil is rotated from a position where the plane of the coil is perpendicular to the magnetic field to a position where the plane is parallel to the field in \( 0.1 \, \text{s} \). What is the magnitude of the average induced emf in the coil?
To calculate the average induced EMF (\( \mathcal{E}_{\text{avg}} \)) in the coil, we use Faraday's law of electromagnetic induction:
\[ \mathcal{E}_{\text{avg}} = \frac{\Delta \Phi}{\Delta t} \times N \]
Here:
- \( N = 100 \): Number of turns in the coil
- \( \Delta t = 0.1 \, \text{s} \): Time interval
- \( \Delta \Phi \): Change in magnetic flux
The magnetic flux (\( \Phi \)) is given by:
\[ \Phi = B \cdot A \cdot \cos \theta \]
Where:
- \( B = 0.5 \, \text{T} \): Magnetic field strength
- \( A = 0.02 \, \text{m}^2 \): Cross-sectional area
- \( \theta \): Angle between the magnetic field and the normal to the plane of the coil
Initially, the plane of the coil is perpendicular to the magnetic field, so:
\[ \theta = 0^\circ \, \Rightarrow \, \cos \theta = \cos 0^\circ = 1 \]
\[ \Phi_{\text{initial}} = B \cdot A \cdot \cos 0^\circ = 0.5 \cdot 0.02 \cdot 1 = 0.01 \, \text{Wb} \]
Finally, the plane of the coil is parallel to the magnetic field, so:
\[ \theta = 90^\circ \, \Rightarrow \, \cos \theta = \cos 90^\circ = 0 \]
\[ \Phi_{\text{final}} = B \cdot A \cdot \cos 90^\circ = 0.5 \cdot 0.02 \cdot 0 = 0 \, \text{Wb} \]
Thus, the change in flux is:
\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = 0 - 0.01 = -0.01 \, \text{Wb} \]
Substitute the values into Faraday's law:
\[ \mathcal{E}_{\text{avg}} = \frac{\Delta \Phi}{\Delta t} \times N = \frac{-0.01}{0.1} \times 100 = 10 \, \text{V} \]
The magnitude of the average induced EMF is:
\[ \mathcal{E}_{\text{avg}} = 10 \, \text{V} \]
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.jpeg&description=A coil with 100 turns and cross-sectional area 0.02 m is placed in a uniform magnetic field of 0.5 T. The coil is rotated from a position where the plane of the coil is perpendicular to the magnetic field to a position where the plane is parallel to the field in 0.1 s. What is the magnitude of the average induced emf in the coil?){kind=link}
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