A coil of area \( A \) and \( N \) turns is rotating with angular velocity \( \omega \) in a uniform magnetic field \( \mathbf{B} \) about an axis perpendicular to \( \mathbf{B} \). Magnetic flux \( \phi \) and induced emf \( \varepsilon \) across it, at an instant when \( \mathbf{B} \) is parallel to the plane of the coil, are:
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When dealing with rotating coils in a magnetic field, use Faraday's Law to calculate the induced emf and remember that the magnetic flux depends on the angle between the magnetic field and the normal to the coil. The induced emf depends on the rate of change of magnetic flux.
The magnetic flux \( \phi \) is defined as the product of the magnetic field \( \mathbf{B} \), the coil's area \( A \), and the cosine of the angle between the magnetic field and the normal to the coil's plane:
\[
\phi = NBA \cos(\theta)
\]
When \( \mathbf{B} \) is parallel to the coil's plane, the angle \( \theta \) is \( 90^\circ \), resulting in \( \cos(90^\circ) = 0 \).
Consequently, the magnetic flux becomes \( \phi = AB \).
The induced electromotive force (emf), \( \varepsilon \), is determined by Faraday's Law of Induction, which states that the induced emf is proportional to the rate of change of magnetic flux:
\[
\varepsilon = -\frac{d\phi}{dt}
\]
Given that the coil rotates at an angular velocity \( \omega \), the rate of change of flux is \( NAB\omega \). Therefore, the induced emf is \( \varepsilon = NAB\omega \).
The final results are \( \phi = AB \) and \( \varepsilon = NAB\omega \).
