Q.Show mathematically that the rotation of a coil in a magnetic field over one rotation induces an alternating emf of one cycle.

Step 1. A coil of N turns, area A, rotates with angular velocity $\omega$ in a uniform field B, starting (t=0) with its plane perpendicular to B (normal along B, maximum flux linkage $\Phi_m=NBA$).

Step 2. After rotating through $\theta=\omega t$, the field component along the coil's normal is $B\cos\omega t$ (the in-plane component contributes nothing), so the flux linkage is $N\Phi_B=NBA\cos\omega t$.

Step 3. Faraday's law gives $\varepsilon=-d(N\Phi_B)/dt = -NBA(-\omega\sin\omega t)=NBA\,\omega\sin\omega t$.

Step 4. When the coil has rotated exactly $90^{\circ}$, $\sin\omega t=1$ and the emf is maximum, $\varepsilon_m=NBA\omega$, so $\varepsilon=\varepsilon_m\sin\omega t$. …

Write the flux linkage as a function of the rotation angle omega*t, differentiate to get the emf, an …

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