Frequency of rotating coil Given a coil initially in the x-y plane, rotating at angular frequency $ \omega $ about the x-axis in a magnetic field in the z-direction. This uniform time varying magnetic field is given by $B_z (t)=B(0)cos(\omega t) $ I am required to show that there is a voltage of frequency $2\omega $ across the loop. Clearly when t=0 the flux is at a maximum, but I dont understand how to relate to the frequency?
If the frequency is just the inverse of the period then $f=\omega / 2\pi $ ?
Clearly I am not understanding something. How does the voltage affect the frequency?
 A: The angle $\theta$ between the normal($\hat n$) to the surface of the coil is given by $\theta=\omega t$ at any instant $t$. Also The magnetic field $B_n$ in the direction of $\hat n$ is given by $$B_n=B_z(t) \cos\omega t$$. calculation of flux through the coil of area $A$ is easy.
$$\Phi = B_n(t)A$$
$$\Phi = B_z(t)cos(wt)A$$
It is given that $B_z(t)=B(0)cos(wt)$
so $$\Phi = B(0)A cos^2(wt)$$
now voltage induced is given by $-\dfrac{d\Phi}{dt}$
$\dfrac{d\Phi}{dt}= \omega B(0)A(-\sin2\omega t)$
So induced emf is $e=\omega B(0)A\sin2\omega t$
Let $\omega B(0)A=E_0$
then $e=E_0\sin(2\omega t)$
We can say that output voltage varies with twice the frequency as that of the input  magnetic field.
A: User31782 gave the right answer, but it's quite hard to read because of formatting. Let me repeat the argument for you:
The coil rotates at $\omega$, and the field is also changing at $\omega$.
At any moment in time, the area of the coil normal to the direction of the field is
$$A = A_0 \cos(\omega t)$$
and the field is
$$B = B_0 \cos(\omega t)$$
And so the instantaneous flux, which is the dot product of field and area, is
$$\Phi = A_0 B_0 \cos^2\omega t$$
The voltage is proportional to the time derivative of flux:
$$V \propto 2 \sin \omega t \cos \omega t = sin(2\omega t)$$
by trig identity.
To persuade yourself why this is so: when the coil has turned 180 degrees, the field is once again pointing in the same direction as the normal to the coil - so while the flux goes to zero when the coil is at 90 degrees to the XY plane, it's positive whenever the coil is in the XY plane.
