Find the emf in the loop by two methods In the Figure below assume the magnetics are shaped such that the magnetic field is in
the $z$ direction and varies as
$$
B_0 = B_m\left(1-\frac{x^2}{a^2}\right)\hat{z}
$$
Find the emf in the loop by two methods, a) the rate of change of flux b) the motional
emf method.
My try:
$$V_{emf} = -\int B \cdot \frac{\partial S}{dt}$$
$S=2a\ell\cos \theta \hat{z}$, $B=B_m\left(1-\frac{a^2\cos^2(\theta)}{a^2}\right)\hat{z}=B_m \sin^2(\theta)\hat{z}$
$$
V_{emf} = 2a\ell B_m\cos \theta  \sin^2(\theta)
$$
I am not sure if my solution is correct.

 A: Ok, we want to calculate the emf induced in the rotating coil.  The B magnitude in the z direction (horizontal) depends on x (vertical position) as shown below:

$$emf = \oint \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}  = - \frac{\mathrm{d}}{\mathrm{d}t} \int \mathbf{B} \cdot \mathrm{d}\mathbf{a}$$
In other words, the emf is generated by the change in total flux linking the coil. $$ emf = - \frac{\mathrm{d\Phi}}{\mathrm{d}t}$$
where,
$$\Phi = \int {\boldsymbol{B} \cdot d \boldsymbol{a}}$$
The total flux linking the coil is time dependent because of the constant rotational speed of the loop.
When the loop is horizontally oriented (θ = zero), no flux passes through the coil. When it is vertically oriented (θ = π/2) the maximum flux possible passes through the coil.
If the B field where constant (Bm) across the region then the problem would simplify to (at t=0 the coil is perfectly vertical),
$$\Phi = B_mA \int cos(\Omega t)$$
where,  $$A = 2al$$
so,
$$ emf = - \frac{\mathrm{d\Phi}}{\mathrm{d}t} = -B_mA\Omega[sin(\Omega t)]$$
Back to this problem...
The rotational speed is Ω rad/s.  The total flux linking the coil is thus time dependent.  The x position of one coil side is positive while the other is negative (except at θ = zero where both are zero).
$$ x = a sin(\Omega t)$$
The angle between the flux density B field and the normal to the coil surface is,
$$ \Theta = \Omega t $$
So, write in the equation for the total flux linkage (time dependent) normal to the coil surface (dot product) using your equation for B and then differentiate.
Please do let us know what the solution actually is (i'd like to know if my result is correct - which i'm holding back for now as this is a homework problem).
