I am having two issues with Maxwell's equations in the following problem:
A thin conducting disc has radius a thickness b and electrical resistivity ρ. It is placed in a uniform time-dependent magnetic induction $B = B_0 \sin ωt$ directed parallel to the axis of the disc. Assuming that ρ is large, find E at a distance r<a from the axis of the disc, in the plane of the disc, and obtain an expression for the induced current density as a function of r.
Issue #1 (finding an expression for $E(r)$):
We can use Faraday's law in two ways to calculate $E(r)$: $$\epsilon = -\frac{∂}{∂t}\int B\cdot dA$$ for induced emf $\epsilon$ at a point on the disc
Evaluating the RHS yields $-\frac{∂}{∂t}\int B\cdot dA =-\frac{∂}{∂t}B\pi r^2 = -\pi r^2 B_0 \omega\cos\omega t$
And then $E = -\nabla V = \nabla \epsilon = \frac{∂}{∂r}\epsilon = -2\pi r B_0 \omega \cos\omega t$
This contradicts the result if you start off with
$$\oint E.dl = -\frac{∂}{∂t}\int B\cdot dA$$
integrating E round a circular path leads to $2\pi r E = -\pi r^2 B_0 \omega\cos\omega t \rightarrow E = - r B_0 \omega\cos\omega t /2$
Which differs by a constant term of $4\pi$. I feel like the first method is wrong because $\epsilon$ and $V$ might not be so whimsically related - is my understanding of $\epsilon$ in this first method incorrect? If so, how could the first method be amended to fix this?
Issue #2 (finding an expression for induced current density $J$):
To obtain induced current density, we do have the formula
$$\rho = E/J$$ (the generalised form of Ohm's law, often written as $J = \sigma E$ for conductivity $\sigma = \rho^{-1}$) which trivially gives the answer $J = E/\rho$ using $E$ obtained in the previous part of the problem.
On the other hand, we could start with
$$\nabla \times B =\mu_0 (J + \epsilon_0 \frac{∂E}{∂t})$$
$B$ has no dependence on any spatial co-ordinates so all of the partial derivates in $\nabla \times B$ are equal to $0$, hence $J + \epsilon_0 \frac{∂E}{∂t} =0$.
Hence $J = -\epsilon_0 \frac{∂E}{∂t}$ which does not have any hint of our parameter $\rho$ included.
Is my understanding of Ampere's law here flawed? How has this second mis-match arisen?
Many thanks.
