A parallel-plate capacitor with circular plates of radius R = 20 mm and gap widt

Question

A parallel-plate capacitor with circular plates of radius

R = 20 mm and gap width d = 5.0 mm has a uniform electric field between the plates. Starting at time t = 0, the potential difference between the two plates is V = (110 V)e-t/?, where the time constant ? = 11 ms. At radial distance r = 0.70R from the central axis, what is the magnetic field magnitude (a) as a function of time for t ? 0 and (b) at time t = 3??  There are 2 parts to this question, please show work. The answer may not be a numerical answer but a derived equation with the above variables.

Explanation / Answer

The changing magnetic field induces a voltage and current in the loop.
I = V/R
I(t) = bt^2
V(t) = Rbt^2
V = d(AB)/dt (Faraday's law, ref.; B is magnetic field strength)
AB(t) = integral(V(t)) = integral(Rbt^2) = Rbt^3/3
B(t) = Rbt^3/(3A)

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