Chapter 29 Problem 78

Problem

A circular-shaped circuit of radius $r$, containing a resistance $R$ and capacitance $C$, is situated with its plane perpendicular to a spatially uniform magnetic field ${\vec {B}}$ directed into the page. Starting at time $t=0$, the voltage difference $V_{ba}=V_{b}-V_{a}$ across the capacitor plates is observed to increase with time $t$ according to $V_{ba}=V_{0}(1-e^{t/\tau }$, where $V_{0}$ and $\tau$ are positive constants. Determine $dB/dt$ , the rate at which the magnetic field magnitude changes with time. Is $B$ becoming larger or smaller as time increases?

Solution

The emf around the loop is equal to the time derivative of the flux. Since the area of the coil is constant, the time derivative of the flux is equal to the derivative of the magnetic field multiplied by the area of the loop.

$I={\frac {dQ}{dt}}={\frac {dCV}{dt}}={\frac {d}{dt}}\left[CV_{0}\left(1-e^{-t/\tau }\right)\right]={\frac {CV_{0}}{\tau }}e^{-t/\tau }$

$I={\frac {V_{0}}{R}}e^{-t/\tau }$

${\mathcal {E}}=IR+V_{c}=\left({\frac {V_{0}}{R}}e^{-t/\tau }\right)R+V_{0}\left(1-e^{-t/\tau }\right)=V_{0}$

$V_{0}={\frac {d\Phi _{B}}{dt}}={\frac {AdB}{dt}}=\pi r^{2}{\frac {dB}{dt}}$

${\frac {dB}{dt}}={\frac {V_{0}}{\pi r^{2}}}$