Answer
a) $Q^{-1}=t=a\ln(\frac{Q_0}{Q_0-Q})$ The inverse is the amount of time taken to recharge the capacitor to a certain amount Q.
b) $t=2\ln10$ seconds
Work Step by Step
a) $1-e^{-\frac{t}{a}}=\frac{Q}{Q_0}$
$e^{-\frac{t}{a}}=1-\frac{Q}{Q_0}=\frac{Q_0-Q}{Q_0}$
$-\frac{t}{a}=\ln(\frac{Q_0-Q}{Q_0})$
$Q^{-1}=t=-a\ln(\frac{Q_0-Q}{Q_0})=a\ln(\frac{Q_0}{Q_0-Q})$.
The inverse is the amount of time taken to recharge the capacitor to a certain amount Q.
b) let $Q=0.9Q_0$ $t=2\ln(\frac{Q_0}{Q_0-0.9Q_0})=2\ln(\frac{Q_0}{0.1Q_0})=2\ln10$ seconds
