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Question
A capacitor of capacitance C is connected to a battery of emf ε at t = 0 through a resistance R. Find the maximum rate at which energy is stored in the capacitor. When does the rate have this maximum value?
Solution
The rate of growth of charge for the capacitor,
\[q = \epsilon C \left(1 − e^\frac{- t}{RC}\right)\]
Let E be the energy stored inside the capacitor. Then,
\[E = \frac{q^2}{2C} = \frac{\epsilon^2 C^2}{2C} \left( 1 - e^{- \frac{t}{RC}} \right)^2 \]
\[ \Rightarrow E = \frac{\epsilon^2 C}{2} \left( 1 - e^{- \frac{t}{RC}} \right)^2\]
Let r be the rate of energy stored inside the capacitor. Then,
\[r = \frac{dE}{dt} = \frac{2 \epsilon^2 C}{2}\left( 1 - e^{- \frac{t}{RC}} \right)\left( - e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right)\]
\[ \Rightarrow r = \frac{\epsilon^2}{R}\left( 1 - e^{- \frac{t}{RC}} \right)\left( e^{- \frac{t}{RC}} \right)\]
\[\frac{dr}{dt} = \frac{\epsilon^2}{R}\left[ \left( - e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right)\left( e^{- \frac{t}{RC}} \right) + \left( 1 - e^{- \frac{t}{RC}} \right)\left( e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right) \right]\]
For r to be maximum,
\[\frac{dr}{dt} = 0\]
\[\Rightarrow \frac{\epsilon^2}{R}\left[ \left( - e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right)\left( e^{- \frac{t}{RC}} \right) + \left( 1 - e^{- \frac{t}{RC}} \right)\left( e^{- \frac{t}{RC}} \right)\left( - \frac{1}{RC} \right) \right] = 0\]
\[ \Rightarrow \left[ \frac{e^{- \frac{2t}{RC}}}{RC} + \frac{e^{- \frac{2t}{RC}}}{RC} - \frac{e^\frac{- t}{RC}}{RC} \right] = 0\]
\[ \Rightarrow 2 e^{- \frac{2t}{RC}} = e^{- \frac{t}{RC}} \]
\[ \Rightarrow e^{- \frac{t}{RC}} = \frac{1}{2}\]
\[ \Rightarrow - \frac{t}{RC} = - \ln2\]
\[ \Rightarrow t = RC\ln2\]
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