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Question
A capacitor of capacitance C is given a charge Q. At t = 0, it is connected to an ideal battery of emf ε through a resistance R. Find the charge on the capacitor at time t.
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Solution
Given:-
Initial charge given to the capacitor = Q
When the capacitor is connected to a battery, it will charge through the battery. The initial charge will also decay.
The growth of charge across the capacitor due to the battery of emf E,
\[q = CE\left( 1 - e^{- \frac{t}{RC}} \right)\]
The decay of charge through the capacitor,
\[q' = Q e^{- \frac{t}{RC}}\]
The net charge on the capacitor at any time t,
\[q_{net} = q + q'\]
\[ = CE\left( 1 - e^{- \frac{t}{RC}} \right) + Q e^{- \frac{t}{RC}}\]
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