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Question
How many time constants will elapse before the current in a charging RC circuit drops to half of its initial value? Answer the same question for a discharging RC circuit.
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Solution
The growth of charge across a capacitor,
\[q = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[q = \frac{Q}{2}\]
\[ \Rightarrow \frac{Q}{2} = Q \left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \Rightarrow \frac{1}{2} = \left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \Rightarrow e^{- \frac{t}{RC}} = \frac{1}{2}\]
\[ \Rightarrow \frac{t}{RC} = \ln 2 \]
Let t = nRC
\[ \Rightarrow \frac{nRC}{RC} = 0 . 69\]
\[ \Rightarrow n = 0 . 69\]
The decay of charge across a capacitor,
\[q = Q e^{- \frac{t}{RC}} \]
\[q = \frac{Q}{2}\]
\[ \Rightarrow \frac{Q}{2} = Q e^{- \frac{t}{RC}} \]
\[ \Rightarrow \frac{1}{2} = e^{- \frac{t}{RC}} \]
Let t = nRC
\[ \Rightarrow \frac{nRC}{RC} = \ln 2\]
\[ \Rightarrow n = 0 . 69\]
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