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Question
How many time constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging RC circuit?
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Solution
The equilibrium value of energy in a capacitor,
\[U = \frac{1}{2}\frac{Q^2}{C},\] where Q is the steady state charge.
Let q be the charge for which energy reaches half its equilibrium. Then,
\[\frac{1}{2}\frac{q^2}{C} = \frac{1}{2}U\]
\[ \Rightarrow \frac{1}{2}\frac{q^2}{C} = \frac{1}{2}\left( \frac{1}{2}\frac{Q^2}{C} \right)\]
\[ \Rightarrow q = \sqrt{\frac{Q^2}{2}}\]
The growth of charge in a capacitor,
\[q = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \because q = \sqrt{\frac{Q^2}{2} ,}\]
\[ \sqrt{\frac{Q^2}{2}} = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \Rightarrow \frac{Q}{\sqrt{2}} = Q\left( 1 - e^{- \frac{t}{RC}} \right)\]
\[ \Rightarrow e^{- \frac{t}{RC}} = \left( 1 - \frac{1}{\sqrt{2}} \right)\]
\[ \Rightarrow - \frac{t}{RC} = \ln\left( 1 - \frac{1}{\sqrt{2}} \right)\]
Let t = nRC
\[So, - \frac{nRC}{RC} = \ln\left( 1 - \frac{1}{\sqrt{2}} \right)\]
\[ \Rightarrow n = 1 . 23\]
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