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?

Answer 1

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

\[So, - \frac{nRC}{RC} = \ln\left( 1 - \frac{1}{\sqrt{2}} \right)\]

\[ \Rightarrow n = 1 . 23\]