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How Many Time Constants Will Elapse before the Energy Stored in the Capacitor Reaches Half of Its Equilibrium Value in a Charging Rc Circuit? - Physics


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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The equilibrium value of energy in a capacitor,

\[U = \frac{1}{2}\frac{Q^2}{C},\] where 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

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

\[ \Rightarrow n = 1 . 23\]

Concept: Energy Stored in a Capacitor
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 10 Electric Current in Conductors
Q 72 | Page 203
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