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

$q=Q(1-e^{-n})$

$\frac{2q^2}{2C}=\text{initial value}$

$\frac{q^2}{2C}=\text{final value}$

$\Rightarrow =\frac{2q^2}{2C}=\frac{q^2}{2C}$

$\Rightarrow q^2=\frac{q^2}{2}\Rightarrow q=\sqrt{\frac{q^2}{2}}$

$\Rightarrow \frac{q}{\sqrt{2}}=q(1-e^{-n})$

$\Rightarrow \frac{1}{\sqrt{2}}=1-e^{-n}$

$\Rightarrow e^{-n}=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}$

$\Rightarrow n=log\left(\frac{\sqrt{2}}{\sqrt{2}-1}\right)=1.22$