Question

An Uncharged capacitor of capacitance $C$ is connected to a battery of emf $\epsilon$ at $t=0$ through a resistance $R$, then
(i) the maximum rate at which energy is stored in the capacitor is:

• A. $\frac{{\epsilon }^2}{4R}$
• B. $\frac{{\epsilon }^2}{2R}$
• C. $\frac{{\epsilon }^2}{R}$
• D. $\frac{2{\epsilon }^2}{R}$

Answer 1

The correct option is C $\frac{{\epsilon }^2}{R}$

The rate of growth of charge for the capacitor

$q=\epsilon C(1-e^{\frac{-t}{RC}})$

Let E be the energy stored inside the capacitor.

Then,

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

$E=\frac{{\epsilon }^2{C}^2}{2C}{(1-e^{\frac{-t}{RC}})}^2$

$E=\frac{{\epsilon }^{2}C}{2}(1-e^{\frac{-t}{RC}})$

Let r be the rate of energy stored inside the capacitor.

Then,

$r=\frac{dE}{dt}$

$r=\frac{2{\epsilon }^{2}C}{2}(1-e^{\frac{-t}{RC}})(-e^{\frac{-t}{RC}})\left(\frac{-1}{RC}\right)$

$r=\frac{{\epsilon }^2}{R}(1-e^{\frac{-t}{RC}})\left(e^{\frac{-t}{RC}}\right)$

Now, at $t=0$

The rate of energy stored is

$r=\frac{{\epsilon }^2}{R}$

Hence, the rate of energy stored inside the capacitor is $\frac{{\epsilon }^2}{R}$