Question

In a charging $RC$ circuit, how many time constants will elapse before the power delivered by the battery drops to half of its maximum value.

• A. $1.23\tau$
• B. $0.69\tau$
• C. $2.23\tau$
• D. $3\tau$

Answer 1

The correct option is B $0.69\tau$

Power delivered by a battery is,
$$P=EI$$Where, $I$ is the current supplied by the battery and $E$ is the emf of the battery.$$\Rightarrow P\propto I$$
So, if power becomes half, current becomes half.

In the case of charging at any instant time $t$,$$I=I_{0}e{}^{-t/\tau }$$
$\Rightarrow \frac{I_0}{2}=I_0{e}^{-t/\tau }$

$\Rightarrow 2=e^{\frac{t}{\tau }}$

Taking $\ln$ on both sides,

$\Rightarrow \ln 2=\frac{t}{\tau }$

$\therefore t=\tau \ln 2=0.69\tau$

Hence, option $\left(b\right)$ is correct.