Question

In the circuit shown in the figure switch $S$ is closed at $t=0$ . Then which of the following statements is WRONG.
$\left(Given:R=\sqrt{\frac{L}{2C}}\right)$

• A. Initial current through the battery is $ϵ/R$ .
• B. After a long time current through battery is $\frac{ϵ}{2R}$ .
• C. The current in the battery is $\frac{5}{8}$ times its initial value at $t=RCln2$ .
• D. The currents in the inductor and the capacitors are equal at $t=RCln3$ .

Answer 1

Answer: (C)

since $RC=\frac{L}{2R}$
$\Rightarrow$ time constants for both branches are equal.
$i_1=\frac{ϵ}{R}{e}^{-t/RC}$
$i_2=\frac{ϵ}{2R}\left(1-e^{-t/RC}\right)$
$i=i_1+i_2=\frac{ϵ}{2R}+\frac{ϵ}{2R}{e}^{-t/RC}$
$i_{max}=\frac{ϵ}{R}$
When, $i=\frac{5ϵ}{8R}=\frac{ϵ}{2R}+\frac{ϵ}{2R}{e}^{-t/RC}$
$\Rightarrow t=RCln4=2RCln2$
$i_1=i_2$
$\Rightarrow \frac{ϵ}{R}{e}^{-t/RC}=\frac{ϵ}{2R}\left(1-e^{-t/RC}\right)$
$\Rightarrow t=RCln3$