Question

Directions :- A voltage source $V=V_{0}sin\left(100 t\right)$ is connected to a black box in which there can be either one element out of $L,C,R$ or any two of them connected in series.

At steady state the variation of current in the circuit and the source voltage are plotted together with time, using an oscilloscope as shown

If $AC$ source is removed, the circuit is shorted and then at $t=0,$ a battery of constant $emf$ is connected across the black box. The current in the circuit will

• A. increase exponentially with constant $=4\times 10^{- 3}s$
• B. decrease exponentially with time constant $=1\times 10^{- 2}s$
• C. oscillate with angular frequency $20s^{- 1}$
• D. first increase and then decrease

Answer 1

Answer: (B)

$\Delta \theta =\omega \Delta t=\frac{\pi }{4}$
$tan\theta =\frac{x}{R}\Rightarrow x=R$
since, current leads the voltage and circuit consists of $R$ and $C$ .
and $i_{0}=\frac{V_{0}}{Z}$
$\therefore Z=\frac{V_{0}}{i_{0}}=\frac{100}{\sqrt{2}}=50\sqrt{2}$
Now, $R\sqrt{2}=50\sqrt{2}\Rightarrow R=X_{C}=50$
$x_{C}=\frac{1}{C \omega }=50$
$\Rightarrow C=\frac{1}{50 \omega }=200μF$
$\tau=RC=50\times 200\times 10^{- 6}=1\times 10^{- 2}s$