Question

Capacitors $C_1=1\mu F$ and $C_2=2\mu F$ are separately charged from the same battery. They are then allowed to discharge separately through equal resistors

• A. the currents in the two discharging circuits at $t=0$ is zero
• B. the currents in the two discharging circuits at $t=0$ are equal but not zero
• C. the currents in the two discharging circuits at $t=0$ are unequal
• D. $C_1$ loses 50% of its initial charge sooner than $C_2$ loses 50% of initial charge

Answer 1

Answer: (B), (D)

The correct options are
B the currents in the two discharging circuits at $t=0$ are equal but not zero
D $C_1$ loses 50% of its initial charge sooner than $C_2$ loses 50% of initial charge

The current for discharging : $I_1=I_{10}{e}^{-t/C_{1}R}$ and $I_2=I_{20}{e}^{-t/C_{2}R}$
At $t=0,I_1=I_{10},I_2=I_{20}$
As they are charged from same battery $\left(V_0\right)$ and discharge through same resistor , so
$I_{10}=I_{20}=V_0/R$
Thus the currents in the two discharging circuits at $t=0$ are equal but not zero.
Now charge during discharging: $Q_1=Q_{10}{e}^{-t/C_{1}R}$ and $Q_2=Q_{20}{e}^{-t/C_{2}R}$
For $50\%$ discharging , $Q_1=\frac{Q_{10}}{2}$
so, $\frac{Q_{10}}{2}=Q_{10}{e}^{-t/C_{1}R}$
$2=e^{t/C_{1}R}\Rightarrow t=C_{1}Rln2=t_1$
similarly, $t_2=C_{2}Rln2$
As $C_1<C_2,t_1<t_2$
Ans : (B), (D)