Question

The instantaneous charges on capacitor in two discharging $RC$ circuits are plotted with respect to time as shown in figure below. Choose the correct statement(s) (where $E_1$ and $E_2$ are emfs of two DC sources in two different charging circuits and capacitors are fully charged initially).

• A. $R_1{C}_1>R_2{C}_2$
• B. $\frac{R_1}{R_2}<\frac{C_2}{C_1}$
• C. $R_1>R_2\text{if}{E}_1=E_2$
• D. $C_2>C_1\text{if}{E}_1=E_2$

Answer 1

Answer: (A), (C)

$R_1>R_2\text{if}{E}_1=E_2$

Since capacitor is discharging, we can write,

$q=q_{\text{max}}{e}^{\left(-\frac{t}{RC}\right)}$

From the graph, we can say circuit $2$ is discharging faster so its time constant is lesser

$\Rightarrow R_2{C}_2<R_1{C}_1----\left(1\right)$

So, option (a) is correct.

Now, for charging circuit,

$q_{\text{max}}=CV$

Since for both $q_{\text{max}}$ is same we can write that, $C_1{V}_1=C_2{V}_2$

If $E_1=E_2$ or $V_1=V_2$

$\Rightarrow C_1=C_2----\left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$,

$R_1>R_2$

$\therefore$ option (c) is also correct.

Hence, options (a) and (c) are the correct answers.