Three identical capacitors are given a charge $Q$ each and they are then allowed to discharge through resistance $R_1, R_2\,$ and $\,R_3$. Their charges, as a function of time shown in the graph below. The smallest of the three resistance is
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(c) During the discharge of a capacitor through a resistance charge at any instant $Q = {Q_0}{e^{ - t/CR}}$
$⇒$ $\frac{{{Q_0}}}{Q} = {e^{t/CR}}$ $⇒$ $t = CR\,{\log _e}\,\frac{{{Q_0}}}{Q}$
If $Q$ $⇒$ constant, then $t$ $\propto$ $R$
Now, draw a line parallel to the time axis as shown. Suppose this line cut the graphs at points $1,\, 2$ and $3$. Corresponding time are $t_1$, $t_2$ and $t_3$ respectively. Hence from graph $t_1 < t_2 < t_3$
$⇒$ $R_1 < R_2 < R_3$
$⇒$ $\frac{{{Q_0}}}{Q} = {e^{t/CR}}$ $⇒$ $t = CR\,{\log _e}\,\frac{{{Q_0}}}{Q}$
If $Q$ $⇒$ constant, then $t$ $\propto$ $R$
Now, draw a line parallel to the time axis as shown. Suppose this line cut the graphs at points $1,\, 2$ and $3$. Corresponding time are $t_1$, $t_2$ and $t_3$ respectively. Hence from graph $t_1 < t_2 < t_3$
$⇒$ $R_1 < R_2 < R_3$
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