There is a circuit consisting of an uncharged capacitor & resistor. A capacitor having capacity $C_{0} & charge Q_{0}$ is connected to the contacts at time $t = 0$ . Graph between charge on this capacitor and time $(t)$ is shown in the figure. Choose the correct statement. Dotted line AB is tangent on graph at $t = 0.$

Value of current just after t = 0 is zero.

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Value of resistance is

\frac{t}{C_{0}} .

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Value of unknown capacitance is

2 C_{0} .

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Total heat loss is

\frac{4 Q_{0}^{2}}{9 C_{0}} .

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Solution

The correct option is C
Value of unknown capacitance is $2 C_{0} .$
Just after $t = 0, i = \frac{d Q}{d t} = slope = \frac{Q_{0}}{t}$

(Charges are from graph)

In parallel, potential difference is same.

$∴ \frac{Q_{0} / 3}{C_{0}} = \frac{2 Q_{0} / 3}{C} \Rightarrow C = 2 C_{0}$

$Heat loss = E_{i} - E_{r} = \frac{Q_{0}^{2}}{2 C_{0}} - (\frac{(Q_{0} / 3)^{2}}{2 C_{0}} + \frac{(2 Q_{0} / 3)^{2}}{2 (2 C_{0})}) = \frac{Q_{0}^{2}}{3 C_{0}}$

As, $i = \frac{Q_{0}}{t}$ and $i R = V_{0}$

We can write that, $\frac{Q_{0} R}{t} = \frac{Q_{0}}{C_{0}}$

$\Rightarrow R = \frac{t}{C_{0}}$

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