Question

There is a circuit consisting of an uncharged capacitor & resistor. A capacitor having capacity $C_0\&\text{charge}{Q}_0$ is connected to the contacts at time $t=0$. Graph between charge on this capacitor and time $\left(t\right)$ is shown in the figure. Choose the correct statement. Dotted line AB is tangent on graph at $t=0.$

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• A.
  Value of current just after t = 0 is zero.
• B.
  Value of resistance is $\frac{t}{C_0}.$
• C.
  Value of unknown capacitance is $2C_0.$
• D.
  Total heat loss is $\frac{4Q_0^2}{9C_0}.$

Answer 1

Answer: (B), (C)

Value of unknown capacitance is $2C_0.$
Just after $t=0,i=\frac{dQ}{dt}=\text{slope}=\frac{Q_0}{t}$


(Charges are from graph)

In parallel, potential difference is same.

$\therefore \frac{Q_0/3}{C_0}=\frac{2Q_0/3}{C}\Rightarrow C=2C_0$

$\text{Heat loss}=E_i-E_r=\frac{Q_0^2}{2C_0}-(\frac{(Q_0/3{)}^2}{2C_0}+\frac{(2Q_0/3{)}^2}{2\left(2C_0\right)})=\frac{Q_0^2}{3C_0}$

As, $i=\frac{Q_0}{t}$ and $iR=V_0$

We can write that,$\frac{Q_{0}R}{t}=\frac{Q_0}{C_0}$

$\Rightarrow R=\frac{t}{C_0}$