Question

In the circuit given in Figure switch $S$ is at position $1$ for long time. Find the total heat generated in resistor of resistance $\left(2r_0\right)$, when the switch $S$ is shifted from position $1$ to position $2$.

• A. $\frac{C_0{E}_0^2}{2}$
• B. $C_0{E}_0^2$
• C. $\frac{C_0{E}_0^2}{3}$
• D. $none$

Answer 1

The correct option is C $\frac{C_0{E}_0^2}{3}$

$StepI:Switchatposition1:$ Since circuit is in the steady state, the current through circuit is zero.According to the loop rule,

$E_0-\frac{q_0}{C_0}=0$ or $q_0=C_0{E}_0$

$StepII:Switchatposition2:$ In this case, the total energy stored in the capacitor appears as heat energy in the resistor.

$\Delta H=I^{2}RT\therefore \Delta H\propto R$

$\therefore \frac{\Delta H_1}{\Delta H_2}=\frac{R_1}{R_2}=\frac{r_0}{2r_0}=\frac{1}{2}or\Delta H_2=2\Delta H_1$

But $\Delta H=\Delta H_1+\Delta H_2=\frac{\Delta H_2}{2}+\Delta H_2=\frac{3}{2}\Delta H_2$

$\therefore \Delta H_2=\frac{2}{3}\Delta H=\frac{2}{3}\times \frac{1}{2}{C}_0{E}_0^2=\frac{1}{3}{C}_0{E}_0^2$