Question

At t = 0 switch $S_1$ and $S_2$ are closed. Then

• A. Ratio of charges on $C_1$ and $C_2$ at t = RC is 2 : 1.
• B. Potential difference across R is zero at t = 0.
• C. Ratio of rate of heat generation across $r_1$ and $r_2$ at t = RC is 9 : 2.
• D. Ratio of energy stored in $C_1\&C_2$ at t = RC is 3 : 2

Answer 1

Answer: (B), (C)

Ratio of rate of heat generation across $r_1$ and $r_2$ at t = RC is 9 : 2.

${\tau }_1={\tau }_2=2RC$
and $q_1=3\epsilon C(1-e^{\frac{-t}{\tau }})$
$q_2=2\epsilon C(1-e^{\frac{-t}{\tau }})$
$i_1=\frac{d{q}_1}{dt}=\frac{3ϵe^{\frac{-t}{\tau }}}{2R}$
$i_2=\frac{d{q}_2}{dt}=\frac{ϵe^{\frac{-t}{\tau }}}{R}$
$\therefore \frac{q_1}{q_2}=\frac{3}{2}$
hence, $\frac{i_1}{i_2}=\frac{3}{2}$
Now, $\frac{(Heat{)}_1}{(Heat{)}_2}=(\frac{i_1}{i_2}{)}^2\frac{r_1}{r_2}=\frac{9}{2}$
The ratio of energies : $\frac{U_1}{U_2}=(\frac{q_1}{q_2}{)}^2\frac{C_2}{C_1}=(\frac{3}{2}{)}^2\times \frac{2C}{C}=\frac{9}{2}$