Question

A cell of emf $50V$, with negligible resistance is connected in parallel through a switch $S$ to two branches: one is R-L branch and other is R-C branch as shown in the figure. Now in Column I, certain numerical values are given as A, B, C and D, which may correspond to one or more physical quantities, given in Column II.
$[R_1=25\Omega ,R_2=5\Omega ,L=10H,C=20\mu F]$. Use $e^x\simeq 1+x,x<<1$.

Answer 1

$i_1=\frac{50}{25}(1-e^{-t\frac{25}{10}})$

$i_2=\frac{50}{5}(1-e^{\frac{-t}{{10}^{-4}}})$

1) at $t\ll 0.4s,i_1=2(1-e^{-2.5t})$

i.e. $2.5t\ll 1\Rightarrow i_1=2(1-1+2.5t)$

$\approx 0$

2) at $t\gg 0.1s,$ current will stop flowing

$\therefore$ whole voltage will drop across capacitor. $\therefore V_C=50V$

3) at $t\ll 0.15,i_2=10(1-e^{-10\frac{3}{t}})\approx 10A$

4) $V_L=L\frac{di}{dt}=(2\times 10\times 2.5\times e^{-2.5t})=50e^{-2.5t}$

at $t\ll 0.4s\Rightarrow V_L\approx 50V$

5) $i_1=i_2\Rightarrow 2(1-e^{-2.5t})=10(1-e^{-{10}^4+})$

$1-e^{-2.5t}=5-5e^{-{10}^{4}t}$

$4e^{-{10}^{4}t}-e^{-2.5t}=4$