JEE Advanced 2021 — Paper 2 — Question 29

${{\text{P}}_{\text{lamp }\!\!~\!\!\text{ }}}=500\text{ }\!\!~\!\!\text{ W}$ $\Rightarrow {{\text{V}}_{\text{lamp }\!\!~\!\!\text{ }}}{{\text{i}}_{\text{lamp }\!\!~\!\!\text{ }}}=500$ $\Rightarrow \left( 100 \right){{\text{i}}_{\text{lamp }\!\!~\!\!\text{ }}}=500$ ${{\text{i}}_{\text{lamp }\!\!~\!\!\text{ }}}=5\text{ }\!\!~\!\!\text{ A}$ Impedance of circuit $\text{Z}=\frac{{{\text{V}}_{\text{source }\!\!~\!\!\text{ }}}}{{{\text{i}}_{\text{source }\!\!~\!\!\text{ }}}}=\frac{200}{5}=40\text{ }\!\!\Omega\!\!\text{ }$ Resistance of lamp $R=\frac{{{\text{V}}_{\text{lamp }\!\!~\!\!\text{ }}}}{{{\text{i}}_{\text{source }\!\!~\!\!\text{ }}}}=20\text{ }\!\!\Omega\!\!\text{ }$ Reactance of capacitance

\begin{array}{*{35}{r}}{{\text{X}}_{\text{C}}} & ~=\sqrt{{{\text{Z}}^{2}}-{{\text{R}}^{2}}}=20\sqrt{3}\text{ }\!\!\Omega\!\!\text{ } \\\frac{1}{\omega \text{C}} & ~=20\sqrt{3} \\\end{array}

$\text{C}=\frac{1}{20\sqrt{3}\times 2\times \pi \times 50}=100\mu \text{ }\!\!~\!\!\text{ F}$