In a circuit, a metal filament lamp is connected in series with a...

Given,

$P = 500W$

${V}_{R} = 100V$

$f = 50Hz$

${V}_{rms} = 200V$

$P = \dfrac{{{V}_{R}}^{2}}{R} ⇒ R = \dfrac{{{V}_{R}}^{2}}{P} ⇒ R = \dfrac{{\left(100\right)}^{2}}{500}$

$∴ R = 20Ω$

Now current in the circuit,

$i = {i}_{rms} = \dfrac{voltageacrossresistor}{resistance}i = \dfrac{100}{20}i = 5A$

Phasor diagram of the given circuit for voltage,

Now,

${V}^{2} = {{V}_{C}}^{2} + {{V}_{R}}^{2} ⇒ {200}^{2} = {{V}_{C}}^{2} + {100}^{2} ⇒ {V}_{C} = 100\sqrt{3}V$

Now,

${V}_{C} = 100\sqrt{3}V = I{X}_{C} ⇒ {X}_{C} = \dfrac{100\sqrt{3}}{5} = 20\sqrt{3}Ω$

and

${X}_{C} = \dfrac{1}{\left(2\pif\right)C} ⇒ 20\sqrt{3} = \dfrac{1}{\left(2\pi \times 100\right)C} ⇒ C = 100\mu F$

From the phasor diagram,

$\tan \left(ϕ\right) = \dfrac{{X}_{C}}{R} = \dfrac{20\sqrt{3}}{20} = \sqrt{3} ⇒ ϕ = 60 °$