Question

In the circuit, a metal filament lamp is connected in series with a capacitor of capacitance $C\mu F$ across a $200V,50Hz$ supply. The power consumed by the lamp is $500W$ while the voltage drop across it is $100V$. Assume that there is no inductive load in the circuit. Take $rms$ values of the voltages. The magnitude of the phase-angle (in degrees) between the current and supply voltage is $\varphi .$ Assume, $\pi \sqrt{3}\approx 5$.
The value of $C$ is _____.

Answer 1

Answer: 100.00

$P_{\text{lamp }}=500W$
$\Rightarrow V_{\text{lamp }}{i}_{\text{lamp }}=500$
$\Rightarrow (100)i_{\text{lamp }}=500$
$i_{\text{lamp }}=5A$
Impedance of circuit $Z=\frac{V_{\text{source }}}{i_{\text{source }}}$
$=\frac{200}{5}=40\Omega$
Resistance of lamp $R=\frac{V_{\text{lamp }}}{i_{\text{source }}}$
$=20\Omega$
Reactance of capacitance
$X_C=\sqrt{Z^2-R^2}=20\sqrt{3}\Omega$
$\frac{1}{\omega C}=20\sqrt{3}$
$C=\frac{1}{20\sqrt{3}\times 2\times \pi \times 50}$
$=100\mu F$