Question

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

Answer 1

$P _{\text {lamp }}=500 \,W$
$\Rightarrow V _{\text {lamp }} i _{\text {lamp }}=500$
$\Rightarrow (100) i _{\text {lamp }}=500$
$i _{\text {lamp }}=5 A$
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
$X _{ C }=\sqrt{ Z ^{2}- R ^{2}}=20 \sqrt{3} \Omega$
$\frac{1}{\omega C }=20 \sqrt{3}$
$C =\frac{1}{20 \sqrt{3} \times 2 \times \pi \times 50}$
$=100 \,\mu F$