Question

A series circuit has a resistance of $60\Omega$ and an impedance of $135\Omega$. When the total potential difference is $120\text{V}$, the power consumed in the circuit is $x\text{W}$. The value of $x$ is ________.

$[$Round off to the nearest odd integer$]$

Answer 1

Impedance is given by $Z^2=R^2+(X_L-X_C{)}^2$

$\therefore {135}^2={60}^2+(X_L-X_C{)}^2$

$\Rightarrow \left|\begin{array}{c}{X}_L-X_C\end{array}\right|\approx 121\Omega$

Also,
$\tan \theta =\frac{\left|\begin{array}{c}{X}_L-X_C\end{array}\right|}{R}=\frac{121}{60}\approx 2$

The power consumed is,

$P=\frac{V^2}{Z}\cos \theta =\frac{{120}^2}{135}\times \frac{1}{\sqrt{1+2^2}}$

$[\because {\sec }^2\theta -{\tan }^2\theta =1\text{and}\cos \theta =\frac{1}{\sec \theta }]$

$\Rightarrow P=47\text{W}$ (the nearest odd integer)