Question

A resistor and a capacitor are connected in series across an AC supply. Which of the following will cause increase in average power utilized by the circuit?

$\left(I\right)$. Increasing the frequency of the AC source.

$\left(II\right)$. Increasing the value of capacitance.

$\left(III\right)$. Increasing the value of peak alternating voltage.

• A. $I$ and $II$
• B. $II$ and $III$
• C. Only $I$
• D. $I$, $II$ and $III$

Answer 1

The correct option is D $I$, $II$ and $III$

For the given circuit,

$X_C=\frac{1}{\omega C}$,

$Z=\sqrt{R^2+X_C^2}$, and,

$I_{\text{rms}}=\frac{E_{\text{rms}}}{Z}$

Now, average power,

$P_{\text{avg}}=E_{\text{rms}}{I}_{\text{rms}}\cos \varphi$

$\Rightarrow P_{\text{avg}}=E_{\text{rms}}\times \frac{E_{\text{rms}}}{Z}\times \frac{R}{Z}=\frac{E_{\text{rms}}^{2}R}{Z^2}$

$\Rightarrow P_{\text{avg}}=\frac{E_{\text{rms}}^{2}R}{R^2+{\left(\frac{1}{\omega C}\right)}^2}=\frac{E_0^{2}R}{2[R^2+{\left(\frac{1}{\omega C}\right)}^2]}$

From the above expression, we can conclude that,

$\left(I\right)$ If frequency is increased, the average power utilized increases.

$\left(II\right)$ If the capacitance is increased, the average power utilized increases.

$\left(III\right)$ If the value of peak alternating voltage $\left(E_0\right)$ is increased, the average power utilized increases.

Hence, option $\left(D\right)$ is the correct answer.