Maharashtra Board Class 12 Physics 2025 – Complete Solutions

Obtain / Derive an expression for average power dissipated in a series LCR circuit

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Published: December 15, 2025 •

Series LCR Circuit

(a) Obtain / Derive an expression for average power dissipated in a series LCR circuit

Answer:

Let an alternating voltage be applied across a series LCR circuit.

Applied voltage:

$$ V = V_0 \sin \omega t $$

Current in the circuit:

$$ I = I_0 \sin (\omega t - \phi) $$

where $\phi$ is the phase difference between the applied voltage and the current.

The phase angle $\phi$ for a series LCR circuit is given by:

$$ \tan \phi = \frac{X_L - X_C}{R} $$

where:

$X_L = \omega L$ is inductive reactance
$X_C = \frac{1}{\omega C}$ is capacitive reactance
$R$ is resistance of the circuit

Instantaneous power:

The instantaneous power supplied to the circuit is:

$$ P = VI $$

Substituting the values of $V$ and $I$:

$$ P = V_0 \sin \omega t \cdot I_0 \sin (\omega t - \phi) $$

Using the trigonometric identity:

$$ \sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)] $$

We get:

$$ P = \frac{V_0 I_0}{2}[\cos \phi - \cos(2\omega t - \phi)] \qquad (1) $$

The second term $\cos(2\omega t - \phi)$ is a periodic function whose average value over a complete cycle is zero.

Average power over one complete cycle:

$$ P_{avg} = \frac{V_0 I_0}{2} \cos \phi $$

Since:

$$ V_{rms} = \frac{V_0}{\sqrt{2}}, \quad I_{rms} = \frac{I_0}{\sqrt{2}} $$

The average power can also be written as:

$$ P_{avg} = V_{rms} I_{rms} \cos \phi $$

For a series LCR circuit:

$$ \cos \phi = \frac{R}{Z} $$

where impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$.

Hence,

$$ P_{avg} = I_{rms}^2 R $$

(b) Define the following terms

(i) Wattless Current

Answer:

In a purely inductive or purely capacitive circuit, the phase difference between voltage and current is:

$$ \phi = 90^\circ $$

Therefore, the power factor:

$$ \cos \phi = 0 $$

Hence, the average power dissipated in the circuit is zero, even though current flows through the circuit. Such a current which does not consume power is called wattless current.

(ii) Q-factor of LCR Circuit

Answer:

The Q-factor (Quality factor) of an LCR circuit is a measure of the sharpness of resonance of the circuit.

It is defined as the ratio of resonant frequency to bandwidth, or equivalently:

$$ Q = \frac{\omega_0 L}{R} $$

where:

$\omega_0$ is the angular resonant frequency
$L$ is the inductance
$R$ is the resistance

A higher Q-factor indicates a sharper resonance and better selectivity of the circuit.

✦✦✦ End of Solution ✦✦✦

Key Takeaways:

Always use curly braces {} around numerator and denominator in fractions
\frac{a}{b} is preferred over a \div b for professional mathematical typesetting
Use \boxed{} to emphasize important formulas in your document
For inline fractions in text, consider using / or \nicefrac{}{} for better readability

Frequently Asked Questions (FAQ)

Is this compatible with MathJax?

Yes, absolutely! All LaTeX expressions shown in this tutorial work perfectly with MathJax, which is the most popular JavaScript library for displaying mathematical notation in web browsers. You can use these commands in any website that supports MathJax rendering.

Can I use this in WordPress?

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What's the difference between \frac and \div?

The \frac{a}{b} command creates a vertical fraction (a over b), which is the standard mathematical notation. The \div command creates the division symbol (÷), which is less commonly used in professional mathematics. In academic and professional contexts, \frac is strongly preferred.

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