Q.A charged $30\ \mu\text{F}$ capacitor is connected to a $27\ \text{mH}$ inductor. What is the angular frequency of free oscillations of the circuit?

The key idea is that the free oscillations in an LC circuit occur at the natural resonant frequency, where energy sloshes between the capacitor and inductor.

Reasoning:

1. For an ideal LC circuit (no resistance), the angular frequency of free oscillations is given by the resonance formula:

2. Given values: $C = 30\ \mu\text{F} = 30 \times 10^{-6}\ \text{F}$, $L = 27\ \text{mH} = 27 \times 10^{-3}\ \text{H}$.

3. Substitute: …

The Concept: Why an LC Circuit Oscillates

When you connect a charged capacitor to an inductor, you create a perfect electrical pendulum. The capacitor stores energy in its electric field; the inductor stores energy in its magnetic field. There is no resistor here, so no energy is lost — the circuit will oscillate forever at a single natural frequency.

The key insight is that this oscillation is analogous to a mass on a spring. In a mechanical system, the angular frequency is $\omega = \sqrt{k/m}$ where $k$ is the spring constant and $m$ is the mass. In an LC circuit, the inductor $L$ plays the role of inertia (mass), and the capacitor $C$ plays the role of stiffness (the reciprocal of the spring constant). So the natural angular frequency is:

This is one of the most fundamental results in AC circuit theory. It tells you that the oscillation frequency depends only on the component values, not on how much charge you started with or what the initial voltage was.

Step-by-Step Solution

1. Identify the circuit type.

We have only a capacitor and an inductor — no resistor. This is an ideal LC circuit (also called a tank circuit). Free oscillations means the circuit is left to itself after the initial energy is supplied (here, by charging the capacitor).

2. Recall the formula for angular frequency.

For an LC circuit, the charge on the capacitor and the current in the inductor both vary sinusoidally with time. The angular frequency $\omega$ (in radians per second) is:

3. Write down the given values with correct units.

• Capacitance: $C = 30\ \mu\text{F} = 30 \times 10^{-6}\ \text{F}$
• Inductance: $L = 27\ \text{mH} = 27 \times 10^{-3}\ \text{H}$

Always convert micro and milli to the base SI units before plugging in.

4. Substitute into the formula.

First, multiply the numbers inside the square root:

So: …

Method: LC Oscillation Frequency Formula (Resonance in an Ideal LC Circuit)

This method uses the fact that in a lossless LC circuit, the energy oscillates between the capacitor and inductor at a natural angular frequency determined solely by $L$ and $C$.

Steps

1. Identify the given quantities

   • Capacitance: $C = 30\ \mu\text{F} = 30 \times 10^{-6}\ \text{F}$
   • Inductance: $L = 27\ \text{mH} = 27 \times 10^{-3}\ \text{H}$

2. Recall the formula for angular frequency of free oscillations

   For an ideal LC circuit (no resistance), the angular frequency $\omega$ is:

3. Substitute the values

4. Simplify inside the square root

5. Take the square root …

Here are the most common mistakes students make when solving this problem, along with the conceptual fixes to avoid them.

Mistake 1: Using the wrong formula for angular frequency

The Mistake:

Students often confuse the formula for the resonant angular frequency in an LC circuit with the formula for frequency ($f$) or time period ($T$). They might write:

• $\omega = \frac{1}{2\pi \sqrt{LC}}$ (this is actually $f$, not $\omega$)
• Or they forget the square root entirely.

Why it happens:

In AC circuit theory, there are three closely related quantities: $\omega$ (angular frequency in rad/s), $f$ (cyclic frequency in Hz), and $T$ (time period in s). Mixing up their formulas is very common.

How to avoid it:

Memorise the exact formula for angular frequency of free oscillations in an LC circuit:

• $\omega$ is in radians per second.
• If the question asks for $f$, then use $f = \frac{1}{2\pi \sqrt{LC}}$.
• Always check the unit asked in the problem — here it says angular frequency, so use $\omega$.

Mistake 2: Forgetting to convert units to SI

The Mistake:

Plugging in values directly without converting:

• $C = 30\ \mu\text{F}$ used as $30$ instead of $30 \times 10^{-6}\ \text{F}$
• $L = 27\ \text{mH}$ used as $27$ instead of $27 \times 10^{-3}\ \text{H}$

Why it happens:

Micro ($\mu$) and milli (m) prefixes are common in exam problems, but students treat them as "just numbers" out of habit.

How to avoid it:

Always write the conversion step explicitly:

Then substitute into the formula. This single step prevents most numerical errors.

Mistake 3: Arithmetic errors in the square root and reciprocal

The Mistake:

After substituting, students make mistakes like:

• Computing $LC$ incorrectly (e.g., multiplying $27 \times 30$ and then handling powers of 10 wrongly)
• Taking square root incorrectly
• Forgetting to take the reciprocal at the end

How to avoid it:

Break the calculation into clear, small steps:

1. Compute $LC$:

2. Take square root:

$\sqrt{8.1} \approx 2.846$, and $10^{-3.5} = 10^{-4} \times 10^{0.5} = 10^{-4} \times \sqrt{10} \approx 3.162 \times 10^{-4}$

So $\sqrt{LC} \approx 2.846 \times 3.162 \times 10^{-4} \approx 9.0 \times 10^{-4}$ …