NEET 2022 Physics Series LCR Circuit & Resonance MCQ Question

Chapter: Alternating Current

Class: 12 Physics Topic: Series LCR Circuit & Resonance Difficulty: 🟡 Medium

✅ Ans: **D — $

u_0= u=\frac{50}{\pi},\text{Hz}$**

Given:

$$L=10\,H,\qquad C=10\,\mu F=10^{-5}F$$

The AC voltage is:

$$V=200\sin(100t)$$

Comparing with:

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

we get:

$$\boxed{\omega=100\ rad/s}$$

Step 1: Find source frequency

$$\omega=2\pi u$$ $$u=\frac{100}{2\pi}$$ $$\boxed{ u=\frac{50}{\pi}\,Hz}$$

Step 2: Find resonant frequency

For a series LCR circuit:

$$\omega_0=\frac{1}{\sqrt{LC}}$$ $$=\frac{1}{\sqrt{10\times10^{-5}}}$$ $$=\frac{1}{\sqrt{10^{-4}}} =100\ rad/s$$

Therefore,

$$u_0=\frac{\omega_0}{2\pi} =\frac{100}{2\pi}$$ $$\boxed{ u_0=\frac{50}{\pi}\,Hz}$$

Hence,

$$\boxed{ u_0= u=\frac{50}{\pi}\,Hz}$$

❌ Why other options are wrong?

• A: $u_0=\frac{50}{\pi},\ u=50$ ❌ Source frequency is not 50 Hz; $100$ is angular frequency, not frequency.
• B: ❌ Both values are interchanged/incorrect.
• C: $u_0= u=50$ ❌ Confuses angular frequency $100\,rad/s$ with frequency.
• D: ✅ Correct.

📌 NCERT Concept

At resonance in a series LCR circuit:

$$\boxed{X_L=X_C}$$

and

$$\boxed{\omega_0=\frac1{\sqrt{LC}}}$$

At resonance, the source frequency equals the resonant frequency:

$$\boxed{ u= u_0}$$

🧠 NEET Trick

If voltage is written as:

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

the number inside $t$ is $\omega$, not $f$.

Always use:

$$\boxed{f=\frac{\omega}{2\pi}}$$

So here:

$$\boxed{ u_0= u=\frac{50}{\pi}\,Hz}$$