Question

An ac circuit contains a resistance of 1 k$\Omega$, a capacitor of 0.1 $\mu$F and an inductor of 1 mH connected in series. The resonance frequency of the circuit is approximately: ____.

• A. 13.5 kHz
• B. 15.9 kHz
• C. 10.1 kHz
• D. 20.7 kHz

Hint:

To speed up calculations involving $2\pi$, remember that $1/2\pi \approx 0.159$. This makes $0.159 \times 10^5$ immediately recognizable as 15.9 kHz.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Resonance in a series RLC circuit occurs when the inductive reactance equals the capacitive reactance ($X_L = X_C$), allowing the maximum possible current to flow.

Step 2: Key Formula or Approach:
The resonance frequency ($f_r$) is given by: \[ f_r = \frac{1}{2\pi\sqrt{LC}} \]

Step 3: Detailed Explanation:
Given: $L = 1\text{ mH} = 10^{-3}\text{ H}$, $C = 0.1\,\mu\text{F} = 10^{-7}\text{ F}$. 1. Calculate $LC$: \[ LC = 10^{-3} \times 10^{-7} = 10^{-10} \] 2. Calculate $\sqrt{LC}$: \[ \sqrt{LC} = \sqrt{10^{-10}} = 10^{-5} \] 3. Calculate $f_r$: \[ f_r = \frac{1}{2\pi \times 10^{-5}} = \frac{10^5}{2\pi} \] \[ f_r \approx \frac{100,000}{6.28} \approx 15,923\text{ Hz} \approx 15.9\text{ kHz} \]

Step 4: Final Answer:
The resonance frequency is approximately 15.9 kHz.