The Question is from neet-physics alternating-current resontant-frequency-l-c-oscillations, and it was created by the author P.NARASIMHA

Question

A variable capacitor is connected to an inductor of inductance 20 mH. For what capacitance will the circuit resonate at a frequency of 1 kHz?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
 **Primary Concept**: Resonance in LCR circuits
 **Related Topics**: Oscillations, AC circuits
 **Prerequisites**: Understanding of inductance, capacitance, frequency, and resonance.
## SOLUTION PATHWAY
**Given Information**:
  * Inductance (L) = 20 mH = 20 × 10⁻³ H
  * Resonance frequency (f) = 1 kHz = 1000 Hz
**Method & Steps**:
 1. The resonance frequency of an LCR circuit is given by the formula:
  Reasoning: At resonance, the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$).
  Formula/Rule: $f = \frac{1}{2\pi\sqrt{LC}}$
 2. Rearrange the formula to solve for capacitance (C):
  Working: $C = \frac{1}{4\pi^2f^2L}$
  Substitute the given values:
  $C = \frac{1}{4\pi^2(1000)^2(20 	imes 10^{-3})} = \frac{1}{7895683.5} \approx 1.267 	imes 10^{-6} F$
  Convert Farads to nanoFarads: $C \approx 1.27 nF$
 3. Final Result:
  Result: The capacitance required for resonance is approximately 1.27 nF.
  Verification: Substitute the calculated capacitance back into the resonance frequency formula to verify the result.
 ## OPTION ANALYSIS
A :x: 12.7 nF is significantly larger than the calculated value.
B ✓: 1.27 nF matches the calculated value.
C :x: 0.127 nF is smaller than the calculated value.
D :x: 127 nF is much larger than the calculated value.
 ## LEARNING AIDS
 **Common Mistakes**:
  1. Incorrect unit conversions (e.g., mixing mH and F without converting): Always ensure consistent units before calculation.
  2. Errors in rearranging the formula: Carefully manipulate the formula to isolate the desired variable.
 **Quick Tips**:
  1. Memorize the resonance frequency formula for LCR circuits.
  2. Use consistent units throughout the calculation to avoid errors.

P.NARASIMHA · Answer

12.7 nF

P.NARASIMHA · Answer

The correct Option is "1.27 nF"

P.NARASIMHA · Answer

0.127 nF

P.NARASIMHA · Answer

127 nF

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