Question

A series LCR circuit with \( R = 20\ \Omega, L = 1.6\text{ H and } C = 40\ \mu\text{F} \) is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is _____ \(\Omega\). 
 

Answer 1

Correct Answer: 200

Step 1: Understanding the Concept:
Resonance in a series LCR circuit occurs when the inductive reactance (\( X_L \)) and capacitive reactance (\( X_C \)) become equal. At this state, the angular frequency is called the resonant frequency.
: Key Formula or Approach:
Resonant frequency \( \omega_o = \frac{1}{\sqrt{LC}} \).
Inductive Reactance \( X_L = \omega_o L \).
Step 2: Detailed Explanation:
Given:
\( L = 1.6 \text{ H} \).
\( C = 40 \times 10^{-6} \text{ F} \).
Calculate resonant angular frequency \( \omega_o \):
\[ \omega_o = \frac{1}{\sqrt{1.6 \times 40 \times 10^{-6}}} = \frac{1}{\sqrt{64 \times 10^{-6}}} \]
\[ \omega_o = \frac{1}{8 \times 10^{-3}} = 125 \text{ rad/s} \].
Calculate inductive reactance \( X_L \):
\[ X_L = \omega_o \cdot L = 125 \times 1.6 = 200\ \Omega \].
Step 3: Final Answer:
The inductive reactance at resonant frequency is 200 \(\Omega\).