Question

In the series LCR circuit shown in figure the impedance is

• A. \(300\,\Omega\)
• B. \(500\,\Omega\)
• C. \(700\,\Omega\)
• D. \(900\,\Omega\)

Hint:

Find omega from the frequency, then XL and XC, then Z = sqrt(R^2 + (XL - XC)^2).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The figure shows a series circuit with an inductor of 1 H, a capacitor of 20 microfarad and a resistor of 300 ohm, driven by a 50 V source of frequency \(\dfrac{50}{\pi}\) Hz.

Step 2: Key Formula or Approach:
\[ \omega = 2\pi f,\quad X_L = \omega L,\quad X_C = \frac{1}{\omega C},\quad Z = \sqrt{R^2 + (X_L - X_C)^2} \]

Step 3: Detailed Explanation:
Angular frequency:
\[ \omega = 2\pi\cdot\frac{50}{\pi} = 100 \text{ rad/s} \]
Inductive reactance:
\[ X_L = 100\times1 = 100\ \Omega \]
Capacitive reactance:
\[ X_C = \frac{1}{100\times20\times10^{-6}} = \frac{1}{2\times10^{-3}} = 500\ \Omega \]
Impedance:
\[ Z = \sqrt{300^2 + (100 - 500)^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500\ \Omega \]
Option (A) 300 Ω would be the resistance alone. Option (C) 700 Ω is \(R + |X_L - X_C|\) minus something, and (D) 900 Ω adds \(R + X_L + X_C\), which wrongly adds the reactances.

Final Answer:
The impedance is 500 Ω, option (B). \[ \boxed{500\ \Omega \text{ (B)}} \]