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:

- $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - Always compute $\omega = 2\pi f$ first

Answer 1

The Correct Option is B

Given:
\[ L = 1\,\text{H}, \quad C = 20\,\mu\text{F} = 20 \times 10^{-6}\,\text{F}, \quad R = 300\,\Omega \] \[ f = \frac{50}{\pi}\,\text{Hz} \]

Step 1: Angular frequency
\[ \omega = 2\pi f = 2\pi \cdot \frac{50}{\pi} = 100\,\text{rad/s} \]

Step 2: Inductive reactance
\[ X_L = \omega L = 100 \times 1 = 100\,\Omega \]

Step 3: Capacitive reactance
\[ X_C = \frac{1}{\omega C} = \frac{1}{100 \times 20 \times 10^{-6}} = \frac{1}{2 \times 10^{-3}} = 500\,\Omega \]

Step 4: Net reactance
\[ X = X_L - X_C = 100 - 500 = -400\,\Omega \]

Step 5: Impedance
\[ Z = \sqrt{R^2 + X^2} = \sqrt{300^2 + 400^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500\,\Omega \]