Question

An AC circuit with \(R=2\pi^2\,\Omega\) and \(L=0.02\pi\) H powered with an a.c. source of frequency 50 Hz has an impedance of

• A. \(2\pi^2\,\Omega\)
• B. \(2\sqrt{2}\pi^2\,\Omega\)
• C. \(2\,\Omega\)
• D. \(2\pi\,\Omega\)
• E. \(\pi\,\Omega\)

Hint:

In an \(RL\) AC circuit: \[ Z=\sqrt{R^2+X_L^2},\qquad X_L=\omega L \] Calculate \(X_L\) first, then substitute into impedance formula.

Answer 1

The Correct Option is B

For an \(RL\) AC circuit: \[ Z=\sqrt{R^2+X_L^2} \] where \[ X_L=\omega L = 2\pi fL \] Given: \[ f=50\text{ Hz} \] \[ L=0.02\pi \text{ H} \] So, \[ X_L=2\pi(50)(0.02\pi)=2\pi^2 \] Also, \[ R=2\pi^2 \] Thus, \[ Z=\sqrt{(2\pi^2)^2+(2\pi^2)^2} \] \[ Z=\sqrt{2(2\pi^2)^2}=2\pi^2\sqrt{2} \]
Hence, the correct answer is: \[ \boxed{(B)\ 2\sqrt{2}\pi^2\,\Omega} \]