Question

A series LCR circuit is connected to an a.c. source of \(220\) V, \(50\) Hz. The circuit contains a resistance \(R = 80\,\Omega\), an inductor of inductive reactance \(70\,\Omega\) and capacitor of capacitive reactance \(130\,\Omega\). The power factor of the circuit is \(\frac{x}{10}\). The value of x is

• A. \(8\)
• B. \(6\)
• C. \(4\)
• D. \(7\)

Hint:

Z = sqrt(R^2 + (XL - XC)^2); power factor = R / Z.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a series LCR circuit, the impedance is \(Z=\sqrt{R^2+(X_L-X_C)^2}\) and the power factor is \(\cos\phi=\dfrac RZ\).

Step 2: Net reactance:
\(X_L-X_C=70-130=-60\ \Omega\). Only its square matters.

Step 3: Impedance:
\[ Z=\sqrt{80^2+60^2}=\sqrt{6400+3600}=100\ \Omega \]

Step 4: Power factor:
\[ \cos\phi=\frac{80}{100}=0.8=\frac{8}{10} \]
So \(x=8\), option (A).

Final Answer:
The power factor is 0.8, so x = 8. \[ \boxed{8} \]