A series LCR circuit is connected to an AC source of 220 V, 50 Hz. The circuit contains a resistance R = 80, an inductor of inductive reactance XL = 70, and a capacitor of capacitive reactance XC =…

Given: A series LCR circuit has R = 80, XL = 70, and XC = 130. Find: The value of x if the power factor is x/10. For a series LCR circuit, the power factor is = RZ where the impedance is Z = R^2 + (XL - XC)^2 Substituting the given values, = 80 80^2 + (70 - 130)^2 Now simplify the reactance difference, 70 - 130 = -60 So, = 80 80^2 + (-60)^2 Evaluating further, = 80 6400 + 3600 = 80 10000 = 80100 = 0.8 Since the power factor is given as x/10, compare directly: x10 = 0.8 = 810 Therefore, x = 8. The numerical answer is 8.

A series LCR circuit is connected to an AC source of 220 V, 50 Hz… — Solution | JEE Physics

NVAEasyJEE 2023LCR Circuits & Resonance

A series LCR circuit is connected to an AC source of $220 \, \text{V}$ , $50 \, \text{Hz}$ . The circuit contains a resistance $R = 80 \, \Omega$ , an inductor of inductive reactance $X_L = 70 \, \Omega$ , and a capacitor of capacitive reactance $X_C = 130 \, \Omega$ . The power factor of the circuit is $x/10$ . The value of $x$ is:

Answer

Correct answer:8

Step-by-step solution

Standard Method

Given: A series LCR circuit has $R = 80 \, \Omega$ , $X_L = 70 \, \Omega$ , and $X_C = 130 \, \Omega$ .

Find: The value of $x$ if the power factor is $x/10$ .

For a series LCR circuit, the power factor is

\cos \phi = \frac{R}{Z}

where the impedance is

Z = \sqrt{R^2 + (X_L - X_C)^2}

Step-by-step Calculation

Substituting the given values,

\cos \phi = \frac{80}{\sqrt{80^2 + (70 - 130)^2}}

Now simplify the reactance difference,

70 - 130 = -60

So,

\cos \phi = \frac{80}{\sqrt{80^2 + (-60)^2}}

Evaluating further,

\cos \phi = \frac{80}{\sqrt{6400 + 3600}} = \frac{80}{\sqrt{10000}} = \frac{80}{100} = 0.8

Relating to x/10

Since the power factor is given as $x/10$ , compare directly:

\frac{x}{10} = 0.8 = \frac{8}{10}

Therefore, $x = 8$ . The numerical answer is 8.

Common mistakes

Using $X_L + X_C$ instead of $X_L - X_C$ for net reactance. In a series LCR circuit, inductive and capacitive reactances oppose each other. Use $X_L - X_C$ or $X_C - X_L$ inside the square term.
Writing the power factor as $Z/R$ instead of $R/Z$ . Power factor is $\cos \phi = R/Z$ , so reversing the ratio gives a value greater than $1$ , which is not possible.
Ignoring the square on the reactance difference. The impedance formula is $Z = \sqrt{R^2 + (X_L - X_C)^2}$ . Do not substitute $X_L - X_C$ without squaring it.

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