A series LCR circuit is connected to an AC source of $220 \mathrm{~V}$, $50 \mathrm{~Hz}$. 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 circuit is $\frac{x}{10}$. The value of x is:

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Alternating Current

A series LCR circuit is connected to an AC source of

220 V

50 Hz

. The circuit contains a resistance

R = 80Ω

, an inductor of inductive reactance

X_{L} = 70Ω

, and a capacitor of capacitive reactance

X_{C} = 130Ω

. The power factor of circuit is

\frac{x}{10}

. The value of x is:

Answer: 8

The power factor is

cos ϕ = \frac{R}{Z}

with

Z = R^{2} + (X_{C} - X_{L})^{2}

. Here

Z = 8 0^{2} + (130 - 70)^{2} = 6400 + 3600 = 10000 = 100 Ω

, so

cos ϕ = \frac{80}{100} = \frac{8}{10}

. Matched against

\frac{x}{10}

, this gives

x = 8

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Integer answer