Question

A series LCR circuit is connected to an ac source of $220\, V , 50\, Hz$ The circuit contain a resistance $R =100\, \Omega$ and an inductor of inductive reactance $X _{ L }=796\, \Omega$ The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be ________$\mu F$

Answer 1

Correct Answer: 40

The goal is to determine the capacitance of the capacitor needed to maximize the average power supplied in a series LCR circuit connected to an AC source. When resonance is achieved, the power factor is maximized because impedance is minimized. The condition for resonance is when the inductive reactance X_L and capacitive reactance X_C are equal.

Given:

• Resistance, R = 100 Ω
• Inductive reactance, X_L = 796 Ω
• Voltage, V = 220 V
• Frequency, f = 50 Hz

For resonance, X_L = X_C. The capacitive reactance can be expressed as:

X_C = 1 / (2πfC)

Equating X_L = X_C gives us:

796 = 1 / (2π × 50 × C)

Rearranging for C:

C = 1 / (2π × 50 × 796)

Calculating the capacitance:

C ≈ 4.00 × 10^-6 F

Converting Farads to microfarads:

C ≈ 4.00 μF

The computed capacitance of 4.00 μF falls perfectly within the specified range.