Question

An inductor of inductance L, a capacitor of capacitance C and a resistor of resistance ‘R’ are connected in series to an ac source of potential difference ‘V’ volts as shown in the figure. The potential difference across L, C and R is 40 V, 10 V and 40 V, respectively. The amplitude of current flowing through LCR series circuit is 10√2 A. The impedance of the course is

• A. 5 Ω
• B. 4√2 Ω
• C. 5√2 Ω
• D. 4 Ω

Answer 1

The Correct Option is A

To find the impedance of the LCR series circuit, let's go through the given information and use relevant formulas.

The LCR series circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series with an AC source of potential difference \( V \) volts. The potential differences across L, C, and R are given as:

• Potential difference across the inductor \( V_L = 40 \, \text{V} \) 
• Potential difference across the capacitor \( V_C = 10 \, \text{V} \)
• Potential difference across the resistor \( V_R = 40 \, \text{V} \)

The amplitude of the current \( I \) flowing through the circuit is \( 10\sqrt{2} \, \text{A} \).

The impedance \( Z \) of the LCR circuit is calculated using:

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

where \( X_L = \omega L \) is the inductive reactance and \( X_C = \frac{1}{\omega C} \) is the capacitive reactance.

The total potential difference in the circuit using impedance can be expressed as:

\(V = I \cdot Z\)

We can calculate the total voltage:

\(V = \sqrt{V_R^2 + (V_L - V_C)^2}\)

Substituting the given values, we get:

\(V = \sqrt{40^2 + (40 - 10)^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \, \text{V}\)

Now, using the formula \( V = I \cdot Z \), we substitute the known values:

\(50 = 10\sqrt{2} \cdot Z\)

\(Z = \frac{50}{10\sqrt{2}} = \frac{5}{\sqrt{2}} = 5 \, \Omega\)

Hence, the impedance of the circuit is \( 5 \, \Omega \), which is the correct answer.