Question

The inductance L, Capacitance C and resistance R are the values of the components connected in series to an ac source of angular frequency \(\omega\). The inductive and capacitive reactances are \(X_L\) and \(X_C\) respectively. If the circuit is purely resistive, then

• A. L = C
• B. \(X_L = X_C\)
• C. \(\omega L = \omega C\)
• D. R = L = C

Hint:

A "purely resistive" AC circuit means that the voltage and current are in phase. In a series LCR circuit, this happens only at resonance, where the effects of the inductor and capacitor cancel each other out. The condition for resonance is always \(X_L = X_C\), which leads to the resonant frequency \(\omega_0 = 1/\sqrt{LC}\).

Answer 1

The Correct Option is B

In a series LCR circuit, the total impedance (Z) is given by the formula:
\( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where R is the resistance, \(X_L\) is the inductive reactance, and \(X_C\) is the capacitive reactance.
The term \( (X_L - X_C) \) represents the total reactance of the circuit.
A circuit is considered "purely resistive" when its impedance is equal to its resistance.
This means the reactive part of the impedance must be zero.
So, we must have \( (X_L - X_C) = 0 \).
This implies that \( X_L = X_C \).
This condition is known as resonance. At resonance, the inductive reactance and capacitive reactance cancel each other out, and the circuit behaves as if it only contains the resistor. The impedance is at its minimum value (\(Z=R\)), and the current is at its maximum.
The other options are incorrect. L=C compares inductance and capacitance, which have different units and cannot be equated. \(\omega L = \omega C\) implies L=C. R=L=C is dimensionally inconsistent.