Question

For resonance condition in any L-C-R circuit, the phase-difference between applied voltage and current is

• A. \(\pi\)
• B. \(\pi/2\)
• C. \(\pi/4\)
• D. zero

Hint:

At resonance, the effects of the inductor and capacitor perfectly cancel each other out in terms of phase. The LCR circuit essentially behaves like a simple circuit with only a resistor. In a purely resistive circuit, voltage and current are always in phase (\(\phi = 0\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Resonance in a series L-C-R circuit is a condition where the inductive reactance and capacitive reactance cancel each other out, leading to minimum impedance and maximum current.
Step 2: Key Formula or Approach:
The impedance (\(Z\)) of a series L-C-R circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where \(X_L = \omega L\) is the inductive reactance and \(X_C = 1/(\omega C)\) is the capacitive reactance.
The phase difference (\(\phi\)) between voltage and current is given by:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \] Step 3: Detailed Explanation:
The resonance condition occurs when the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \] Substituting this condition into the formula for the phase difference:
\[ \tan(\phi) = \frac{X_L - X_L}{R} = \frac{0}{R} = 0 \] If \(\tan(\phi) = 0\), then the phase angle \(\phi\) must be zero.
When the phase difference is zero, the applied voltage and the current are in the same phase. This means the circuit behaves as a purely resistive circuit at resonance.
Step 4: Final Answer:
At resonance, the phase difference between the applied voltage and current is zero. Therefore, option (D) is correct.