Question

In a series LCR circuit alternating e.m.f. and current are given by the equations \(V = V_0sin(ωt)\) and \(I = I_0sin(ωt+\frac{π}{3})\) respectively. The average power dissipated in the circuit over one cycle of a.c. is \((cos60^{\circ} = 0.5)\)

• A. zero
• B. \(\frac{V_0I_0}{2}\)
• C. \(\frac{\sqrt{3}}{2}V_0I_0\)
• D. \(\frac{V_0I_0}{4}\)

Hint:

At resonance the reactances cancel, so the circuit behaves as a pure resistance.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
At resonance \(X_L = X_C\), so the net reactance is zero and \(Z = R\).

Step 2: Check the options:
(A) The impedance is a minimum (equal to R), not a maximum, so (A) is false.
(B) The current is a maximum \(\frac VR\), not a minimum, so (B) is false.
(C) The current leads the voltage only in a capacitive circuit. At resonance the phase angle is zero, so (C) is false.
(D) The phase difference is zero, so the current and voltage are in phase. (D) is true.

Final Answer:
At resonance the current and voltage are in phase, option (D). \[ \boxed{\text{(D)}} \]