Question

In an A.C. circuit, containing inductance and a capacitor in series, the current is found to be maximum when the value of the inductance is 0.5 Henry and capacitance is \[8\mu F\] . The angular frequency of the input A.C. voltage must be equal to
(A) \[500\dfrac{{rad}}{s}\]
(B) \[4000\dfrac{{rad}}{s}\]
(C) \[5 \times {10^5}\dfrac{{rad}}{s}\]
(D) \[5000\dfrac{{rad}}{s}\]

Answer 1

Hint: In this question, we are given that the circuit is an LC circuit. For the maximum current to occur, this impedance should be equal to 0. This means that the inductive resistance will become equal to the capacitive resistance. Solving this we can get the value of angular frequency.

Complete step by step solution
The impedance of an LC circuit is given as:
 \[Z = {X_L} - {X_C}\]
For the current to be maximum, we will have the impedance of the circuit equal to 0, therefore:
 \[
  {X_L} = {X_C} \\
  \omega L = \dfrac{1}{{\omega C}} \\
  {\omega ^2} = \dfrac{1}{{LC}} \\
  \omega = \dfrac{1}{{\sqrt {0.5 \times 8 \times {{10}^{ - 6}}} }} \\
  \omega = 500\dfrac{{rad}}{s} \\
 \]

Therefore the option with the correct answer is option A

Note
The condition when the current is maximum in an LC circuit is known as resonance. The angular frequency of both the inductor and capacitor becomes equal. This angular frequency is also known as the resonating frequency of the circuit.