Question

An rlc circuit is driven by a sinusoidal emf with angular frequency. If angular frequency is increased without changing the amplitude of the emf, the current amplitude increases, it means,
$\begin{aligned} & 1)\omega L>\dfrac{1}{\omega C} \\\ & b)\omega L<\dfrac{1}{\omega C} \\\ & c)\omega L=\dfrac{1}{\omega C} \\\ & d)none \\\ \end{aligned}$

Answer 1

Find the effective resistance of an rlc circuit which is in terms of inductance, capacitance, angular frequency and the resistance. Given, the voltage angular frequency isn’t changed but current angular frequency is increased. It means the resistance of the circuit is decreased.
Formula used:
$\begin{aligned} & i={{i}_{0}}\sin \omega t \\\ & R=\sqrt{{{R}_{0}}^{2}+{{(\omega L-\dfrac{1}{\omega C})}^{2}}} \\\ \end{aligned}$

Complete answer:
Given, the current amplitude is increased with the emf amplitude constant. We know, the emf voltage is the product of current and the resistance of the circuit.
As the voltage or emf is constant, the current in the circuit is inversely proportional to the net resistance in the circuit. It means,
$R=\sqrt{{{R}_{0}}^{2}+{{(\omega L-\dfrac{1}{\omega C})}^{2}}}$is decreased.
For the effective resistance to decrease, the term $\sqrt{{{(\omega L-\dfrac{1}{\omega C})}^{2}}}$must be zero as the resistance can’t be zero.
Therefore, we can get,
$\begin{aligned} & R=\sqrt{{{R}_{0}}^{2}+{{(\omega L-\dfrac{1}{\omega C})}^{2}}} \\\ & \Rightarrow R=0 \\\ & \Rightarrow (\omega L-\dfrac{1}{\omega C})=0 \\\ & \Rightarrow \omega L=\dfrac{1}{\omega C} \\\ \end{aligned}$

So, the correct answer is “Option C”.

Additional Information:
An rlc circuit is a combination of all the three components resistor, inductor and capacitor. The three components can be connected either in series or parallel connection. The name of the circuit is simply derived from the letters of the components used in the circuit. In this oscillating rlc circuit, the capacitor is charged initially, the voltage of this charged capacitor causes a current to flow in the inductor to discharge the capacitor after some time. When the components are connected in series, we use impedance as the effective resistance in the circuit. When the components are connected in parallel, admittance is used instead of impedance. The power factor is simply the cosine of phase angle between voltage and current.

Note:
Pay attention while you calculate for the impedance in the case of series rlc circuit and admittance in the case of parallel rlc circuit. The current in an open circuit is not zero because the voltage across it is zero. Most of us assume it the same way.