Animated Solution for Physics - Electromagnetic Induction: Match List I with List II. Choose the correct answer from the options given below.

The beauty of an L-C-R series circuit lies in the constant tug-of-war between the inductor and the capacitor. While the resistor simply dissipates energy, the inductor and capacitor store and release it, but they do so completely out of sync with each other. This matrix match question is a fantastic exercise in understanding who wins this tug-of-war under different frequency conditions.

Let's break down the physics behind each condition using the elegant tool of phasor diagrams.

The Inductive Dominance

In the first condition, we are given that $\omega L>\frac{1}{\omega C}$.

We know that the inductive reactance is $X_L=\omega L$ and the capacitive reactance is $X_C=\frac{1}{\omega C}$. Therefore, this condition simply states that $X_L>X_C$. The inductor is dominating the circuit!

Because the voltage across an inductor leads the current by $90^{\circ }$, and the voltage across a capacitor lags by $90^{\circ }$, a larger $X_L$ means the net reactive voltage ($V_L-V_C$) points upwards in our phasor diagram. The overall voltage phasor is pulled ahead of the current phasor.

Conclusion: The voltage leads the current, which is exactly the same as saying the current lags behind the applied EMF. Thus, A matches with (ii).

The Perfect Harmony of Resonance

Next, we look at the condition where $\omega L=\frac{1}{\omega C}$.

This is the magical state where $X_L=X_C$. The opposition offered by the inductor perfectly cancels out the opposition offered by the capacitor. The circuit is no longer reactive; it behaves as if it is purely resistive.

In a purely resistive circuit, there is no phase difference. The voltage and current phasors lie flat on top of each other, marching perfectly in step.

Conclusion: The current is in phase with the EMF. Thus, B matches with (i).

The Capacitive Shift

Now, let's flip the script. What if $\omega L<\frac{1}{\omega C}$?

Here, $X_L<X_C$. The capacitor has taken control. Because the capacitor's voltage lags the current, the net reactive voltage ($V_L-V_C$) is negative, pointing downwards in the phasor diagram. The overall voltage phasor is dragged behind the current phasor.

Conclusion: The voltage lags the current, meaning the current leads the EMF. Thus, C matches with (iv).

The Climax of Resonance

Finally, we are asked about the Resonant frequency.

As we discovered in Case B, resonance is the exact frequency where $X_L=X_C$. But what is the physical consequence of this harmony?

The total impedance of an L-C-R circuit is given by the master equation:

At resonance, the term $(X_L-X_C)$ becomes zero. The impedance drops to its absolute minimum value, $Z=R$. According to Ohm's Law for AC circuits ($I=\frac{V}{Z}$), when the opposition (impedance) is at its minimum, the flow must be at its maximum.

Conclusion: At the resonant frequency, the maximum current occurs. Thus, D matches with (iii).

By systematically analyzing the tug-of-war between the inductor and capacitor, we have successfully decoded the matrix: - A $\to$ (ii) - B $\to$ (i) - C $\to$ (iv) - D $\to$ (iii)

This perfectly aligns with the first option provided in the question. Mastering phasor diagrams is the ultimate key to unlocking the secrets of Alternating Current!