Animated Solution for Physics - Alternating Current: In a series $L-C-R$ resonance circuit, if we change the resistance only, from a lower to higher value,

How Resistance Shapes the L-C-R Circuit

Imagine tuning a radio to your favorite station. You are essentially adjusting an $L-C-R$ circuit to resonate at a specific frequency. But what happens when you swap out the resistor for one with a higher value? Does the station change? Does the signal get clearer? Let's dive into the physics of resonance to find out.

In a series $L-C-R$ circuit, resonance occurs when the inductive reactance ($X_L$) perfectly cancels out the capacitive reactance ($X_C$). At this magical point, the circuit behaves as if it's purely resistive. The frequency at which this happens is called the resonance frequency, denoted by ${\omega }_0$.

Mathematically, it is given by:

Look closely at this equation. Do you see the resistance $R$ anywhere? No! The resonance frequency is determined entirely by the inductor $L$ and the capacitor $C$. Therefore, changing the resistance has absolutely zero effect on the resonance frequency. It remains constant.

The Spread of Frequencies

While the resonance frequency tells us where the peak is, the bandwidth ($\Delta \omega$) tells us how wide the peak is. It represents the range of frequencies over which the circuit absorbs at least half of its maximum power.

The formula for bandwidth is beautifully simple:

This equation reveals a direct proportionality. If we increase the resistance $R$ from a lower to a higher value, the numerator grows, and consequently, the bandwidth $\Delta \omega$ increases. Visually, the sharp, narrow peak of the resonance curve flattens out and becomes wider.

The Sharpness of the Peak

The Quality Factor ($Q$) is the ultimate measure of a circuit's selectivity. A high $Q$ means the circuit is highly selective (a sharp peak), while a low $Q$ means it's less selective (a flat peak). It is defined as the ratio of the resonance frequency to the bandwidth:

Because $R$ sits in the denominator, the Quality Factor is inversely proportional to the resistance. As we increase the resistance, the Quality Factor decreases.

By analyzing the fundamental equations of the $L-C-R$ circuit, we can confidently conclude that increasing the resistance will flatten the resonance curve, thereby increasing the bandwidth. The resonance frequency will stubbornly remain the same, and the quality factor will drop. Thus, the correct statement is that the bandwidth of the resonance circuit will increase.