To determine the bandwidth of a resonant circuit, it is essential to know both the resonant frequency and the quality factor.

The bandwidth of a resonant circuit refers to the range of frequencies around the resonant frequency at which the circuit responds with a specific amplitude level. This is quantified by measuring the difference between the two frequencies at which the amplitude falls to approximately $70.7\%$ of its maximum value. Importantly, the bandwidth is directly proportional to the resonant frequency and inversely proportional to the quality factor.

To calculate the bandwidth, you begin by finding the resonant frequency using the formula:

where $L$ represents the inductance and $C$ denotes the capacitance of the circuit. Next, you can compute the quality factor with the formula:

where $R$ is the resistance of the circuit. Finally, you can calculate the bandwidth using the equation:

For instance, consider a resonant circuit with a resonant frequency of $1\, \text{MHz}$ and a quality factor of $100$. The bandwidth in this case would be:

This indicates that the circuit will effectively respond with a certain amplitude between the frequencies of $995\, \text{kHz}$ and $1.005\, \text{MHz}$.

A-Level Physics Tutor Summary: To calculate the bandwidth of a resonant circuit, first find the resonant frequency using the formula:

Then, determine the quality factor with:

Finally, compute the bandwidth using:

This method illustrates the relationship between bandwidth, resonant frequency, and quality factor, which is crucial for understanding how circuits respond to various frequencies.