A series \(R L C\) circuit with \(R=47 \Omega, L=250 \mathrm{mH},\) and \(C=\) \(4.0 \mu \mathrm{F}\) is connected across a sine-wave generator whose peak output voltage is independent of frequency. Find the frequency range over which the peak current will exceed half its value at resonance.

The transfer of energy in a series RLC circuit reaches its optimal efficiency at a specific frequency called the resonance frequency. This is the frequency at which the circuit naturally oscillates due to the inductive and capacitive reactances canceling each other out, thus minimizing the overall impedance. The formula to calculate this special frequency is given by \( f_0 = \frac{1}{2 \pi \sqrt{LC}} \).

For instance, consider a circuit with an inductor of \( 250 \, \text{mH} \) and a capacitor of \( 4.0 \, \text{\textmu F} \). Using our formula, the resonance frequency can be determined. It's where the circuit would allow the greatest current to flow if driven by an external AC voltage source with a frequency equal to \( f_0 \). The concept of resonance is crucial in various applications such as radios and telecommunication systems, where it's used to select specific frequencies from a broad spectrum of signals.

Impedance, symbolized by \( Z \), is a measure of opposition that a circuit presents to the flow of alternating current. It is a complex quantity that combines the effects of resistance \( R \), inductive reactance \( X_L \), and capacitive reactance \( X_C \). The impedance of a series RLC circuit is calculated by the formula \( Z = \sqrt{R^2 + (X_L - X_C)^2} \).

At resonance, the inductive and capacitive reactances are equal in magnitude but opposite in phase, resulting in their cancelation: \( X_L = X_C \). This minimizes the total impedance to just the resistance \( R \), allowing for the maximum current to flow. However, away from the resonant frequency, the impedance increases, and less current can flow for a given voltage. Understanding how impedance varies with frequency is essential in designing circuits with desired electrical characteristics and in troubleshooting.

When dealing with current in resonant circuits, it's interesting to observe how the current behaves at and around the resonance frequency. At resonance, the current reaches its maximum value because the impedance is at its minimum – equivalent to the resistance of the circuit. The current is only limited by the resistance and not by the reactive effects of the inductor or capacitor.

As the frequency deviates from the resonance frequency, the current starts to decrease due to the increasing impedance. If we want to find the frequencies at which the current is, for example, half of its maximal value at resonance, we set the impedance \( Z \) to twice the resistance \( 2R \), signifying that the current has halved. Solving this gives us the frequencies that define our band of interest around the resonance frequency. These frequencies are important for designing filters and understanding bandwidth in communication systems.