An AC voltage of fixed amplitude is applied across a series \(R L C\) circuit. The components are such that the current at half the resonant frequency is half the current at resonance. Show that the current at twice the resonant frequency is also half the current at resonance.

Impedance is a critical concept in understanding how RLC circuits, consisting of a resistor (R), inductor (L), and capacitor (C) in series, behave with alternating current (AC).

Consider impedance as the AC equivalent to resistance in DC circuits. It's not just a simple measure of how much a circuit resists the flow of electricity; it also takes into account how inductors and capacitors oppose current changes.

Mathematically, impedance (\(Z\)) can be calculated using the formula \(Z = \sqrt{R^2 + (X_L - X_C)^2}\), where \(R\) represents the resistance, \(X_L = ωL\) is the inductive reactance, and \(X_C = \frac{1}{ωC}\) is the capacitive reactance. Since \(ω\) represents the angular frequency of the AC source, changes in frequency have a direct impact on the impedance of a circuit.

A resonant frequency is when an RLC circuit is exactly in tune with the AC supply, causing a maximum or peak current to flow through the circuit. This phenomenon happens because the inductive reactance equals the capacitive reactance (\(X_L = X_C\)), which causes the impedance to be at its minimum and is purely resistive.

The formula to calculate the resonant frequency (\(f_{res}\)) of a series RLC circuit is \(f_{res} = \frac{1}{2\pi\sqrt{LC}}\). At this specific frequency, the effects of the inductor and capacitor cancel each other out, and the circuit behaves as if it only has a resistor in it. Understanding resonant frequency is vital for a range of applications, including tuning radio transmitters and receivers, and designing efficient power systems.

Ohm's law is a fundamental principle in the field of electronics, relating voltage (\(V\)), current (\(I\)), and resistance (\(R\)). For direct current (DC) circuits, it’s simply \(I = \frac{V}{R}\); however, for alternating current (AC) circuits, especially ones with inductors and capacitors, we use impedance instead of resistance.

Thus, Ohm's law in the context of AC circuits with components like RLC circuits is expressed as \(I = \frac{V}{Z}\). It helps us calculate the current based on the voltage applied across the circuit and its impedance. This form of Ohm's law lets us delve deeper into how RLC circuits behave under various frequencies of the AC supply.

Inductive reactance (\(X_L\)) represents the opposition to current change by an inductor in an AC circuit. It is directly proportional to the frequency of the AC source (\(ω\)) and the inductance (\(L\)) of the inductor, given by the formula \(X_L = ωL\).

The higher the frequency or the larger the inductance, the greater the inductive reactance. Unlike resistance, which is constant irrespective of frequency, inductive reactance changes with frequency. At low frequencies, inductors offer little opposition to current, but at high frequencies, they significantly impede the flow of current.

Capacitive reactance (\(X_C\)) is the opposition to current change presented by a capacitor. Inversely proportional to the frequency of the AC source (\(ω\)) and the capacitance (\(C\)), it is calculated with \(X_C = \frac{1}{ωC}\).

This means that at high frequencies, capacitors offer very little opposition, allowing the current to pass through easily, while at low frequencies, they resist current flow more strongly. Understanding capacitive reactance is crucial in designing circuits that filter out certain frequencies, such as in audio equipment or signal processing systems.