A coil of self-inductance \(L\) and resistance \(R\) is connected in series with a switch and the combination is placed in parallel with a capacitor of capacitance C. An alternating voltage is applied across \(C\), and the current supplied by this source is measured by a milliammeter. The value of \(C\) is adjusted until the milliammeter reading remains the same whether the switch is open or closed. Show that \(2 \omega^{2} L C=1\) (Turner's method for measuring a large \(L\) )

Alternating current (AC) circuits are fundamental in electrical engineering, with applications ranging from household appliances to industrial machinery. Unlike direct current (DC) circuits, where the flow of electric charge is unidirectional, AC circuits have a current that varies in magnitude and periodically reverses direction. The alternating nature is typically sinusoidal due to its ease of generation and energy efficiency.

In an AC circuit, the voltage across components also varies with time, usually in sync with the current. When analyzing AC circuits, engineers use complex numbers to represent the circuit parameters like resistance, inductance, and capacitance, which together determine the circuit's overall behavior. The concept of impedance, denoted as Z, is introduced as a complex extension of resistance to encapsulate not only the opposition to current that resistors provide but also the phase shift between voltage and current caused by inductors and capacitors.

Understanding how these elements interact in an AC circuit is crucial for designing reliable and efficient electrical systems. The response of AC circuits to different frequencies is a defining characteristic, making the analysis significant for a variety of applications, such as filtering and tuning in communication systems.

The RLC circuit is a classic example of an AC circuit that includes a resistor (R), an inductor (L), and a capacitor (C). RLC circuit analysis revolves around studying how these three components influence the overall impedance and, consequently, the circuit's current and voltage profile at different frequencies.

In the context of the provided exercise, the RLC circuit analysis becomes vital when the switch is closed, as both the resistor-inductor (RL) branch and the capacitor contribute to the total impedance of the circuit. The behavior of the RLC circuit is typically characterized by the resonant frequency at which the circuit naturally oscillates. This is the frequency where the inductive and capacitive reactances cancel each other out, leading to the phenomenon called resonance, which is crucial in many applications like radio receivers and oscillators.

Understanding RLC Resonance

In resonance, the impedance of the RLC circuit is purely resistive, meaning that the circuit behaves as if it only has resistance without any reactance. This makes it easier for the alternating current to flow through the circuit, often resulting in a peak in the current amplitude. The calculation of resonant frequency and impedance is a significant aspect of RLC circuit analysis and is key for understanding the conditions under which the milliammeter reading would remain unchanged upon opening or closing the switch, as per the exercise.

Impedance plays a central role in the operation of AC circuits. It can be thought of as the total opposition a circuit presents to the flow of alternating current. Impedance extends the concept of resistance to AC circuits and is a complex quantity encompassing both magnitude and phase difference between voltage and current.

Components of Impedance

• Resistive Component (R): Represents traditional resistance, with no phase shift between current and voltage.
• Inductive Reactance (X_L): Caused by inductors in a circuit, it causes the current to lag behind the voltage by a quarter of a cycle (90 degrees).
• Capacitive Reactance (X_C): Caused by capacitors, it leads to the current leading the voltage by a quarter of a cycle.

In the given exercise, when the switch is closed, the inductor and resistor are in series forming an RL circuit, which together with the shunt capacitor, forms a parallel RLC circuit. The total impedance is a combination of the RL circuit's impedance and the capacitor's impedance.

Calculating Total Impedance

For components in parallel, the total impedance is found by the inverse sum of the individual impedances. Given that capacitors and inductors have opposing effects on the phase of the current, their reactances can cancel out under certain conditions – precisely the phenomenon being taken advantage of in the provided exercise to measure the coil's self-inductance.

A thorough understanding of impedance is required to solve complex AC circuit problems and is foundational in fields such as telecommunications, signal processing, and power systems.