1. Electromagnetic Induction Basics: Faraday and Lenz Laws

Electromagnetic induction is the phenomenon where a changing magnetic field induces an electromotive force (EMF) in a conductor. Faraday's Law of Induction quantifies this: the induced EMF in any closed circuit is equal to the negative of the time rate of change of the magnetic flux ($\Phi_B$) through the circuit, $\mathcal{E} = -\frac{d\Phi_B}{dt}$. Lenz's Law determines the direction of the induced current, stating that the induced current flows in a direction that opposes the change in magnetic flux that produced it, a consequence of the conservation of energy.

2. Eddy Currents and Inductance

When a conductor is exposed to a changing magnetic field, circulating currents, known as eddy currents, are induced within it. These currents can cause resistive heating and energy loss. Eddy currents are minimized in transformers and motors by using laminated cores. Inductance ($L$) is a property of a circuit element (inductor) that opposes changes in current flowing through it. It is defined by the relationship between the induced EMF and the rate of change of current: $\mathcal{E} = -L \frac{dI}{dt}$.

3. AC Generators

An AC generator converts mechanical energy into alternating current (AC) electrical energy. It typically consists of a coil rotating in a magnetic field. As the coil rotates, the magnetic flux through it changes, inducing an EMF according to Faraday's Law. The induced EMF and current vary sinusoidally with time, producing alternating current. AC generators are the primary source of electricity in most countries, including India, for powering homes and industries.

4. AC Circuits with Individual Components (R, L, C)

In AC circuits, resistors ($R$), inductors ($L$), and capacitors ($C$) exhibit different behaviors. A resistor dissipates energy as heat, following Ohm's Law. An inductor opposes changes in current, with its opposition characterized by inductive reactance ($X_L = \omega L$). A capacitor opposes changes in voltage, with its opposition called capacitive reactance ($X_C = \frac{1}{\omega C}$). These reactances depend on the angular frequency ($\omega$) of the AC supply.

5. Series LCR Circuits and Resonance

In a series LCR circuit, the components are connected in series with an AC source. The total opposition to current flow is the impedance ($Z$), which depends on resistance, inductive reactance, and capacitive reactance. Resonance occurs in an LCR circuit when the inductive reactance equals the capacitive reactance ($X_L = X_C$). At resonance, the impedance is minimal, and the current is maximum. The resonant frequency is given by $\omega_0 = \frac{1}{\sqrt{LC}}$. This principle is used in tuning radios and other electronic circuits.

6. Power in AC Circuits and LC Oscillations

The power consumed in an AC circuit is related to the resistance and the current. The average power dissipated in an AC circuit is given by $P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi$, where $\cos\phi$ is the power factor, and $\phi$ is the phase difference between voltage and current. In an ideal inductor or capacitor, the power factor is zero, meaning they do not dissipate power. LC oscillations involve the exchange of energy between the inductor's magnetic field and the capacitor's electric field, analogous to energy exchange in a simple harmonic oscillator.

7. Transformers

A transformer is a device that transfers electrical energy from one AC circuit to another, typically changing the voltage and current levels. It works on the principle of mutual induction. In an ideal transformer, the ratio of voltages is equal to the ratio of the number of turns in the coils: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$. For power to be conserved, $\frac{I_s}{I_p} = \frac{N_p}{N_s}$. Transformers are essential for stepping up voltage for long-distance power transmission (reducing current and thus resistive losses) and stepping down voltage for safe use in homes.

8. Additional: Energy Stored in Inductors

An inductor stores energy in its magnetic field when current flows through it. The energy stored in an inductor ($U_L$) is given by $U_L = \frac{1}{2}LI^2$, where $L$ is the inductance and $I$ is the current. This stored energy can be released back into the circuit when the current changes. This principle is utilized in various electronic circuits, including power supplies and oscillators, where energy is stored and released cyclically.