Unit 3: Electromagnetic Induction & Maxwell’s Equations (PHYDSM252T)

1. Faraday’s Laws of Electromagnetic Induction

Electromagnetic induction is the process where a conductor placed in a changing magnetic field (or a conductor moving through a stationary magnetic field) causes the production of a voltage across the conductor.

The Two Laws:

• First Law: Whenever the magnetic flux linked with a circuit changes, an induced electromotive force (e.m.f.) is produced in the circuit.
• Second Law: The magnitude of the induced e.m.f. is directly proportional to the rate of change of magnetic flux linked with the circuit.

2. Lenz’s Law and Conservation of Energy

Lenz’s Law states that the direction of the induced current is such that it opposes the change that produced it. The negative sign in Faraday's law represents this opposition.

Lenz’s Law is a consequence of the Law of Conservation of Energy. If the induced current did not oppose the change, it would create a runaway energy gain, violating thermodynamic principles.

3. Self and Mutual Inductance

Self-Inductance (L): The property of a coil by which it opposes any change in the current flowing through it by inducing an e.m.f. in itself.

Mutual Inductance (M): The property of two coils kept near each other such that a change in current in one coil (primary) induces an e.m.f. in the neighboring coil (secondary).

4. Growth and Decay of Current (LR and CR Circuits)

In circuits containing inductors (L) or capacitors (C) along with resistors (R), the current does not reach its maximum value or drop to zero instantaneously.

LR Circuit:

• Growth: i = I0 * [1 - exp(-Rt/L)]
• Decay: i = I0 * exp(-Rt/L)
• Time Constant (τ): L / R. It is the time taken for the current to grow to 63.2% of its maximum value.

5. Maxwell’s Equations and Displacement Current

James Clerk Maxwell unified electricity and magnetism into a single theory. He introduced the concept of Displacement Current to explain how magnetic fields are produced even in a vacuum (like between capacitor plates) where no conduction current flows.

Maxwell’s Equations (Differential Form):

1. Gauss's Law for Electricity: div E = ρ / ε₀
2. Gauss's Law for Magnetism: div B = 0
3. Faraday’s Law: curl E = - dB / dt
4. Ampere-Maxwell Law: curl B = μ₀(J + ε₀ * dE/dt)

6. Poynting Vector and Energy Flow

The Poynting Vector (S) represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field.

The unit of the Poynting vector is Watts per square meter (W/m²). It shows that electromagnetic waves carry energy in the direction of wave propagation.