Chapter 6: Electromagnetic Induction | Summary

Magnetic flux is the measure of the total magnetic field passing through a given surface. For a plane of area A placed in a uniform magnetic field B, the magnetic flux is defined as ΦB = B·A = BA cosθ, where θ is the angle between the magnetic field vector B and the area vector A. The SI unit of magnetic flux is the weber (Wb), which is equivalent to tesla metre squared (T m²). Magnetic flux is a scalar quantity.

For non-uniform magnetic fields or curved surfaces, the flux is obtained by summing contributions from each area element: ΦB = ΣBi·dAi. The flux is maximum when the field is perpendicular to the surface (θ = 0°) and zero when the field is parallel to the surface (θ = 90°). Understanding flux is essential for applying Faraday's law, as it is the rate of change of flux that determines the induced emf.

Faraday's law states that the magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit: ε = −dΦB/dt. The negative sign indicates the direction of the induced emf, which opposes the change in flux. For a closely wound coil of N turns, the total induced emf is ε = −N(dΦB/dt), since each turn contributes equally to the induced emf.

The flux through a coil can be changed by varying the magnetic field strength B, the area A of the coil, or the angle θ between B and A. All three methods induce an emf in the coil. For example, moving a magnet toward a coil changes B; shrinking or stretching a coil changes A; rotating a coil in a magnetic field changes θ. In each case, the induced emf is determined by the rate at which the flux changes.

Lenz's law, deduced by German physicist Heinrich Lenz in 1834, states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it. This is represented by the negative sign in Faraday's law. When the north pole of a magnet approaches a coil, the induced current flows in such a direction that the coil's face near the magnet acts as a north pole, repelling the incoming magnet. When the magnet is withdrawn, the induced current reverses to attract it.

Lenz's law is a consequence of the law of conservation of energy. If the induced current aided the change in flux rather than opposing it, the magnet would accelerate indefinitely without any energy input, creating a perpetual-motion machine — which is impossible. In reality, work must be done against the repulsive force to move the magnet. This mechanical energy is converted into electrical energy, which is then dissipated as Joule heat in the coil's resistance.

When a conductor of length l moves with velocity v perpendicular to a uniform magnetic field B, an emf is induced across its ends given by ε = Blv. This is called the motional emf. It arises because the Lorentz force (qvB) acts on the free charge carriers in the moving conductor, pushing them toward one end and creating a potential difference. The work done in separating the charges, W = qvBl, divided by the charge q, gives the emf ε = Blv.

Motional emf can also be understood through Faraday's law: as the conductor moves, the area of the closed circuit changes, altering the flux, which induces an emf. For a stationary conductor in a time-varying magnetic field, the force on charges is purely electric (F = qE), implying that a time-varying magnetic field generates an electric field. This demonstrates the deep connection between electricity and magnetism, and hints at the relevance of the principle of relativity.

Inductance is the property of a coil by which it opposes any change in the current flowing through it. It arises because a changing current produces a changing magnetic flux, which induces an emf opposing the change. Inductance is defined as the ratio of flux linkage to current: L = NΦB/I. It is a scalar quantity with SI unit henry (H), named after Joseph Henry. Like capacitance, inductance depends only on the geometry of the coil and the permeability of the medium, not on the current.

There are two types of inductance. Mutual inductance (M) refers to the emf induced in one coil due to a changing current in a nearby coil: ε1 = −M(dI2/dt). For two coaxial solenoids, M = μ0n1n2πr1²l. It can be shown that M12 = M21 = M, a general equality useful in complex geometries. Self-inductance (L) refers to the emf induced in a coil due to its own changing current: ε = −L(dI/dt). For a long solenoid, L = μ0n²Al. With a core of relative permeability μr, L = μrμ0n²Al. The energy stored in an inductor is W = (1/2)LI², analogous to kinetic energy (1/2)mv² in mechanics.

An AC generator converts mechanical energy into electrical energy using electromagnetic induction. It consists of a coil (armature) mounted on a rotor shaft and mechanically rotated in a uniform magnetic field. The ends of the coil are connected to an external circuit through slip rings and brushes. As the coil rotates with constant angular speed ω, the flux changes as ΦB = BA cos(ωt), and from Faraday's law, the instantaneous induced emf is ε = NBAω sin(ωt) = ε0 sin(ωt), where ε0 = NBAω is the peak emf.

The emf alternates between +ε0 and −ε0 periodically, giving an alternating current (AC). The emf is maximum when the plane of the coil is parallel to the magnetic field (θ = 90° or 270°) and zero when the coil is perpendicular to the field. Commercial generators in India rotate at 50 Hz. They may be hydro-electric, thermal, or nuclear, depending on the energy source used to rotate the armature. Modern generators can produce up to 500 MW of power.

Electromagnetic induction, discovered by Faraday and Henry around 1830, is governed by Faraday's law (ε = −NdΦB/dt) and Lenz's law. Key phenomena include motional emf (ε = Blv), self-inductance (ε = −LdI/dt), mutual inductance (ε1 = −MdI2/dt), and the energy stored in an inductor (W = LI²/2). The AC generator, which converts mechanical energy to electrical energy, is the most significant technological application of these principles. Together, these concepts establish that electricity and magnetism are deeply inter-related.

For CBSE board examinations, this chapter is of high importance. Students must be able to state and apply Faraday's law and Lenz's law, derive the expression for motional emf using both Faraday's law and the Lorentz force approach, calculate self-inductance and mutual inductance for solenoids, determine the energy stored in an inductor, and derive the expression for instantaneous emf in an AC generator. Numerical problems involving induced emf, current, flux change, and inductance calculations are regularly asked in board examinations.