Class 12 Physics · Chapter 6 NotesElectromagnetic Induction

Magnetic flux is a measure of the number of magnetic field lines passing through a given area. For a uniform magnetic field B through a plane surface of area A, the flux is Φ = B·A = BA cos θ, where θ is the angle between the magnetic field vector B and the area vector A. The area vector is perpendicular to the surface. If the field is non-uniform, the total flux is obtained by summing over small area elements: Φ = Σ B_i · dA_i. Magnetic flux is a scalar quantity and its SI unit is the weber (Wb) or tesla metre squared (T m²).

• Φ = BA cos θ
• SI unitweber (Wb) = T m²
• Magnetic flux is a scalar.

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. For a coil of N turns, the induced emf is e = –N (dΦ/dt). The negative sign indicates the direction of the induced emf, which is given by Lenz's law. The flux can be changed by varying the magnetic field B, the area A, or the angle θ. This law explains the experiments of Faraday and Henry: when the magnetic flux through a coil changes, an emf is induced, causing a current if the circuit is closed.

• e = –dΦ/dt for a single turn
• e = –N dΦ/dt for N turns
• Flux can change by changing B, A, or θ.

Lenz's law gives the direction of the induced emf: the polarity of the induced emf is such that it tends to produce a current that opposes the change in magnetic flux that produced it. This is represented by the negative sign in Faraday's law. Lenz's law is a consequence of the law of conservation of energy. If the induced current aided the change, it would lead to a perpetual motion machine, which is impossible. The work done in moving a magnet against the opposing force is dissipated as Joule heating in the circuit.

• Induced current opposes the change in flux.
• Negative sign in Faraday's law indicates Lenz's law.
• Lenz's law follows from energy conservation.

When a conductor of length l moves with velocity v perpendicular to a uniform magnetic field B, an emf is induced across its ends. This is called motional emf and is given by e = Blv. It can be derived from Faraday's law by considering the change in area of a closed loop, or from the Lorentz force acting on free charges in the moving conductor. The Lorentz force on a charge q is qvB, which separates charges and creates an emf. Motional emf is the basis of many practical generators.

• e = Blv
• Derived from Faraday's law or Lorentz force.
• Direction given by right-hand rule or Lenz's law.

Eddy currents are circulating currents induced in bulk pieces of metal when they are subjected to a changing magnetic flux. They can cause unwanted heating and energy loss in transformers and electric motors. To reduce eddy currents, laminated cores are used. However, eddy currents have useful applications: magnetic braking in trains, induction furnaces, and speedometers. The direction of eddy currents is again given by Lenz's law, and they oppose the change producing them.

• Eddy currents are induced in bulk conductors.
• Reduced by using laminated cores.
• Used in magnetic braking and induction heating.

Self-inductance is the property of a coil due to which a changing current in the coil induces an emf in the same coil. The flux linkage NΦ is proportional to the current I: NΦ = LI, where L is the self-inductance. The induced emf is e = –L (dI/dt). The self-induced emf opposes any change in current, acting like electrical inertia. For a long solenoid of length l, area A, and n turns per unit length, L = μ₀ n² A l. If the core has relative permeability μ_r, then L = μ_r μ₀ n² A l. The energy stored in an inductor is U = ½ L I².

• NΦ = LI
• e = –L dI/dt
• L = μ₀ n² A l for a solenoid
• Energy storedU = ½ L I²

Mutual inductance is the phenomenon in which a changing current in one coil induces an emf in a nearby coil. For two coils, the flux linkage in coil 1 due to current I₂ in coil 2 is N₁Φ₁ = M₁₂ I₂, where M₁₂ is the mutual inductance of coil 1 with respect to coil 2. The induced emf is e₁ = –M₁₂ (dI₂/dt). For long co-axial solenoids, M₁₂ = μ₀ n₁ n₂ π r₁² l. It can be shown that M₁₂ = M₂₁ = M. Mutual inductance depends on the geometry, separation, and relative orientation of the coils, and on the permeability of the medium.

• N₁Φ₁ = M₁₂ I₂
• e₁ = –M₁₂ dI₂/dt
• M₁₂ = M₂₁ = M
• M = μ₀ n₁ n₂ π r₁² l for co-axial solenoids

An ac generator converts mechanical energy into electrical energy using electromagnetic induction. It consists of a coil (armature) rotated in a uniform magnetic field. The flux through the coil changes as it rotates, inducing an emf. For a coil of N turns and area A rotating with angular speed ω, the instantaneous emf is e = NBAω sin ωt = e₀ sin ωt, where e₀ = NBAω is the maximum emf. The direction of the current reverses every half rotation, producing alternating current. In commercial generators, the coil is stationary and the electromagnets rotate. The frequency of ac in India is 50 Hz.

• e = NBAω sin ωt
• e₀ = NBAω
• AC frequency in India50 Hz
• Mechanical energy → electrical energy