Electromagnetic Induction

SYLLABUS

Faraday’s laws; Induced electric fields, Lenz’s law, Motional electromotive force, A.C. generators; Eddy currents, Self-inductance and mutual inductance, Energy stored in an inductor, Transformer.

What is This Chapter About?

In 1831, Michael Faraday made one of the most consequential discoveries in the history of science. He found that a changing magnetic field induces an electric current in a nearby conductor — even without any physical connection between them. This phenomenon — electromagnetic induction — is the principle behind every electrical generator, every transformer, every electric motor running in reverse, and every wireless charger. It is the discovery that made the modern electrical world possible.

Chapter 18 of NEB Grade 12 Physics explores electromagnetic induction in depth, carrying 6 teaching hours. It covers Faraday’s and Lenz’s laws, AC generators, eddy currents, self and mutual inductance, energy stored in inductors, and transformers — a comprehensive treatment of one of the most practically important chapters in the entire Physics course.

Key Concepts You Will Learn

Faraday’s Laws of Electromagnetic Induction

Faraday’s First Law states that whenever the magnetic flux through a circuit changes, an EMF is induced in the circuit. The EMF exists only while the flux is changing — a constant magnetic flux produces no EMF.

Faraday’s Second Law states that the magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux:

[latex]\mathcal{E} = -\frac{d\Phi}{dt}[/latex]

For a coil of N turns: [latex]\mathcal{E} = -N\frac{d\Phi}{dt}[/latex].

The negative sign is crucial — it leads directly to Lenz’s Law.

Lenz’s Law

Lenz’s Law states that the induced current always flows in a direction such that it opposes the change in magnetic flux that caused it. If the flux through a coil is increasing, the induced current creates a magnetic field opposing the increase. If the flux is decreasing, the induced current creates a field trying to maintain the flux.

Lenz’s Law is a consequence of conservation of energy — if the induced current aided the change in flux rather than opposing it, it would amplify itself indefinitely, creating energy from nothing. The opposition ensures that work must be done to change the flux, and this work appears as electrical energy in the induced current.

Motional EMF

When a conductor of length L moves with velocity v perpendicular to a magnetic field B, the free charges in the conductor experience a Lorentz force that separates positive and negative charges, creating a potential difference — a motional EMF:

[latex]\mathcal{E} = BLv[/latex]

This is the most direct way to understand electromagnetic induction microscopically — the magnetic force on moving charges creates the EMF. The macroscopic Faraday’s Law and the microscopic motional EMF are fully equivalent descriptions.

AC Generators

An AC generator (alternator) converts mechanical energy into electrical energy using electromagnetic induction. A rectangular coil of N turns and area A rotates with angular velocity [latex]\omega[/latex] in a uniform magnetic field B. As it rotates, the magnetic flux through it changes sinusoidally:

[latex]\Phi = NBA\cos(\omega t)[/latex]

By Faraday’s Law, the induced EMF is:

[latex]\mathcal{E} = NBA\omega\sin(\omega t) = \mathcal{E}_0\sin(\omega t)[/latex]

where [latex]\mathcal{E}_0 = NBA\omega[/latex] is the peak EMF. This sinusoidal variation of EMF with time is the origin of alternating current — the standard form of electrical power delivered to homes and businesses worldwide. The frequency of the AC supply (50 Hz in Nepal) corresponds to the rotational speed of generators in power plants.

Eddy Currents

When a solid conducting material is placed in a changing magnetic field, induced EMFs drive currents in closed loops within the material itself — these are eddy currents. By Lenz’s Law, eddy currents oppose the change in flux and therefore oppose motion through the field, creating a braking effect. Eddy currents also dissipate energy as heat.

Eddy currents are both useful and problematic. Useful applications include electromagnetic braking (smooth, contactless braking in trains and roller coasters), induction heating (cooking on induction hobs, melting metals), and eddy current testing (detecting cracks in metal structures). Problematic effects include energy losses in transformer cores and motors — minimised by laminating the core with thin insulated sheets of metal, which break up the eddy current loops and drastically reduce their magnitude.

Self-Inductance

When the current in a coil changes, the magnetic flux through the coil itself changes, inducing an EMF in the coil that opposes the change in current. This property is called self-inductance (L), defined by:

[latex]\mathcal{E} = -L\frac{dI}{dt}[/latex]

The SI unit of inductance is the henry (H). A coil with high self-inductance strongly resists changes in current — it acts as an electrical inertia. For a solenoid of N turns, length l, and cross-sectional area A:

[latex]L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 V[/latex]

where n = N/l is the turn density and V is the volume.

Mutual Inductance

When two coils are placed near each other, a changing current in one (the primary) induces an EMF in the other (the secondary) through the changing magnetic flux linking them. This is mutual inductance (M):

[latex]\mathcal{E}_2 = -M\frac{dI_1}{dt}[/latex]

The mutual inductance depends on the geometry of both coils and their relative positions and orientations. It is the fundamental principle of transformers, wireless chargers, and inductively coupled circuits. For two coaxial solenoids:

[latex]M = \frac{\mu_0 N_1 N_2 A}{l}[/latex]

Energy Stored in an Inductor

When current builds up in an inductor, work is done against the back EMF, and this work is stored as magnetic energy in the magnetic field of the inductor:

[latex]U = \frac{1}{2}LI^2[/latex]

This is the magnetic analogue of [latex]U = \frac{1}{2}CV^2[/latex] for a capacitor (which stores energy in an electric field). The energy stored per unit volume in a magnetic field is [latex]u = \frac{B^2}{2\mu_0}[/latex].

Transformers

A transformer is a device that changes the voltage of an AC supply using mutual inductance. It consists of a primary coil of [latex]N_1[/latex] turns and a secondary coil of [latex]N_2[/latex] turns wound on a common soft iron core. The changing current in the primary creates a changing magnetic flux in the core, which links the secondary coil and induces an EMF in it.

The transformer equation gives the ratio of secondary to primary voltage:

[latex]\frac{V_2}{V_1} = \frac{N_2}{N_1}[/latex]

If [latex]N_2 > N_1[/latex], the transformer steps up voltage (and steps down current proportionally). If [latex]N_2 < N_1[/latex], it steps down voltage (and steps up current). For an ideal transformer, power is conserved: [latex]V_1 I_1 = V_2 I_2[/latex].

Energy Losses in Practical Transformers

Real transformers are not 100% efficient. Energy is lost through: copper losses ([latex]I^2R[/latex] heating in the coil windings), iron losses (eddy current losses and hysteresis losses in the core — minimised by lamination and soft magnetic materials respectively), flux leakage (not all flux from the primary links the secondary), and magnetostriction (mechanical vibration of the core produces sound and wastes energy). High-quality transformers achieve efficiencies of 95–99%, making them among the most efficient energy conversion devices in existence.

Why This Chapter Matters for NEB Exams

Electromagnetic Induction is one of the most thoroughly examined chapters in NEB Grade 12 Physics. Faraday’s Laws and Lenz’s Law — stating them and explaining their physical basis — appear in every board paper. The AC generator — derivation of the EMF equation [latex]\mathcal{E} = \mathcal{E}_0\sin(\omega t)[/latex] and explanation of its working — is a classic long-answer question. Transformer equation derivation, energy losses in transformers, self-inductance of a solenoid, and energy stored in an inductor are all standard examination topics. Eddy currents — explaining their origin, applications, and how to minimise unwanted effects — are popular short and medium-answer questions.

Important Topics to Focus On

• Faraday’s First and Second Laws: [latex]\mathcal{E} = -N\frac{d\Phi}{dt}[/latex]
• Lenz’s Law — statement, physical basis, and conservation of energy argument
• Motional EMF: [latex]\mathcal{E} = BLv[/latex] — derivation and applications
• AC generator — construction, working, and derivation of [latex]\mathcal{E} = \mathcal{E}_0\sin(\omega t)[/latex]
• Eddy currents — origin, Lenz’s Law explanation, useful applications, minimisation by lamination
• Self-inductance — definition, formula for solenoid, and unit (henry)
• Mutual inductance — definition and formula
• Energy stored in inductor: [latex]U = \frac{1}{2}LI^2[/latex]
• Transformer — working principle, turns ratio equation, ideal transformer
• Energy losses in transformers — copper losses, eddy current losses, hysteresis losses, flux leakage
• Numerical problems on induced EMF, self-inductance, mutual inductance, transformer ratio, and energy storage

Connection to Other Chapters

Electromagnetic Induction builds directly on Chapter 16 (Magnetic Field) — magnetic flux, Lorentz force, and field of solenoids are all prerequisites. It connects forward to Chapter 19 (Alternating Currents) — the AC generator produces the sinusoidal EMF studied in that chapter, and inductors (characterised by self-inductance L) are key components of AC circuits. Chapter 17 (Magnetic Properties of Materials) connects through transformer core materials — soft magnetic materials minimise hysteresis losses. The energy stored in an inductor [latex]\frac{1}{2}LI^2[/latex] is directly analogous to energy stored in a capacitor [latex]\frac{1}{2}CV^2[/latex] from Grade 11 Chapter 22 (Capacitor) — this parallel reveals a deep symmetry between electric and magnetic energy storage. The transformer’s role in long-distance power transmission connects to Grade 11 Chapter 23 (DC Circuits) through Joule’s Law — high-voltage transmission reduces [latex]I^2R[/latex] losses.