Faraday’s Law of Electromagnetic Induction

Faraday’s law of electromagnetic induction states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in that loop. Discovered by Michael Faraday in 1831, it’s the principle behind generators, transformers, electric motors, induction cooktops, microphones, and most of the electrical infrastructure of the modern world. Together with Ampère’s law and Maxwell’s equations, it links electricity and magnetism into a single unified theory.

The Law

The induced EMF in a circuit equals the negative rate of change of magnetic flux through the circuit:

$$ \varepsilon = -\frac{d\Phi_B}{dt} $$

For a coil with \( N \) turns, the EMFs add:

$$ \varepsilon = -N \frac{d\Phi_B}{dt} $$

The magnetic flux \( \Phi_B \) through a surface is the integral of the magnetic field \( \vec{B} \) over the surface area:

$$ \Phi_B = \int \vec{B} \cdot d\vec{A} $$

For a uniform field perpendicular to a flat area: \( \Phi_B = BA \). Units: webers (Wb = T·m²).

Lenz’s Law: The Minus Sign

The minus sign in Faraday’s law isn’t decoration — it encodes Lenz’s law: the induced current flows in the direction that opposes the change in flux. If the flux through a coil is increasing, the induced current creates a magnetic field pointing the opposite way to fight that increase. If the flux is decreasing, the induced current tries to maintain it. This is a consequence of conservation of energy: if induced currents reinforced the change, you’d get free energy.

Three Ways to Change the Flux

Flux through a loop changes if any of three things changes:

1. The magnetic field \( B \). Moving a magnet toward or away from the loop changes \( B \) at the loop’s location. This is the most common textbook setup.
2. The area \( A \). A loop expanding, contracting, or moving partly out of a magnetic field region changes the area enclosing flux.
3. The angle between \( B \) and the loop’s normal. Rotating a loop in a uniform field changes the dot product \( B \cos\theta \). This is the basis of every electrical generator.

Worked Example: A Coil in a Changing Field

A circular coil with 50 turns and radius 0.10 m sits in a magnetic field that changes from 0.20 T to 0.50 T over 0.10 s. The field is perpendicular to the coil. Find the induced EMF.

Area \( A = \pi r^2 = \pi (0.10)^2 = 0.0314 \) m². Flux change per turn: \( \Delta \Phi = (0.50 – 0.20) \cdot 0.0314 = 0.00942 \) Wb. EMF: \( \varepsilon = -N \Delta\Phi / \Delta t = -50 \cdot 0.00942 / 0.10 = -4.71 \) V. Magnitude: 4.71 V.

Applications

• Generators. A coil rotating in a magnetic field has a sinusoidally varying flux, producing an AC EMF. Every power plant in the world — coal, gas, nuclear, hydro, wind — is a turbine spinning a generator.
• Transformers. An AC current in a primary coil produces a changing flux that induces an EMF in a secondary coil. The turns ratio determines the voltage ratio. Transformers are why long-distance power transmission is feasible.
• Induction cooktops. An AC current in a coil under the cooktop induces eddy currents in a ferromagnetic pan. The pan’s resistance turns those currents into heat.
• Microphones. A diaphragm moving in response to sound vibrates a coil attached to it within a magnetic field, inducing an EMF that mirrors the sound waveform.
• RFID and wireless charging. A changing magnetic field from a transmitter induces a current in a tuned receiver coil — used for contactless cards, key fobs, and phone charging pads.

Related study notes: Electromagnetic Induction, Lenz’s Law, Maxwell’s Equations, Ohm’s Law.

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