Faraday’s Law of Induction

Faraday's Law of Induction is one of the foundational principles of electromagnetism, revealing that a changing magnetic environment around a conductor is enough to drive an electric current — no battery required. Discovered experimentally by Michael Faraday in 1831, the law states that whenever the magnetic flux through a conducting loop changes, an electromotive force (emf) is induced in that loop. The faster the flux changes, the greater the induced emf. This single idea underpins the operation of every electrical generator, transformer, and induction motor on the planet. To understand Faraday's Law intuitively, imagine pushing a bar magnet in and out of a coil of wire connected to a sensitive galvanometer. Each time the magnet moves, the needle deflects — electricity flows even though there is no battery. Stop the magnet, and the needle returns to zero instantly. The key insight is that it is not the presence of a strong magnetic field that matters, but its rate of change. A massive static field induces nothing; a tiny but rapidly changing field induces a measurable emf. This is analogous to how it is not the height of a hill, but the steepness of its slope, that determines how fast you accelerate down it. Faraday's Law sits alongside Gauss's Law, Ampère's Law, and Gauss's Law for Magnetism as one of Maxwell's four equations, which together form the complete theoretical description of classical electromagnetism. Mastering Faraday's Law is therefore not just about understanding one experiment — it is the gateway to understanding all of AC circuit theory, electromagnetic waves, and modern electrical technology.

Key takeaways
  • What is Magnetic Flux?
  • Three Ways to Change Flux
  • The Law Itself
  • Units and Scale

What is Magnetic Flux?

Magnetic flux (Φ_B) measures how much of a magnetic field passes through a given surface area. Think of field lines as rain falling on a window: flux is the total amount of rain that actually passes through the glass. Mathematically, for a uniform field, Φ_B = B A cos θ, where B is the magnetic field strength (in tesla), A is the area of the loop (in m²), and θ is the angle between the field direction and the normal (perpendicular) to the surface. Flux is maximised when the field is perpendicular to the loop (θ = 0°) and zero when the field is parallel to the loop (θ = 90°).

Three Ways to Change Flux

Flux can change in three distinct ways, all of which induce an emf: (1) Change B — make the magnetic field stronger or weaker over time; (2) Change A — squeeze or expand the area of the loop (like stretching a wire loop); (3) Change θ — rotate the loop relative to the field. This third method is exactly what happens inside every AC generator: a coil spins in a magnetic field, causing θ to vary sinusoidally with time, producing a sinusoidal emf output.

The Law Itself

Faraday's Law states that the induced emf (ε) in a loop equals the negative rate of change of magnetic flux through that loop: ε = −dΦ_B/dt. The negative sign is Lenz's Law in disguise — it tells us the induced emf acts in a direction that opposes the change causing it (more on Lenz's Law separately). For a coil of N turns, each turn experiences the same flux change, so the effects add: ε = −N (dΦ_B/dt). This is the full, practical form of Faraday's Law used in engineering calculations.

Units and Scale

Since emf is measured in volts (V) and time in seconds (s), flux must be measured in volt-seconds, which is the unit called the weber (Wb). We can verify: 1 Wb = 1 T·m². So a flux change of 1 Wb per second induces exactly 1 V of emf in a single-turn loop. A transformer with 500 turns of secondary winding experiencing the same flux change would produce 500 V — which is precisely the principle behind voltage step-up transformers in the electricity grid.

The Key Equation

ε=−NdΦBdt\varepsilon = -N \frac{d\Phi_B}{dt}
  • \varepsilon — Induced emf (V (volt)): The electromotive force induced in the coil; acts like a voltage source driving current around the circuit.
  • N — Number of turns (dimensionless): The number of loops in the coil; each loop contributes equally to the total induced emf.
  • \Phi_B — Magnetic flux (Wb (weber)): The total magnetic field passing through one turn of the coil; Φ_B = BA cos θ for a uniform field.
  • \frac{d\Phi_B}{dt} — Rate of change of flux (Wb/s = V): How quickly the flux is changing with time; this is the driving quantity for the induced emf.
  • - — Negative sign (Lenz's Law) (—): Indicates that the induced emf opposes the change in flux that created it, consistent with energy conservation.

Worked Example

Problem. A circular coil of 200 turns has a radius of 5.0 cm. It is placed in a uniform magnetic field directed perpendicular to the plane of the coil. The field increases uniformly from 0.10 T to 0.35 T in 0.40 s. Calculate the magnitude of the induced emf.

∣ε∣=NΔΦBΔt=200×1.96×10−30.40≈0.98 V|\varepsilon| = N\frac{\Delta\Phi_B}{\Delta t} = 200 \times \frac{1.96 \times 10^{-3}}{0.40} \approx 0.98 \text{ V}

Answer. The magnitude of the induced emf is approximately 0.98 V.

Figure
Diagram
Faraday's Law: Changing Flux Induces emfWeak field (dots)Low Φ_BStrong field (crosses)High Φ_Btime →ΔΦ_B / Δt ≠ 0ε = −N dΦ_B/dtGB (out of page)B (into page, growing)
Origins · 1 minMichael Faraday and how this was discovered
MF
Michael Faraday
1791–1867 · British

Faraday discovered electromagnetic induction in 1831, demonstrating that a changing magnetic field induces an electric current. He also invented the first electric motor and electric generator, and introduced the concept of field lines — the visual tool we still use today to represent electric and magnetic fields. His work was later given precise mathematical form by James Clerk Maxwell. Faraday had virtually no formal education and began his career as a bookbinder's apprentice. He taught himself science by reading the books he was binding, and eventually became one of history's greatest experimental physicists — proof that curiosity and persistence can outpace privilege.

  • Lenz's Law — Lenz's Law explains the negative sign in Faraday's equation, specifying that the induced emf always opposes the change in flux that caused it — a direct consequence of energy conservation.
  • Magnetic Flux — Magnetic flux (Φ_B = BA cos θ) is the central quantity in Faraday's Law; you cannot apply the law without first being able to calculate flux correctly.
  • AC Generators — An AC generator works by rotating a coil in a magnetic field, continuously changing the flux angle θ — a direct, large-scale application of Faraday's Law that produces sinusoidal emf.
  • Transformers — Transformers use Faraday's Law with multiple turns (N) to step AC voltages up or down; the turn ratio directly controls the ratio of induced emfs in the primary and secondary coils.