Electromagnetic induction & Maxwell's equations
The idea
Faraday's law is the bridge between electricity and magnetism: a changing magnetic flux through a circuit induces an emf equal to the rate of change of that flux, emf = −N dΦ/dt for an N-turn coil. Flux Φ = BA cosθ can change because the field strengthens, the area changes, or the orientation rotates — generators exploit the last. Lenz's law fixes the sign: the induced current flows so as to oppose the change creating it, which is energy conservation enforcing itself; an aiding current would be a free-energy machine.
Maxwell completed the picture by adding the symmetric partner — a changing electric field creates a magnetic field — and the resulting four equations (Gauss's laws for E and for B, Faraday's law, and the Ampere-Maxwell law) form a closed system. Their crowning prediction: self-sustaining electromagnetic waves traveling at 1/√(μ₀ε₀), which evaluates to the measured speed of light. Light is an electromagnetic wave; optics is a chapter of electromagnetism.
Note carefully that flux alone induces nothing — only its rate of change does. A coil sitting in an enormous steady field has zero induced emf, while a tiny but rapidly changing field can drive large currents. When the flux varies nonlinearly in time, you must differentiate, not divide by elapsed time.
Worked example
A flat coil of 150 turns and area 4.0 × 10⁻³ m² lies with its plane perpendicular to a magnetic field that grows as B(t) = 0.20 t² (tesla, t in seconds). The coil's total resistance is 2.0 Ω. Find the induced emf and current at t = 3.0 s.
- Write the flux through one turn: with the field perpendicular to the coil plane, Φ(t) = B(t)A = 0.20 t² × 4.0 × 10⁻³ = 8.0 × 10⁻⁴ t² webers.
- Differentiate — the field grows nonlinearly, so dividing by time would be wrong: dΦ/dt = 1.6 × 10⁻³ t volts per turn.
- Multiply by the turn count for the total emf magnitude: emf = N dΦ/dt = 150 × 1.6 × 10⁻³ t = 0.24 t, so at t = 3.0 s the emf is 0.24 × 3.0 = 0.72 V.
- Apply Ohm's law to the coil circuit: I = emf/R = 0.72/2.0 = 0.36 A.
- Fix the direction with Lenz's law: the outward flux is increasing, so the induced current circulates to make its own magnetic field oppose the growth — viewed from the side the field points toward, the current runs clockwise. Note also that this emf grows linearly in time, so at t = 6.0 s it would be twice as large.
Answer. At t = 3.0 s the induced emf is 0.72 V, driving a 0.36 A current that opposes the growing flux.
Check your understanding
- Why does Lenz's law have to hold — what would a flux-aiding induced current allow you to build?
- What are the three independent ways to change magnetic flux through a loop, and which one do power-station generators use?
- How did adding the changing-electric-field term to Ampere's law lead to the prediction that light is an electromagnetic wave?
- Why does a constant 10 T field through a coil induce nothing while a millitesla field oscillating at high frequency can induce a large emf?
Build the foundations first
Electromagnetic induction & Maxwell's equations builds on these concepts. If any feel shaky, start there.