Faraday's Law of Induction

The Core Idea: Changing Flux Induces EMF

Faraday’s experiments showed that an electromotive force (emf) is induced in a circuit only when the magnetic flux through that circuit changes with time. A steady magnetic flux produces no induced emf. The key physical quantity is the time rate of change of magnetic flux.

Understanding the Experiments

• Moving a magnet towards or away from a coil changes the magnetic field ($B$) at the coil, thus changing the flux.
• Moving a current-carrying coil towards or away from another coil changes the magnetic field linking the second coil.
• Switching a current on/off in a nearby coil causes the magnetic field to rise from zero (or fall to zero), creating a change in flux.

In all cases, the induced emf exists only while the flux is changing. Once the flux becomes constant (e.g., key held pressed, magnet stationary), the induced emf drops to zero.

Faraday’s Law (Quantitative Statement)

The magnitude of the induced emf ($\varepsilon$) in a circuit is equal to the rate of change of magnetic flux through the circuit.

For a single loop:

Where:

• $\varepsilon$ = induced emf (in volts, V)
• $\Phi_B$ = magnetic flux through the loop (in weber, Wb)
• $\frac{d\Phi_B}{dt}$ = time derivative of flux (Wb/s)
• The negative sign indicates the direction of the induced emf (Lenz’s law, discussed in the next section).

For a coil of $N$ turns (closely wound, so flux through each turn is the same):

This shows that the induced emf can be increased by using more turns in the coil.

How to Change Magnetic Flux

From the definition of magnetic flux for a uniform field:

Where:

• $B$ = magnitude of magnetic field (T)
• $A$ = area of the loop (m²) …