Magnetic flux, Faraday's law and Lenz's law

Magnetic flux

• Magnetic flux measures how much magnetic field passes through a given area.

$$\Phi = BA$$

• $\Phi$ — magnetic flux (webers, Wb)

• $B$ — magnetic flux density, or field strength (tesla, T)

• $A$ — area perpendicular to the field (m²)

• If the field is at an angle to the surface, only the perpendicular component counts, so $\Phi = BA\cos\theta$ where $\theta$ is measured from the normal to the surface.

• Flux linkage for a coil of $N$ turns is $N\Phi$ — each turn links the same flux, so the effect is multiplied.

Faraday's law

• An emf is induced whenever the magnetic flux linking a circuit changes.

$$\varepsilon = -N\frac{\Delta \Phi}{\Delta t}$$

• $\varepsilon$ — induced emf (V)

• $N$ — number of turns

• $\dfrac{\Delta\Phi}{\Delta t}$ — the rate of change of flux (Wb s⁻¹)

• The essential point: it is the rate of change of flux that matters, not the flux itself. A coil sitting in a huge steady field has no induced emf.

• The flux can be changed by:

  • moving the magnet or the coil,
  • changing the field strength $B$,
  • changing the area $A$ of the circuit,
  • rotating the coil so the angle changes.

Lenz's law and the minus sign

• Lenz's law: the induced emf (and any current it drives) acts in the direction that opposes the change producing it.

• This is what the minus sign in Faraday's law expresses.

• Applying it to a magnet approaching a coil:

  • the flux through the coil is increasing,
  • the induced current flows so as to oppose that increase, creating a magnetic field that repels the approaching magnet,
  • so the magnet is pushed away, and work must be done to keep moving it closer.

• Pulling the magnet away reverses everything: the induced current now attracts the magnet, opposing its removal.

• Lenz's law is required by conservation of energy. If the induced current attracted an approaching magnet instead, the magnet would accelerate on its own while simultaneously generating electrical energy — creating energy from nothing. The opposition is what forces you to do work to generate electricity.

Move a magnet through a coil below and watch the induced emf reverse as the motion reverses:

Using Lenz's law in practice

• Identify whether the flux through the circuit is increasing or decreasing.

• Determine the direction of the induced field needed to oppose that change:

  • flux increasing → induced field opposes the original field,
  • flux decreasing → induced field reinforces the original field.

• Apply the right-hand grip rule to find the current direction that produces that induced field.

• Eddy currents are a direct application: a solid conductor moving in a changing field has circulating currents induced in it, which oppose the motion. This is the basis of electromagnetic braking in trains and gym equipment, and it is why transformer cores are laminated — to break up the eddy current paths and reduce wasted heat.