Magnetic field flux

To formulate Faraday’s law we first need the concept of flux of the magnetic field through a surface. Intuitively, if you draw the lines of $\vv{B}$, the flux is proportional to the number of lines crossing the surface: many lines piercing it head-on give a large flux, lines sliding parallel to the surface give zero flux.

Principle — Magnetic flux

The flux of $\vv{B}$ through a flat surface $S$ is: $\ev{\Phi_B = \vv{B}\cdot\vv{S} = |\vv{B}|\,|\vv{S}|\,\cos\theta}$ where $\theta$ is the angle between $\vv{B}$ and the normal to the surface.

If $\vv{B}$ is not uniform, the flux is the sum (integral) of the contributions $\vv{B}\cdot d\vv{S}$ over the small surface elements. The unit of measurement is the weber: $1\;\text{Wb} = 1\;\text{T}\cdot\text{m}^2$.

The factor $\cos\theta$ tells the geometric story of the flux: it depends on how the surface is oriented relative to the field.

Key formula

$\Phi_B = B\,A\,\cos\theta$ Maximum when $\vv{B}\parallel\uv{n}$ (surface perpendicular to the field, $\theta = 0$); zero when $\vv{B}\perp\uv{n}$ (surface parallel to the field, $\theta = 90^\circ$).

It is the variation of this flux over time — due to a change in $B$, in the area, or in the orientation — that produces the induced electromotive force.

Links

Topics: Electromagnetic induction Concepts: Magnetic flux Skills: Flux balance

Related exercises: Problem — Ranking the EMF of four square loops · EMF from flux B = 10t · Concentric rings with variable flux