Magnetic Flux: MDCAT Physics notes
Magnetic Flux MDCAT notes: Φ = B·A = BA cosθ, the weber, maximum and zero flux orientations, flux as a scalar, and flux density versus flux explained.
Definition
Magnetic flux through a surface is a measure of the number of magnetic field lines passing through it. It is the scalar (dot) product of the flux density and the vector area:
$$\Phi = \vec{B}\cdot\vec{A} = BA\cos\theta$$
$\theta$ is the angle between $\vec{B}$ and the vector area $\vec{A}$. The vector area points along the normal (perpendicular) to the surface.
- Flux is a scalar quantity, although both $\vec{B}$ and $\vec{A}$ are vectors.
- SI unit: weber (Wb). 1 Wb = 1 T m$^2$ = 1 N m A$^{-1}$.
Effect of orientation
| Angle between $\vec{B}$ and $\vec{A}$ | Surface relative to field | Flux |
|---|---|---|
| $0^\circ$ | Surface perpendicular to field lines | Maximum, $\Phi = BA$ |
| $60^\circ$ | Tilted | $\Phi = BA/2$ |
| $90^\circ$ | Field parallel to surface | Zero |
| $180^\circ$ | Surface perpendicular, reversed | $-BA$ (maximum magnitude) |
Take care with wording. "Angle between the field and the area vector" uses $\cos\theta$ directly. "Angle between the field and the plane of the surface" is the complement: if the plane makes $\alpha$ with $B$, then $\Phi = BA\sin\alpha$. A surface lying along the field has no lines crossing it.
Shape does not matter
For a uniform field, only the area and its orientation enter the formula. A circle, square or rectangle of the same area, held the same way in the same field, has the same flux.
Flux density from flux
Rearranging with the surface perpendicular to the field:
$$B = \frac{\Phi}{A_\perp}$$
So flux density is flux per unit area (number of lines per unit area), measured in Wb m$^{-2}$ = tesla.
Worked examples
1. A 0.4 T field passes through a 3 m$^2$ area whose normal makes $60^\circ$ with the field. $\Phi = 0.4\times3\times\cos60^\circ = 0.6$ Wb.
2. Which gives the same flux as a 2 m$^2$ loop in a 0.3 T field (normal along field, 0.6 Wb)? A 1 m$^2$ loop in 0.6 T at $0^\circ$ also gives 0.6 Wb. A loop at $90^\circ$ always gives zero, whatever $B$.
3. A coil of $N$ turns has flux linkage $N\Phi$. With 50 turns and 0.02 Wb per turn, the linkage is 1 Wb-turn.
Why flux matters
A change in magnetic flux through a circuit induces an emf (Faraday's law). The flux can be changed by changing $B$, the area, or the angle $\theta$, for example by rotating a coil in a generator.
Key formulas
- $\Phi = BA\cos\theta$
- $B = \Phi/A$ (area perpendicular)
- Flux linkage $= N\Phi$
- 1 Wb = 1 T m$^2$
Common MDCAT traps
- Tesla is the unit of flux density; weber is the unit of flux.
- Maximum flux is at $0^\circ$ between $B$ and the area vector, not $90^\circ$.
- Flux is a scalar, even though it is built from two vectors.
- Shape of the surface is irrelevant; only area and orientation matter.
- Check whether the angle given is to the plane or to the normal.
Quick revision
- $\Phi = \vec{B}\cdot\vec{A}$ is magnetic flux.
- Field parallel to a surface gives zero flux.
- Flux density = field lines per unit area.
- Weber is the SI unit of magnetic flux.