Magnetic flux measures how much magnetic field passes through an oriented surface.

For a surface $S$,

$$\Phi_B=\int_S\mathbf B\cdot d\mathbf A.$$

For a uniform field crossing a flat area $A$ at angle $\theta$ to the surface normal,

$$\Phi_B=BA\cos\theta.$$

Orientation

The sign depends on the chosen surface normal. Field passing in the direction of the normal contributes positive flux; field passing oppositely contributes negative flux.

Changing flux

Magnetic flux can change because the field strength changes, the surface area changes, or the angle between field and surface changes.

This is important because Faraday's law responds to the change of flux, not merely to the presence of a magnetic field.

Closed surfaces

For a closed surface the outward orientation is conventional. Unlike electric flux, the net magnetic flux through every closed surface is zero, reflecting the absence of observed isolated magnetic monopoles.