Describe the magnetic flux through an arbitrary area or geometric shape.

For a magnetic field \(\vec{B}\) that is constant across an area \(A\), the magnetic flux through the area is defined as \[\phi = B \cdot A\]

The area vector is defined as perpendicular to the plane of the surface and outward from a closed surface.

The sign of flux is given by the dot product of the magnetic field vector and the area vector.

The total magnetic flux passing through a surface is defined by the surface integral of the magnetic field over the surface area. \[\phi = \int B \cdot dA\]