I know flux is a scalar (slightly confused how this can be negative, as scalars don't take direction into account) but if flux density is flux*Area, how can multiplying be another scaler yield a vector?
[Physics] How to magnetic flux be a scalar but magnetic flux density is a vector
magnetic fieldsvectors
Best Answer
The area of a plane surface $A$ can by convention be considered to be a vector whose magnitude is $A$ and whose direction $\hat n$ is the direction of the perpendicular to the area. $\vec A= A\,\hat n$.
If the area is a curved surface then one must consider small elements $dA$ which can be considered planar so $d\vec A =dA\,\hat n$.
The magnetic flux $\Phi$ is then defined as the product of the component of the magnetic flux density $\vec B$ which is perpendicular to surface $B_\perp$ and the area.
$\Phi = B_\perp \,dA = B \cos \phi\, dA=\vec B\cdot d\vec A$.
The direction of the magnetic flux is defined by the choice of the direction of the normals to the surface.
In the diagram the magnetic flux is positive.
You can think of this as the magnetic field flowing through the surface in the arbitrarily chosen direction of the normals.
If the magnetic field was reversed in direction or the normals to the surface were in the opposite direction then the magnetic flux would be negative.