Magnetic Flux

Defining magnetic flux. Just as the flow of water through a net can be measured by how much water crosses it per second, the 'amount' of a magnetic field $\vec{B}$ passing through a given surface is measured by a quantity called the magnetic flux, $\Phi_B$. For a flat surface of area $A$ placed in a UNIFORM field $\vec{B}$, the flux is defined as

where $\vec{A}$ is the area vector -- a vector of magnitude $A$ drawn perpendicular (normal) to the surface -- and $\theta$ is the angle between $\vec{B}$ and this normal. Flux is a SCALAR quantity (the dot product of two vectors is always a scalar), with SI unit the weber (Wb), where $1\ \text{Wb} = 1\ \text{T}\,\text{m}^2$.

Two extreme cases worth remembering. When the plane of the surface is PERPENDICULAR to $\vec{B}$, the normal is parallel to $\vec{B}$, so $\theta=0^\circ$ and the flux is MAXIMUM: $\Phi_B=BA$. When the plane of the surface is PARALLEL to $\vec{B}$ instead, the normal is perpendicular to $\vec{B}$, so $\theta=90^\circ$ and the flux is exactly ZERO -- no field lines cross the surface at all in this orientation, however strong the field is.

Flux linkage for a coil of many turns. A real coil almost always has $N$ turns of wire rather than just one loop, and each turn independently links the same flux $\Phi_B$ (assuming all turns lie in essentially the same plane and area). The TOTAL flux linkage of the coil is then $N\Phi_B$, and it is this total flux linkage, not the flux through a single turn alone, that determines the emf induced across the coil's terminals (Section 6.3). …