Magnetic Flux

Consider the figure below.


The surface is divided into elements with area dA. For each element, \(B_{\perp}\) is determined as the component of the magnetic field \(\vec B\) normal to the surface at the position of the indicated element. Based on the figure, the component of the magnetic field \(B_{\perp} = B cos \phi\), where \(\phi\) is the angle between the direction of \(\vec B\) and a line perpendicular to the surface. This component varies from point to point on the surface and is called as the magnetic flux denoted by the symbol \(\Phi_B\). The magnetic flux, \(\Phi_B\)  through this area can be expressed mathematically as

\(d\Phi_B = B_{\perp}dA = B\cos \phi dA = \vec B \cdot d\vec A\)     (1)

the sum of the contributions from the individual area elements through the surface is referred to as the total magnetic flux. It can be expressed mathematically as

\(\Phi_{B} = \int B \cos \phi dA = \int B_{\perp} dA = \int \vec B \cdot d\vec A\)     (2)

If the magnetic field, \(\vec B\) is uniform over a plane surface and given that the total area is A, the magnetic flux can be solved as

\(\Phi_B = B_{\perp}A = BA \cos \phi\)     (3)

Equation (3) implies that the component of the magnetic field \(B_{\perp}\) and the angle \(\phi\) are the same at all points in the surface.

If the magnetic field \(\vec B\) is perpendicular to the given surface, the cosine of the angle is equal to 1, that is, \(\cos \phi = 1\). This leads to another equation, simplifying equation (3) as

\(\Phi_B = BA\)     (4)

The unit for magnetic flux is weber, denoted as Wb. It is named after the German physicist Wilhelm Weber.

From equation (4), we know that the unit for the magnetic field is Tesla (T) and for the area is square meter (m2), this implies that \(\text{weber} = \text{Tesla} \cdot \text{square meter}\) or using their abbreviations, \(Wb = T \cdot m^2\).