Last updated: 2026-07-01

Magnetic Flux Through a Surface

Magnetic flux measures how much of a magnetic field passes through a given surface. In a uniform field through a flat surface, the flux is:

$\Phi = B A \cos\theta$

where $B$ is the magnetic flux density in tesla (T), $A$ is the area in square metres, and $\theta$ is the angle between the field direction and the surface normal — the line drawn perpendicular to the surface. The result, $\Phi$, is measured in webers (Wb), where $1\ \text{Wb} = 1\ \text{T·m}^2$.

Flux is a scalar quantity. It can be pictured as the number of field lines threading the surface: the denser the field and the larger the area, the more lines pass through and the greater the flux.

Why the Angle Matters

Only the component of the field perpendicular to the surface contributes to the flux. When the surface is tilted, the perpendicular component shrinks by a factor of $\cos\theta$, where $\theta$ is measured from the normal.

At $0°$ the field is perpendicular to the surface and every field line passes straight through. At $90°$ the field skims along the surface and none of it threads through, so the flux vanishes.

Formula

Worked Example

A flat coil with an area of $0.01\ \text{m}^2$ sits in a uniform field of $0.5\ \text{T}$, with the field perpendicular to the coil ($\theta = 0°$). Find the magnetic flux.

If the same coil is tilted so the field makes a $60°$ angle with the normal, $\cos 60° = 0.5$ and the flux halves to $2.5\ \text{mWb}$. Entering these values in the calculator reproduces the same results.

Lead-in to Faraday's Law

Magnetic flux is the central quantity in electromagnetic induction. Faraday's law states that a changing flux through a loop induces an electromotive force (EMF) equal to the negative rate of change of flux: $\mathcal{E} = -N\,\dfrac{d\Phi}{dt}$, where $N$ is the number of turns in the coil.

This is why the flux formula matters in practice. A coil can experience a changing flux in three ways: the field strength $B$ can change, the area $A$ can change, or the angle $\theta$ can change as the coil rotates. Each is the basis of a real device — transformers vary $B$, and electric generators rotate a coil so that $\cos\theta$ sweeps continuously, producing an alternating EMF.

Beyond Uniform Fields

The formula $\Phi = B A \cos\theta$ assumes the field is uniform across the surface and the surface is flat. When the field varies from point to point, the flux is found by integrating the perpendicular field component over the surface. For a coil with $N$ turns, the total flux linkage is $N$ times the flux through a single turn, which is the quantity that appears in Faraday's law and in the definition of inductance.