= Magnetic flux
{title2=$\Phi_B$}
{wiki}
<Magnetic flux> measures the signed amount of <magnetic field> crossing an <oriented surface> $S$. Choose a <unit normal> $\widehat{\mathbf n}$ and use the <oriented surface element> $d\mathbf S=\widehat{\mathbf n}\,dA$. The defining <surface integral> is
$$
\Phi_B=\int_S\mathbf B\cdot d\mathbf S
=\int_S\mathbf B\cdot\widehat{\mathbf n}\,dA.
$$
For a uniform <magnetic field> across a flat <surface> of <area> $A$, the <dot product> reduces this to
$$
\Phi_B=BA\cos\theta,
$$
where $\theta$ is the <angle> between the <magnetic field> and the chosen <unit normal>, rather than the <plane> of the <surface>. The <SI> unit is the <weber (unit)>, $1\,\mathrm{Wb}=1\,\mathrm{T\,m^2}$. https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law[OpenStax, magnetic flux and its orientation]{ref}.
For a stipulated <magnetic field> $B=2\,\mathrm T$ and <area> $A=3\,\mathrm{m^2}$, the <magnetic flux> is $6\,\mathrm{Wb}$ at $\theta=0$, $3\,\mathrm{Wb}$ at $\theta=\pi/3$, and zero at $\theta=\pi/2$. In the last case the <magnetic field> lies in the <plane> of the <surface>, so it does not cross it. Reversing the chosen <unit normal> reverses the sign of <magnetic flux> without changing the physical <magnetic field>; at $\theta=\pi$ the same example gives $-6\,\mathrm{Wb}$.
\Image[/magnetic-flux-orientation.png]
{title=<Magnetic flux> as a flat <surface> rotates in a fixed <magnetic field>}
{description=Left: an edge-on <surface> and its <unit normal> at three <angles> to the fixed <magnetic field>. Right: the signed <magnetic flux> $\Phi_B=6\cos\theta\,\mathrm{Wb}$, including reversal of the <unit normal>. The stipulated <magnetic field> is $2\,\mathrm T$ and the <area> is $3\,\mathrm{m^2}$.}
The geometric idea is projected <area>: tilting a flat <surface> reduces the <area> seen perpendicular to the <magnetic field> by the factor $\cos\theta$. This explains how rotation alone can change <magnetic flux> even when the <magnetic field> is constant; <Faraday's law of induction> turns such changes into an <electromotive force>. https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law[OpenStax, induction from changing magnetic flux]{ref}.
Back to article page
