Magnetic flux

The magnetic flux $\Phi$ through a surface $S$ is the flux integral of the magnetic flux density $\mathbf{B}$ over $S$:

$\Phi ={\int }_S\mathbf{B}\cdot d\mathbf{s}\text{(Wb, weber)}.$

One weber equals one tesla-square-meter (Wb = T·m²). The flux through a loop is “how many field lines pierce it,” weighted by the angle between the field and the surface normal.

Gauss’s law for magnetism

For any closed surface:

${\oint }_S\mathbf{B}\cdot d\mathbf{s}=0.$

Magnetic field lines have no sources or sinks: every line that enters a closed region must leave. Equivalently, $\nabla \cdot \mathbf{B}=0$, no magnetic monopoles.

So the flux through any open surface depends only on the boundary loop, not on the surface itself. If two surfaces $S_1$ and $S_2$ share the same boundary curve $C$, then ${\Phi }_1={\Phi }_2$, because gluing them together forms a closed surface with zero net flux.

That’s what makes “flux through a loop” well-defined: the loop fixes the flux, regardless of which surface you spread across it.

Flux linkage

For an inductor with $N$ turns of wire, each turn encloses an area through which the same $\Phi$ flows (in a tightly-wound solenoid). The flux linkage is the total flux summed over all turns:

$\Lambda =N\Phi \text{(Wb-turns or just Wb)}.$

This is the quantity that appears in Faraday’s law: the induced emf is $V_{\text{emf}}=-d\Lambda /dt=-Nd\Phi /dt$. Adding more turns to a coil multiplies the induced voltage proportionally.

Connection to inductance

The Inductance $L$ of a structure is defined as flux linkage per ampere:

$L=\frac{\Lambda }{I}\text{(H, henry)}.$

A 1 H inductor produces 1 Wb of flux linkage for every 1 A of current. Geometry-only — depends on shape, size, number of turns, permeability of the core. Doesn’t depend on $I$.

Examples

Single-turn loop of area $A$ in uniform $\mathbf{B}$ perpendicular to the loop:

$\Phi =BA.$

If the loop tilts so $\hat{\mathbf{n}}$ makes angle $\theta$ with $\mathbf{B}$:

$\Phi =BA\cos \theta .$

Solenoid of length $l$, $N$ turns, cross-section $S$, current $I$, vacuum core:

The interior field is $B={\mu }_{0}NI/l$ from Ampère’s law. Flux through one turn: $\Phi =BS={\mu }_{0}NIS/l$. Flux linkage:

$\Lambda =N\Phi =\frac{{\mu }_0{N}^{2}IS}{l}.$

Inductance:

$L=\frac{\Lambda }{I}=\frac{{\mu }_0{N}^{2}S}{l}.$

Quadratic in $N$ — doubling the turns quadruples the inductance.

Coaxial cable, inner radius $a$, outer $b$, length $l$:

The field between conductors is $B_{\varphi }=\mu I/(2\pi r)$. The flux through a rectangular strip of width $dr$ and length $l$ at radius $r$ is $d{B}_{\varphi }ldr$. Integrate from $a$ to $b$:

$\Phi ={\int }_a^b\frac{\mu I}{2\pi r}ldr=\frac{\mu Il}{2\pi }\ln \frac{b}{a}.$

Per unit length: $L^{\prime }=(\mu /2\pi )\ln (b/a)$, one of the parameters of a coaxial Transmission line.

In the time-varying case

A changing flux through a loop generates an electromotive force (Faraday’s law):

$V_{\text{emf}}=-\frac{d\Phi }{dt}.$

This is what makes generators work, couples primary and secondary coils in a transformer, and makes a moving magnet near a coil produce current. Three ways $\Phi$ can change:

• Loop is stationary, $\mathbf{B}$ varies in time (transformer emf).
• Loop moves through static $\mathbf{B}$, changing the area linked (motional emf).
• Both at once.

In each case it’s $d\Phi /dt$ that drives the induced emf.