Magnetic Flux (Φ = BA cos θ)

Flux is the amount of magnetic field passing through a surface — in Faraday's own picture, the number of field lines that thread the loop. Two things control it. The first is how much surface there is and how strong the field is, which gives the product $BA$. The second is orientation, and that is where the cosine comes in: only the component of the field perpendicular to the surface gets through. Hold the loop face-on to the field and it catches everything; tilt it edge-on and the lines slide past without crossing it at all. Written out, $\Phi = BA\cos\theta$, with the flux in webers when $B$ is in tesla and $A$ in square metres.

A 0.4 T field through a rectangular loop 150 mm by 200 mm — an area of 0.030 m² — held face-on threads $0.4 \times 0.030 = 12\ \text{mWb}$. Tilt that loop by 60° and it drops to $12 \cos 60° = 6\ \text{mWb}$. Spin it steadily and the flux traces out a cosine in time, which is precisely how a generator makes a sine-wave voltage: not by varying $B$ and not by varying $A$, but by rotating $\theta$ through 360° every revolution.

Flux exists as a named quantity for one reason, which is that its rate of change is what induces voltage. Faraday's law, on the next page of this shard, is $\varepsilon = N\,\Delta\Phi/\Delta t$, and every generator, transformer, induction motor, metal detector and transformer-coupled sensor on Earth is a machine built to make $BA\cos\theta$ change. It is also the quantity in Gauss's law for magnetism: the net flux through any closed surface is exactly zero, because magnetic field lines have no beginning and no end, which is a compact way of saying nobody has ever found a magnetic monopole.

The angle is where nearly everyone goes wrong, and the cost is a swapped sine and cosine. $\theta$ is measured from the normal to the loop — the line sticking out perpendicular to its face — and not from the plane of the loop itself. A loop lying flat in a vertical field has $\theta = 0$ and maximum flux, even though the field lies at 90° to the plane. If a textbook or a diagram gives you the angle to the plane, you must take its complement before entering it here. Second, keep flux and flux density apart: $B$ is a density in tesla, which is webers per square metre, while $\Phi$ is the total in webers — they are different quantities with different units and the words are used loosely almost everywhere. Third, flux by itself induces nothing. A loop sitting motionless in the strongest field you can buy has enormous flux through it and not one volt across it; only the change matters. And for a coil of $N$ turns, what the induction law uses is the flux linkage $N\Phi$, not the $\Phi$ this page returns for a single loop.