15.1.1 Magnetic Flux

In 1831, Michael Faraday discovered that turning on or off the current in one coil induces a transient electric current in another coil. Faraday thought that the induction was caused by a changing magnetic flux. So what is magnetic flux?

Imagine a coil with cross sectional area A directly facing a uniform magnetic field B. Does it look like some kind of magnetic flow is captured by the coil? That thingy captured by the coil is called the magnetic flux f. Quantitatively, it has the formula

$\displaystyle \phi =BA$

The SI unit for f is the weber (symbol Wb). 1 Wb is 1 T m².

Notice that $\displaystyle B=\frac{\phi }{A}$ . That’s actually where the name magnetic flux density came from.

If the coil is rotated 90° so its plane is parallel to B, then the area A will not be facing the magnetic field at all. So the magnetic flux (of the coil) is now zero.

If the axis of the coil is oriented at an angle a to B (so the plane of the coil makes an angle b with B), then the coil is partially facing the magnetic field. We can think of the coil as directly facing $\displaystyle {{B}_{\bot }}$ (the component of B perpendicular to its area A), but completely dodging $\displaystyle {{B}_{\parallel }}$ (the component of B parallel to A). So the magnetic flux (of the coil) can be expressed as

$\displaystyle \begin{aligned}\phi &={{B}_{\bot }}A\\&=(B\cos \alpha )A\text{ or (}B\sin \beta )A\\&=BA\cos \alpha \text{ or }BA\sin \beta \end{aligned}$

Now, if the coil has N turns (instead of a single turn), then each turn will be capturing the same magnetic flux $\displaystyle \phi ={{B}_{\bot }}A$ . Logically, the total flux “captured” will be $\displaystyle N{{B}_{\bot }}A$ . We have a term specially to denote the total flux captured by a coil of many turns. It is called the magnetic flux linkage F.