14.1.2 Permeability

To obtain an even stronger magnetic field, we can wrap the turns of a solenoid around a piece of iron (instead of empty space).

When a current passes through the solenoid, the iron will be magnetized (very strongly, since iron has very high permeability) by the current’s magnetic field. Since the iron’s magnetic field is in the same direction as the current’s, the resultant magnetic field is (very dramatically) increased.

So how does iron (and permanent magnets) produce a magnetic field? It turns out that each iron atom is a tiny magnet (something to do with this thing called electron spin). And a region where atoms are magnetized in the same direction is called a magnetic domain. An unmagnetized piece of iron is basically segmented into a large number of individual magnetic domains which are randomly oriented, so the resultant field is negligible. In the presence of an external magnetic field (such as that of the solenoid), these magnetic domains will align themselves in the direction of the external magnetic field. When aligned in the same direction, the resultant magnetic field of all these magnetic domains is substantial.

Materials which can be magnetized in this manner are said to be ferromagnetic. There are not many ferromagnetic materials. The common ones are iron, nickel, cobalt and their alloys, and some compounds of rare earth metals (e.g. neodymium).

Magnetic permeability is a measure of how susceptible a substance is to the formation of a magnetic field. Compared to vacuum and most substances, ferromagnetic materials have significantly higher permeability. Check out the table below.

MaterialPermeability m / H m-1Relative permeability \displaystyle \frac{\mu }{{{{\mu }_{0}}}}
Vacuum\displaystyle {{\mu }_{0}}=4\pi \times {{10}^{{-7}}}1
Carbon Steel\displaystyle 1.26\times {{10}^{{-4}}}100
Iron (99.8% pure)\displaystyle 6.3\times {{10}^{{-3}}}5,000
Cobalt-iron\displaystyle 2.3\times {{10}^{{-2}}}18,000
Permalloy (80% nickel, 20% iron)\displaystyle 1.25\times {{10}^{{-1}}}100,000
Most other materials, e.g. air, water, copper\displaystyle \sim1.26\times {{10}^{{-6}}}~1

Permeability is often used to casually explain why the magnetic field of a solenoid with an iron core is stronger compared to an air core one. In fact, the strength of an iron-core solenoid can be calculated using the same solenoid formula, but replacing m0 with m of iron.

\displaystyle B=\mu nI

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