Soft Iron Sphere in Uniform Magnetic Field

The opposite extreme to a ``hard'' ferromagnetic material, which can maintain a large remnant magnetization in the absence of external fields, is a ``soft'' ferromagnetic material, for which the remnant magnetization is relatively small. Let us consider a somewhat idealized situation in which the remnant magnetization is negligible. In this situation, there is no hysteresis, so the ${\bf B}$ -${\bf H}$ relation for the material reduces to

$${\bf B} = \mu(B) \,{\bf H},$$ (736)

where $\mu(B)$ is a single valued function. The most commonly occurring ``soft'' ferromagnetic material is soft iron (i.e., annealed, low impurity, iron).

Consider a sphere of soft iron placed in an initially uniform external field ${\bf B}_0 = B_0 \,{\bf e}_z$ . The $\mu_0\,{\bf H}$ and ${\bf B}$ fields inside the sphere are most easily obtained by taking the solutions (734) and (735) (which are still valid), and superimposing on them the uniform field ${\bf B}_0$ . We are justified in doing this because the equations that govern magnetostatic problems are linear. Thus, inside the sphere we have

$$\mu_0\,{\bf H} = {\bf B}_0 - \frac{1}{3}\,\mu_0 \,{\bf M},$$ (737)

$${\bf B} = {\bf B}_0 + \frac{2}{3} \,\mu_0\, {\bf M}.$$ (738)

Combining Equations (737), (738), and (739) yields

$$\mu_0\,{\bf M} = 3\left(\frac{\mu-\mu_0}{\mu+2\,\mu_0}\right) {\bf B}_0,$$ (739)

with

$${\bf B} = \left(\frac{3\,\mu}{\mu+2\,\mu_0}\right) {\bf B}_0,$$ (740)

where, in general, $\mu=\mu(B)$ . Clearly, for a highly permeable material (i.e., $\mu/\mu_0\gg 1$ , which is certainly the case for soft iron) the magnetic field strength inside the sphere is approximately three times that of the externally applied field. In other words, the magnetic field is amplified inside the sphere.

The amplification of the magnetic field by a factor three in the high permeability limit is specific to a sphere. It can be shown that for elongated objects (e.g., rods), aligned along the direction of the external field, the amplification factor can be considerably larger than three.

It is important to realize that the magnetization inside a ferromagnetic material cannot increase without limit. The maximum possible value of ${\bf M}$ is called the saturation magnetization, and is usually denoted ${\bf M}_s$ . Most ferromagnetic materials saturate when they are placed in external magnetic fields whose strengths are greater than, or of order, one tesla. Suppose that our soft iron sphere first attains the saturation magnetization when the unperturbed external magnetic field strength is $B_s$ . It follows from Equations (739) and (740) (with $\mu\gg \mu_0$ ) that

$$B = B_0 + 2 \,B_s$$ (741)

inside the sphere, for $B_0 > B_s$ . In this case, the field amplification factor is

$$\frac{B}{B_0} = 1 + 2\,\frac{B_s}{B_0}.$$ (742)

Thus, for $B_0\gg B_s$ the amplification factor approaches unity. We conclude that if a ferromagnetic material is placed in an external field that greatly exceeds that required to cause saturation then the material effectively loses its magnetic properties, so that $\mu\simeq \mu_0$ . Clearly, it is very important to avoid saturating the soft magnets used to channel magnetic flux around transformer circuits. This sets an upper limit on the magnetic field-strengths that can occur in such circuits.