Boundary value problems with ferromagnets

Consider a ferromagnetic sphere, of uniform permeability $\mu$, placed in a uniform $z$-directed magnetic field of magnitude $B_0$. Suppose that the sphere is centred on the origin. In the absence of any true currents, we have $\nabla\times{\bf H} = {\bf0}$. Hence, we can write ${\bf H} = -\nabla\phi_m$. Given that $\nabla\cdot {\bf B}=0$, and ${\bf B} = \mu {\bf H}$, it follows that $\nabla^2\phi_m=0$ in any uniform magnetic medium (or a vacuum). Hence, $\nabla^2\phi_m=0$ throughout space. Adopting spherical polar coordinates, $(r, \theta, \varphi)$, aligned along the $z$-axis, the boundary conditions are that $\phi_m \rightarrow - (B_0/\mu_0) r \cos\theta$ at $r\rightarrow \infty$, and that $\phi_m$ is well-behaved at $r=0$. At the surface of the sphere, $r=a$, the continuity of $H_\parallel$ implies that $\phi_m$ is continuous. Furthermore, the continuity of $B_\perp=\mu H_\perp$ leads to the matching condition

$$\left.\mu_0 \frac{\partial \phi_m}{\partial r}\right\vert _... ...ft.\mu \frac{\partial\phi_m}{\partial r}\right\vert _{r=a-}.$$ (875)

Let us try separable solutions of the form $r^m \cos\theta$. It is easily demonstrated that such solutions satisfy Laplace's equation provided that $m=1$ or $m=-2$. Hence, the most general solution to Laplace's equation outside the sphere, which satisfies the boundary condition at $r\rightarrow \infty$, is

$$\phi_m(r,\theta) = - (B_0/\mu_0) r \cos\theta + (B_0/\mu_0) \alpha \frac{a^3 \cos\theta}{r^2}.$$ (876)

Likewise, the most general solution inside the sphere, which satisfies the boundary condition at $r=0$, is

$$\phi_m(r,\theta) = - (B_1/\mu) r \cos\theta.$$ (877)

The continuity of $\phi_m$ at $r=a$ yields

$$B_0 - B_0 \alpha = (\mu_0/\mu) B_1.$$ (878)

Likewise, the matching condition (875) gives

$$B_0 + 2 B_0 \alpha = B_1.$$ (879)

Hence,

$$\alpha = \frac{\mu-\mu_0}{\mu+2 \mu_0},$$ (880)

$$B_1 = \frac{3 \mu B_0}{\mu+2 \mu_0}.$$ (881)

Note that the magnetic field inside the sphere is uniform, parallel to the external magnetic field outside the sphere, and of magnitude $B_1$. Moreover, $B_1 > B_0$, provided that $\mu>\mu_0$.

Figure 50:

As a final example, consider an electromagnet of the form sketched in Fig. 50. A wire, carrying a current $I$, is wrapped $N$ times around a thin toroidal iron core of radius $a$ and permeability $\mu\gg \mu_0$. The core contains a thin gap of width $d$. What is the magnetic field induced in the gap? Let us neglect any leakage of magnetic field from the core, which is reasonable if $\mu\gg \mu_0$. We expect the magnetic field, $B_c$, and the magnetic intensity, $H_c$, in the core to be both toroidal and essentially uniform. It is also reasonable to suppose that the magnetic field, $B_g$, and the magnetic intensity, $H_g$, in the gap are toroidal and uniform, since $d\ll a$. We have $B_c = \mu H_c$ and $B_g=\mu_0 H_g$. Moreover, since the magnetic field is normal to the interface between the core and the gap, the continuity of $B_\perp$ implies that

$$B_c = B_g.$$ (882)

Thus, the magnetic field-strength in the core is the same as that in the gap. However, the magnetic intensities in the core and the gap are quite different: $H_c = B_c/\mu = B_g/\mu = (\mu_0/\mu) H_g$. Integration of Eq. (871) around the torus yields

$$\oint {\bf H}\cdot d{\bf l} = \int{\bf j}_t\cdot d{\bf S} = N I.$$ (883)

Hence,

$$(2\pi a-d) H_c + d H_g = N I.$$ (884)

It follows that

$$B_g = \frac{N I}{(2\pi a-d)/\mu + d/\mu_0}.$$ (885)