hard ferromagnet

From Physics wiki

Jump to: navigation, search

Consider a hard ferromagnetic object with some permanent magnetization \mathbf{M}\, "frozen in", i.e., the magnetization varies little with the magnetic field \mathbf{B}\,. In the absence of any free currents and time-varying fields,

\boldsymbol\nabla \times \mathbf{H} = 0\,.

Furthermore,

\boldsymbol{\nabla}\cdot\mathbf{H} = \boldsymbol{\nabla}\cdot\left( \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} \right) = -\frac{1}{\mu_0}\boldsymbol{\nabla}\cdot \mathbf{M}\,.

Thus we need to solve the equations

\boldsymbol\nabla \times \mathbf{H} = 0\,,

\boldsymbol{\nabla}\cdot\mathbf{H} = -\boldsymbol{\nabla}\cdot \mathbf{M}\,.

We can therefore make use of a magnetic potential \Phi_m\, such that:

\mathbf{H} = -\boldsymbol\nabla \Phi_m\,,

from which

\nabla^2 \Phi_m = \boldsymbol{\nabla}\cdot \mathbf{M}\,,

which we can write as

\nabla^2 \Phi_m = -\rho_m\,,

where

\rho_m = -\boldsymbol{\nabla}\cdot \mathbf{M}\,,

is a an effective magnetic charge density. The solution to Poisson's equation for \Phi_m\, is simply

\Phi_m(\mathbf{r}) = -\frac{1}{4\pi}\int\!d^3r'\, \frac{\nabla \cdot \mathbf{M}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\,.

It is sometimes a good idealization to let \mathbf{M}\, go to zero discontinuously at the boundary of the object. In that case, \boldsymbol{\nabla}\cdot \mathbf{M}\, is singular on the boundary. On the other hand, as with bound charges, we can replace -\boldsymbol\nabla\cdot\mathbf{M}\, its average across the interface, i.e., with a "surface charge" term -\hat\mathbf{n}\cdot (\mathbf{M}_{out}- \mathbf{M}_{in}) = \hat\mathbf{n}\cdot\mathbf{M}\,, so that

\Phi_m(\mathbf{r}) = -\frac{1}{4\pi}\int_\mathcal{V}\!dV\,\frac{\nabla \cdot \mathbf{M}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} + \frac{1}{4\pi}\int_\mathcal{\partial V}\!dS\,\frac{\hat\mathbf{n} \cdot \mathbf{M}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\,.

on to hard ferromagnetic sphere

References

[1]

  1. ↑ Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. 
Retrieved from "http://www.physics.thetangentbundle.net/wiki/Classical_electrodynamics/hard_ferromagnet"
Personal tools