In this post, some basics about the stray and demagnetization field will be covered. The post originates from reading the book by Stöhr – Ref. [1].

**Figure. 1. **Demonstration of materials magnetization – reproduced from Fig. 2.6 in Ref. [1].

To get an idea about the stray and demagnetization field, one needs to go back to the Stoke’s theorem,

\[\oint \vec{H} \cdot d\vec{l} = \iint_S (\nabla \times \vec{H})_n dS\]The integrand on the right hand side is the current density, according to,

\[\nabla \times \vec{H} = \vec{j}\]
In the case of no current flow, we have \(\vec{j} = 0\) and therefore we should accordingly have the left hand side being 0 as well. There then comes the requirement that we have the magnetic field inside and outside of the material being opposite to each other – inside the material, we have the `demagnetization field`

(since it is opposite to the external magnetic field and therefore is trying to cancel it out) and outside the material, we have the `stray field`

.

To get an idea about the relation between magnetization field inside the material and the magnetic field, one then needs to turn to the Gauss’s theorem,

\[\iint_S \vec{B} \cdot \vec{n} dS = \iiint_V \nabla \cdot \vec{B} dV\]Since we don’t have magnetic monopole, we should have the left hand side of the equation being 0, which then indicates \(\nabla \cdot \vec{B} = 0\).

**Figure. 2. **Demonstration of materials magnetization – reproduced from Fig. 2.7 in Ref. [1].

Combined with the relation between magnetic induction field, magnetic field and magnetization field,

\[\vec{B} = \mu_0\vec{H} + \vec{M}\]we then have,

\[\mu_0\nabla \cdot \vec{H} = -\nabla \cdot \vec{M}\]
Mathematically, it is just a matter of reorganization of expression. However, physically, it infers a clear picture as presented in Fig. 2 – outside the material, it is just the stray field and inside the material, the end result of the interaction between the magnetic induction (\(\vec{B}\)) and the demagnetization (\(\vec{H}_d\)) field is the magnetization field (\(\vec{M}\)). Furthermore, we can imagine positive and negative magnetic ‘charge’ on the surface of the material, just like what we would have for electric field – though, in practice, we know that the magnetic ‘charge’ does not even exist. According to the equation above, we can say that the ‘positive charge’ on the right hand side of Fig. 2 is the source of the external stray field and is the sink of the internal magnetization field, and vice versa for the ‘negative charge’ on the left hand side.

**N. B.** In the equation above, when we say \(\vec{H}\), we mean both \(\vec{H}_d\) inside the material and \(\vec{H}_s\) outside the material.

**References**

[1] J. Stohr and H. C. Siegmann. Magnetism - from fundamentals to nanoscale dynamics. Springer. 2006. New York.