# 2.7.11 Magnetic field of cylinder domains

Let us calculate the magnetic field generated by a cylindrical domain of radius
and height
along the longitudinal axis of symmetry **(Fig. 1)**. Assume that domain magnetization
is uniform over its volume and is directed along axis
.

**Fig. 1. Schematic representation of a cylindrical domain.**

Dividing a cylindrical domain into elementary annular domains with magnetic moment
, where
– distance from domain center to annular domain (ring radius), we can calculate, according to formula (5) in **chapter 2.7.1**, the magnetic field created by this elementary domain at distance
:

(1)

where . It is obvious that due to the cylinder symmetry, the magnetic field of the elementary domain is directed only along the -axis. Integrating over the cylinder volume we obtain:

(2)

According to formula (2) in **chapter 2.7.1**. the signal registered by MFM is determined by the second derivative of the magnetic field. Let us analyze the field and its derivatives dependence on distance
. The qualitative plot of magnetic field and its first derivative vs. distance
in cases
and
calculated in accordance with (2) is shown in **Fig. 2-5**.

**Fig. 2. plot in case .
Solid line shows asymptote.**

**Fig. 3. plot in case .
Solid line shows asymptote.**

**Fig. 4. plot in case .
Solid line shows asymptote.**

**Fig. 5. plot in case .
Solid line shows asymptote.**

As can be seen from **Fig. 2-5**, the magnetic field decays at distances determined by domain dimensions. Far from domain surface both magnetic field and its derivative have asymptotic behavior. In case
the second derivative dependence on
for
fits a
law. In case
the second derivative for
obeys a
law and for
– a
law.

Thus, if during the qualitative study of magnetic field distribution in the scanning mode a magnetic domain is revealed to be cylindrical, then the analysis of magnetization dependence on distance
over the domain center permits to determine magnetic film thickness
and cylinder radius
. The qualitative estimation of ratio *h/R* can be obtained from the general characteristic of the
dependence.

## Summary.

- Calculated is the magnetic field (2) generated by cylindrical domain of radius and height along the longitudinal axis of symmetry.
- If during the qualitative study of magnetic field distribution in the scanning mode it is revealed that magnetic domain is cylindrical, then the analysis of magnetization dependence on distance over domain center permits to determine magnetic film thickness and cylinder radius .
- The qualitative estimation of ratio
*h/R*can be obtained from the general characteristic of the dependence.