**Diamagnetic, paramagnetic, or weakly ferromagnetic materials***χ*

*< 1*), including most surgical stainless steels and non-ferrous metals such as tin, aluminum, copper, or lead are only minimally displaced by an external magnetic field and pose no dangers as projectiles. Any shape effects on their magnetization are largely irrelevant and may be disregarded.

Moderately to strongly ferromagnetic materialsObjects made of iron, nickel, and/or chromium with larger susceptibilities (
χ >> 1) become progressively magnetized when placed in an external magnetic field (Bo). As the external field increases, so does this internal polarization/ magnetization (M) as electron spins begin to align and magnetic domain boundaries are reshaped. This process continues until M reaches the magnetic saturation point of the material, beyond which no further magnetization can occur. (Note that the magnetic polarization M arises from electron spins in microscopic currents and has nothing to do with nuclear magnetization M discussed elsewhere). |

**M**goes from a positive value inside the object to zero immediately outside). An internal

**demagnetizing field (D)**is thereby created, running from the object's north to south poles opposite to the direction both

**M**and

**Bo**. Provided the object is homogeneous and responds linearly to small magnetic field changes, the demagnetizing field (

**D**) is generally assumed to be proportional to

**M**in each direction, related by a constant of proportionality between 0 and 1 known as the

*. Specifically,*

**demagnetization factor (N)****D**=

**−**

*N*

**M.**

**Bo**,

**D**reduces the effective magnetic field inside the object, but because of its relatively small size, does not significantly affect

**M**. The

*, defined as the ratio between*

**volume magnetic susceptibility (***χ*)*M*and the field intensity, is correspondingly reduced to an

**apparent susceptibility (***χapp**given by*

**)**__strongly__ferromagnetic materials (those with

*χ*>> 1), the dependence on

*χ*disappears, leaving

*χapp*

*= 1/N*as an approximation.

*N*• Bsat = (⅓)(2.0) = 0.67 T. Surprisingly, the sphere has reached its full (2.0T) saturation magnetization while still in the fringe field of the 1.5T scanner! Actually this should not come as too big a surprise because the common layman's description of ferromagnetic materials is that they tend to "concentrate" the lines of external magnetic fields thereby increasing magnetic flux density within them.

*N*and

*χ*are not a single numbers, but 2nd order tensors (arrays of values corresponding to different directions). For symmetric 3-D forms like ellipsoids, it is possible to select a coordinate frame aligned with the principal axes of the body resulting in

*three*demagnetizing factors,

*Nx*,

*Ny*, and

*Nz*, corresponding to each cardinal direction. These are all dimensionless numbers between 0 and 1, having the additional property

*Nx*+

*Ny*+

*Nz = 1.*

**Predicting Demagnetizing Factor(s)****D**) is strongly dependent on the shape and orientation of the ferromagnetic object in the external field. Calculation of

**D**for a given object typically requires extensive computer simulations, but a few exact solutions are possible for simple geometric shapes. Below are computed

*in the*

**demagnetization factors (N)****-direction. What trends do you notice?**

*Bo**N*) along the direction of the main field for various simple homogeneous objects.

Note that objects with the highest

*N*-values are those that have a large relative surface area facing in the direction of

**Bo**. Such an object is typified by the thin vertical plate perpendicular to

**Bo**, which has

*N*≈ 1.00 (the highest possible value). Conversely, when the plate is turned by 90º,

*N*≈ 0.00 (the lowest possible value). The same principle is demonstrated by the various ellipsoids and cylinders illustrated.

**Bo**, the virtual north and south poles on their surfaces are close together; hence the demagnetizing field (

**D**) is very strong. Conversely, when the virtual poles are far apart (as in the light blue cylinder pointing parallel to

**Bo**), the poles at the ends are very far apart. In this scenario, the interior N-S pairs cancel each other out leaving widely separated magnetic charge densities only at the two ends. This orientation of the cylinder thus results in a very weak demagnetizing field (

**D**) and correspondingly small value for the demagnetizing factor (

*Nz*).

### Advanced Discussion (show/hide)»

By definition, the magnetization **M** is related to the internal magnetic field (**H**) through the dimensionless quantity (*χ*) known as the magnetic susceptibility:

**M** = *χ* **H**

In diagmagnetic and weakly paramagnetic materials, **H** = **H _{o}**, where

**H**is the external field.

_{o}In ferromagnetic materials shape plays a critical role because the internal field **H** is reduced by an opposing demagnetizing field (**D**), which for homogeneous linear materials can be written as **D** = − *N***M**, where *N* is a direction specific demagnetizing factor.

So in ferromagnetic materials

**H** = **H _{o}** −

**D**=

**H**−

_{o}*N*

**M**

But since **H** = **M** / *χ* we can write

**M** / *χ* = **H _{o}** −

*N*

**M**

or

**H _{o}** =

**M**/

*χ*+

*N*

**M**=

**M**

*χ*+

*N*

**M**(1 +

*N*

*χ*

*χ*

The apparent susceptibility *χ _{app}* is therefore

*χ _{app}* =

**M**/

**H**=

_{o}*χ*/

*N*

*χ*

Dividing the numerator and denominator by *χ* we obtain

*χ _{app}* = 1 /

*χ*+

*N*

So when *χ* → ∞, 1/*χ* → 0, and we are left with

*χ _{app}* = 1 /

*N*

Thus for large values of *χ* (as are typically found in ferromagnetic materials), the dependence on *χ* disappears, with the apparent susceptibility dependent only on the reciprocal (1/*N*) of the demagnetizing factor.

The strength of the external field (**H _{ext}**) needed to cause magnetic saturation (

**M**) can be derived as

_{sat}**H _{ext}** =

**M**/

_{sat}*χ*=

_{app}**M**/ (1/

_{sat}*N*) =

*N*

**M**

_{sat}Converting to SI units by multiplying each side of this equation by the permeability of free space (*μ _{o} = 4π x 10^{−7} T m A^{−1}*), with

**B**=

_{ext}*μ*

_{o}**H**and

_{ext}**B**=

_{sat}*μ*

_{o}**M**we obtain

_{sat}**B _{ext}** =

*N*

**B**

_{sat}**References**

Jackson DP. Dancing paperclips and the geometric influence on magnetization: a surprising result. Am J Phys 2006; 74:272-279.

McRobbie DW. Essentials of MRI Safety. Wiley-Blackwell, 2020. (Excellent recently released book. Chapter 2 on Fields and Forces as well as Appendix is very pertinent to this topic). [Link to purchase]

Osborn JA. Demagnetizing factors of the general ellipsoid. Phys Rev 1945; 67:351-357. [DOI link]

Panych LP, Kimbrell VK, Mukundan Jr S, Madore B. Relative magnetic force measures and their potential role in MRI safety practice. J Magn Reson Imaging 2020; 51:1260-1271. [DOI Link]

Prozorov R, Kogan VG. Effective demagnetizing factors of diamagnetic samples of various shapes. arXiv:1712.06037v2 (2 July 2018)

Sato M, Ishii Y. Simple and approximate expressions of demagnetizing factors of uniformly magnetized rectangular rod and cylinder. J App Phys 1989; 66:983-5. [DOI Link]

Skomski R, Hadjipanayis GC, Sellmyer DJ. Effective demagnetizing factors of complicated particle mixtures. IEEE Trans Magnetics 2007; 43:2956-8. [DOI Link]

Snelling EC. Soft Ferrites. Properties and Applications. Iliffe Books:London, 1969: selected pages

Solivérez CE. Electrostatics and magnetostatics of polarized ellipsoidal bodies: The depolarization tensor method. Bariloche: Rio Negro, Argentina: 2016.

Wysin GM. Demagnetization fields. 2012:1-16. Public teaching notes, downloaded from this link, 20 Oct 2020.

**Related Questions**

*Which types of metal are the most dangerous around a magnetic field?*

**What is magnetic susceptibility?******

*What is ferromagnetism?*