Effect of Shape

I don't understand how an object's shape affects its magnetization.  

You are not alone. The effect of object shape is far more complex than most people have ever imagined!

Diamagnetic, paramagnetic, or weakly ferromagnetic materials

Objects composed of materials with small susceptibilities (i.e., 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 materials

Objects 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).

Ferromagnetic materials, when placed in an external field (B_o), develop a strong internal magnetization (M), behaving like little magnets themselves. Their edges exposed to the external field become like north and south poles, with a resultant internal demagnetizing field (D), that opposes M.

As a result of this process, an object composed of moderate to strongly ferromagnetic material essentially becomes its own "little magnet" whose ends can be thought of as north and south "poles".  These "poles" represent an accumulation of imputed magnetic charge density at the ends of the object where there is an abrupt magnetic field discontinuity (i.e., where 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 demagnetization factor (N).  Specifically, D = −NM. 

By opposing Bo, D reduces the effective magnetic field inside the object, but because of its relatively small size, does not significantly affect M. The volume magnetic susceptibility (χ), defined as the ratio between M and the field intensity, is correspondingly reduced to an apparent susceptibility (χapp) given by

Note that for strongly ferromagnetic materials (those with χ >> 1), the dependence on χ disappears, leaving χapp = 1/N ​as an approximation. 

An important corollary of this result is that for a ferromagnetic material, the external field strength (Bext) needed to reach critical magnetic saturation (Bsat) is much smaller than expected.  Specifically,

Consider, for example, a sphere of silicon steel with Bsat = 2.0 T brought near a 1.5 T MR scanner. The demagnetization factor for a sphere is ⅓ in each direction, so the sphere will become completely magnetized when the external field (Bext) reaches 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.

In general 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)

The demagnetizing field (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 demagnetization factors (N) in the Bo-direction. What trends do you notice?

Demagnetizing factors (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.

So why does this occur? The reason is that for relatively thin objects with their largest areas facing 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). 

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Further information about demagnetizing factors and derivation of the equations above can be found in the Advanced Disccusion.

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_o is the external field.

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 = − NM, where N is a direction specific demagnetizing factor.

So in ferromagnetic materials

H = H_o − D = H_o − NM

But since H = M / χ we can write

M / χ = H_o − NM

or

H_o = M / χ + NM = M (1/χ + N) = M (1 + Nχ) / χ

The apparent susceptibility χ_app is therefore

χ_app = M / H_o = χ / (1 + Nχ)

Dividing the numerator and denominator by χ we obtain

χ_app = 1 / (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_sat) can be derived as

H_ext = M_sat / χ_app = M_sat / (1/N) = NM_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_oH_ext and B_sat = μ_oM_sat we obtain

B_ext = N B_sat

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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.

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Related Questions
     Which types of metal are the most dangerous around a magnetic field? 
     What is magnetic susceptibility? 
     What is ferromagnetism? ​

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