force on an ellipsoid or cylinder aligned to B0 (z‐axis) at an angle θ, made from a soft ferromagnetic material with χ >> 1, e.g. nickel, iron, or martensitic or ferritic stainless steel, is
(2.11)
The force is proportional to the product of B and dB/dz. It is striking that it does not depend upon magnetic susceptibility as long as χ >> 1, a consequence of the demagnetizing fields. As we have seen, most strongly ferromagnetic objects will saturate close to the scanner bore entrance. Figure 2.17 shows the relative forces on cylinders with length to diameter (l /d) ratios ranging from 0 (a flat disk) to 50 (like a knitting needle) and a sphere in the region where the metal is unsaturated. For a long cylinder aligned with z, the maximum force with the object aligned to B0 is (from Equation A1.31)
(2.12)
Figure 2.17 Predicted translational force (logarithmic scale) on spherical and cylindrical 0.1 kg unsaturated objects at distances remote from the iso‐centre along the z‐axis. The bore entrance is at 0.8 m. The objects have density of 8000 kg m−3, χ = 1000 and Bsat = 1.6 T. The force due to gravity is approximately 1 N.
This formula is handy for a quick worst case estimation if you do not know the demagnetization factor or the saturation field.
Figure 2.18 shows the effect on force of different angulations with respect to B0. For objects with a length‐diameter ratio greater than one the maximum force occurs with the greatest alignment to the field (θ = 0°). For flatter objects, the greatest force occurs for an angle of 90°, that is with the planar surface perpendicular to B.
Example 2.4 Force on an unsaturated ferromagnetic object
What is the maximum force on an iron rod of length 10 cm, diameter 2 cm in the fringe field of a MRI magnet with B = 100 mT and dB/dz = 0.6 T m−1? The material is unsaturated.
The material is unsaturated (see Example 2.2), so use Equation 2.12
By contrast the gravitational force is
The magnetic force is approximately eight times the force due to gravity at this point.
Figure 2.18 Influence of angulation of ferromagnetic objects with respect to the B0 direction. The objects have density of 8000 kg m−3 weighing 0.1 kg with χ = 1000 and Bsat = 1.6 T. B = 0.1 T and dB/dz = 0.5 T m−1. The effect of saturation at around 30° is evident for the longest object. Simulation for illustration only.
MYTHBUSTER:
For strongly ferromagnetic objects, the translational force does not depend upon its magnetic susceptibility, but upon its shape, the external field B0 (or Bsat), and the spatial gradient dB/dz.
Soft saturated ferromagnetic material
For a saturated metal the magnetization within the material is at a maximum so once saturation occurs B0 becomes irrelevant, and the maximum force is
(2.13)
Here, for a given object, the only variable is the gradient of the B0 fringe field, which itself changes over distance from the magnet. The shape of the object is no longer a significant factor as it has been completely magnetized. Figure 2.19a shows the relative forces from a 1.5 and 3 T shielded magnet on 0.1 kg ferromagnetic objects which saturate at 1.6 T. State of saturation is more significant than field strength. The object’s shape is a key factor. You will get closer to the magnet holding a sphere without it being wrenched from your grasp than you would with an elongated object of the same mass. Figure 2.19b, plotted on a logarithmic scale, shows the force on each object at greater distances from the iso‐centre, compared to the gravitational force of around 1N. A length‐diameter ratio of 50 corresponds to the geometry of a Birmingham gauge 21 hypodermic needle. The force on the needle exceeds that of gravity around two meters from iso‐centre, half a meter further than for a spherical object of the same material and mass. Needles and scissors constitute two of the most hazardous objects around MRI scanners.
Figure 2.19 Predicted translational force on spherical and cylindrical 0.1 kg objects along the z‐axis: (a) linear plot; (b) logarithmic scale. The bore entrance is at 0.8 m. The objects have density of 8000 kg m−3 and χ = 1000 with Bsat = 1.6 T.
Permanent magnet
What if the object is already magnetized, i.e. is a permanent magnet? Such a situation may arise in MR if it is your institution’s policy to scan patients with cochlear implants. To start, we are not reliant on the external B0 to magnetize the object. Whether an increase occurs close to the magnet will depend upon the hysteresis properties of the object – it may even become demagnetized‐ so it is virtually impossible to predict. We can say, however, that the initial translational (attractive or repulsive depending upon orientation) and twisting forces are likely to exceed those of a soft ferromagnetic object. Cochlear implants are discussed in Chapter 10.
Example 2.5 Force on a ferromagnetic object aligned with B0
What is the maximum force on a ferromagnetic steel cylinder of length 2 cm, diameter 2 mm in the fringe field of a MRI magnet with B = 50 mT and dB/dz = 0.25 Tm−1? The material saturates at 1 T.
The maximum force occurs when the object’s long axis is aligned with
