Figure 2.14 Reciprocal demagnetization factors for a cylinder: 1/d1 (axial, red line), 1/d2 (radial, blue line). For a cylinder d3 = d2.
If the object is rotated 90° to B0
Figure 2.15 shows the theoretical internal B for ferromagnetic cylindrical objects of various length‐diameter (l/d) ratios and a sphere as they approach 1.5 and 3 T shielded magnets. In each case the saturation value Bsat (=1.6 T) is reached at greater distances from the magnet for the more elongated objects.
Figure 2.15 Predicted internal B for a ferromagnetic sphere and cylinders of differing length/diameter ratios in the approach to 1.5 and 3 T MRI magnets. The material saturates at 1.6 T. The bore entrance is at 0.8 m. The dotted gray line indicates the B0 field strength.
Demagnetizing factors and Bsat are crucial for determining the force and torque on different shaped objects within the scanner’s fringe field.
MYTHBUSTER:
The internal magnetic field or degree of magnetization of a ferromagnetic object is not determined by its magnetic susceptibility but by its demagnetization factors and saturation status.
Example 2.1 Magnetization of a nickel coin
A nickel coin (length = 1 mm, diameter = 1 cm) is inadvertently brought into the MRI examination room. If the external field is 100 mT what is the field within the coin if it is: (a) lying face on to the magnet; (b) edge on to the magnet? Will the coin’s metal saturate?
1 (a) The ratio l/d = 0.1, so from Figure 2.14 or Equation A1.31 and 2.9a
Bsat for nickel is 0.62 T (Table 2.2) so, in this orientation it will be unsaturated.
1 (b) For the end‐on orientation use d2 and Equation 2.9b
This exceeds Bsat so the internal field will saturate at 0.62 T.
Example 2.2 Iron rod in the fringe field
An iron rod of length 10 cm, diameter 2 cm is brought within the fringe field of a MR magnet with B = 100 mT. Will it be saturated if its length is aligned with the field? Iron saturates at around 2 T.
From Figure 2.14 or Equation A1.31 and 2.9a
At this point the metal will not saturate.
FORCES AND TORQUE
The forces upon ferromagnetic objects are paramount for MRI safety. “Magnet safety” should be ingrained into our behavior and consciousness. In this section we consider forces and torques on objects, conducting wires, and electrical circuits. The former is relevant for all implants, MR accessories and objects brought into the MR environment; the latter is relevant for active implants.
Translational force: non‐ferromagnetic materials
For diamagnetic and paramagnetic materials where |χ| is very small, we do not have to consider the demagnetizing factors. If we assume only the z‐axis component, then the magnetic force on a paramagnetic object is
(2.10)
For a diamagnetic material the force will be negative, i.e. repulsive. Figure 2.16 shows plots of B, dB/dz and their product B.dB/dz along the z‐axis for simulations of 1.5 and 3 T shielded magnets. Note that the locations of the maximum values of dB/dz and B.dB/dz do not necessarily coincide. For a paramagnetic or diamagnetic object the maximum force is exerted at the location of the maximum field‐gradient product, near to the bore entrance. The figure shows B‐field values on‐axis, but in general the spatial gradient and product values are greatest around the edge of the bore circumference.
Figure 2.16 B, dB/dz and product B.dB/dz along the z‐axis for: (a) a shielded 3 T magnet (b) a shielded 1.5 T magnet with bore length 1.6 m. The horizontal axis is distance from the iso‐centre. The bore entrance is at 0.8 m. Simulation for illustration only.
MYTHBUSTER:
The translational force within the bore is not a maximum but is close to zero.
Example 2.3 Force on a diamagnetic object
What is the force on a 1 litre bag of saline brought towards the bore of a magnet with B = 1 T and dB/dz = 5 T m−1?
This is a repulsive force but is 2700 times less than the force due to gravity, so is negligible.
Translational force: ferromagnetic objects
The situation for a ferromagnetic object is complicated by two additional factors: demagnetization factors which depend strongly upon geometry, and saturation: the degree of magnetization sustainable by the metal. In this section we quote the final results as applied to a cylinder or ellipsoid with equal minor axes (1). Appendix 1 provides a full derivation.
Force on a soft unsaturated ferromagnetic
