Forces on Metal Objects
How do you calculate the magnetic force pulling a piece of metal toward the scanner?
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A metallic object may experience two types of forces when placed in an external magnetic field: translation and rotation. Translational force is that which pulls the object toward the magnet, while rotational force (also known as torque) is that which seeks to align the long axis of the object with the lines of the magnetic field.
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Translational (Ftrans) and rotational (Frot) forces on a magnetizable object placed in an external magnetic field (Bo)
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The Ellipsoidal Model
Predicting the effects of a magnetic field on irregularly-shaped objects generally requires computer simulation, so most analytical equations for force and torque are based on simple geometric forms having exact mathematical solutions. The most popular of these (used in equations below) is the magnetically homogeneous ellipsoid of revolution. By varying length (a), diameter (d), and angulation (θ) of the ellipsoid's axis of symmetry with respect to B, a wide range of representative objects in different orientations (e.g. spheres, footballs, pancakes, rods/needles) can be created and analyzed.
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Additional complexity exists for objects made of metals that are moderately to strongly ferromagnetic as the externally applied field increases.
- Demagnetizing fields must be considered, as such objects begin to act like little magnets. These are described by directionally-specific demagnetizing factors, which for ellipsoids are denoted Na and Nd.
- Magnetic saturation may occur, meaning the object's internal magnetization will plateau when it exceeds a certain critical value (Bsat).
- Permanent magnetization of the object (at value Brem) may occur, even when the external field is removed.
General Equations for Force and Torque on an Ellipsoid
Equations for force (Fz) and torque (T) on an ellipsoid have the general forms below:
Translational force (Fz) in the z-direction (parallel to B),
where 
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Torque (T) on an object immersed in a magnetic field (B),
where 
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Here f(•) and g(•) are functions that depend on the object's intrinsic susceptibility (χ), shape-dependent demagnetizing factors (Na and Nd), and angle the object makes with the external magnetic field (θ). The term μo ≈ 4π x 10−7 N/A² is the magnetic constant, also known as the permeability of free space. For objects other than ellipsoids, the form of f(•) and g(•) will vary, but several common features should be noted:
- Both Fz and T scale directly with volume (Vol), so doubling the size on an object doubles the force or torque upon it.
- Fz is proportional to B, while T is proportional to B². So doubling the local field doubles the translational force but quadruples the torque on an object.
- Translational force is strongest near the edges of the magnet opening and zero in the center of the bore. Translational force is proportional to the local field (B) multiplied by its spatial rate of change (dB/dz), a combined entity known as the spatial gradient product (SGP). The SGP is strongest near the edges of the magnet bore opening, making this the most powerful place for translational forces. At the magnet isocenter, however, dB/dz ≈ 0, so surprisingly there is no translational force once the object reaches the center of the magnet.
- In the center of the bore, torque is extremely powerful while translational force is zero. This is because B² is (nearly) maximal here while dB/dz is negligible.
- Torque is maximal at 45° and minimal at 0°. As seen in the expression for g(•), torque has an angular dependence (sin 2θ), being maximal when the object is at 45° to B and lowest when the object is parallel to B (corresponding to sin 90° = 1 and sin 0° = 0).
- For strongly ferromagnetic objects, dependence on χ disappears and demagnetizing factors (Ni) become important.
Special Cases for Force and Torque
Paramagnetic and Weakly Ferromagnetic Metals
Materials (such as most surgical stainless steels, copper, and lead) that have small magnetic susceptibilities (χ < 1) experience only trivial forces compared to gravity when placed in an external field. Demagnetizing fields and factors are insignificant, so the equations simplify to:
Translational force (Fz) in the z-direction for an object with small magnetic susceptibility (χ < 1)
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Torque (T) of an elongated object with small magnetic susceptibility (χ < 1) making an angle θ with B
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Unsaturated Moderately and Stongly Ferromagnetic Metals
For such materials with high susceptibilities (χ >>1), explicit dependence on the value of χ disappears and demagnetization factors become important. Na and Nd values for ellipsoids can be direcly calculated or obtained from tables. The equations for translational force and torque become
Translational force (Fz) in the z-direction for an unsaturated ferromagnetic ellipsoid (with χ >> 1)
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Torque (T) of an unsaturated ferromagnetic ellipsoid (with χ>>1) making angle (θ) with the main magnetic field
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Saturated Ferromagnetic Metals
As the ambient external field (B) increases, magnetic saturation of the object may occur, as described in a prior Q&A. The internal field where saturation occurs (Bsat) is material-specific, and lying in the range of 0.5-2.0 T for common commercial alloys. Taking demagnetizing fields into account, magnetic saturation occurs whenever an object is brought close enough to the scanner such that the fringe field exceeds N•Bsat, where N is the demagnetizing factor. Thus for a spherical object (with N=⅓) made of nickel (with Bsat = 0.62T), saturation will occur whenever the external magnetic field exceeds (⅓)(0.62) = 0.21T. The force and torque equations are similar to the unsaturated case, with B replaced by Bsat.
Translational force (Fz) in the z-direction for a fully saturated ferromagnetic ellipsoid (with χ >> 1)
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Torque (T) of a fully saturated ferromagnetic ellipsoid (with χ>>1) making angle (θ) with the main magnetic field
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Interestingly, as long as the external field is sufficient to saturate the object, both the force and torque become independent of B and the force depends only on the field spatial gradient (dB/dz).
Permanent Magnets
In some cases an object that has been permanently magnetized (to value Brem) previously is reintroduced into the external field. Clinical examples include patients with cochlear implants and dental magnetic keepers. The actual force and torque may be difficult to predict and depend on the hysteresis characteristics of the magnet material. In some cases the force may be repulsive instead of attractive or the external field may even demagnetize the magnet. In other cases the magnetization of the permanent magnet may increase (up to Bsat.) The force and torque equations are different because the magnetized object has a permanent intrinsic magnetization (Brem) that is constant in both size and direction independent of the external field (B). Assuming the permanent magnet is brought into the field at a fringe field well below N•Brem and is magnetized along its principal axis, we can write
Translational force (Fz) in the z-direction for a permanently magnetized ellipsoid (with B << Brem )
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Torque for a permanently magnetized ellipsoid in an external field (B) (with B << Brem )
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References
Abbott JJ, Ergeneman O, Kummer MP, et al. Modeling magnetic torque and force for controlled manipulation of soft-magnetic bodies. IEEE Trans Robotics 2007; 23:1247-1252. [DOI Link] (excellent paper explains the angle for maximum torque is not always 45°, but depends on several factors, including whether the magnetization (M) and applied field (B or H) are aligned.)
Anonymous. Force and torque on a small magnetic dipole. Downloaded from this link, Nov 2020. (Simplified model using a square current loop to explain how factors such as B and dB/dz affect force and torque.)
Jackson DP. Dancing paperclips and the geometric influence on magnetization: a surprising result. Am J Phys 2006; 74:272-279.
Lowes FJ. The torque on a magnet. Proc R Soc Lond A 1974; 337:555-567. [DOI Link]
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]
Panych LP, Madore B. The physics of MRI safety. J Magn Reson Imaging 2018; 47:28-43. [DOI link]
Snelling EC. Soft Ferrites. Properties and Applications. Iliffe Books:London, 1969: selected pages
Schneider ML, Walker GB, Dormer KJ. Effects of magnetic resonance imaging on implantable permanent magnets. Am J Otol 1995; 16:687-9.
Wysin GM. Demagnetization fields. 2012:1-16. Public teaching notes, downloaded from this link, 20 Oct 2020.
Abbott JJ, Ergeneman O, Kummer MP, et al. Modeling magnetic torque and force for controlled manipulation of soft-magnetic bodies. IEEE Trans Robotics 2007; 23:1247-1252. [DOI Link] (excellent paper explains the angle for maximum torque is not always 45°, but depends on several factors, including whether the magnetization (M) and applied field (B or H) are aligned.)
Anonymous. Force and torque on a small magnetic dipole. Downloaded from this link, Nov 2020. (Simplified model using a square current loop to explain how factors such as B and dB/dz affect force and torque.)
Jackson DP. Dancing paperclips and the geometric influence on magnetization: a surprising result. Am J Phys 2006; 74:272-279.
Lowes FJ. The torque on a magnet. Proc R Soc Lond A 1974; 337:555-567. [DOI Link]
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]
Panych LP, Madore B. The physics of MRI safety. J Magn Reson Imaging 2018; 47:28-43. [DOI link]
Snelling EC. Soft Ferrites. Properties and Applications. Iliffe Books:London, 1969: selected pages
Schneider ML, Walker GB, Dormer KJ. Effects of magnetic resonance imaging on implantable permanent magnets. Am J Otol 1995; 16:687-9.
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?
So then, exactly where is the most dangerous place for force or torque near an MRI scanner?
Which types of metal are the most dangerous around a magnetic field?
So then, exactly where is the most dangerous place for force or torque near an MRI scanner?