is saturated. From Figure 2.14 or Equations A1.31 and 2.9a
This exceeds Bsat so use Equation 2.13
By contrast the gravitational force is
The magnetic force is 2.5 times the force due to gravity at this point.
Example 2.6 Force on a ferromagnetic object side on to B0
If the object in Example 2.5 has its long axis perpendicular to B0. What is the attractive force on it?
Firstly determine if the object is saturated. From Figure 2.14 or Equations A1.31 and 2.9b
In this orientation the object is not saturated so use Equation 2.11 with θ = 90°
(We could also have used Equation 2.13 using B = 0.1 T in place of Bsat).
This is one quarter of the gravitational force. The orientation of the object matters! Twisting it towards the magnet is potentially dangerous.
Projectile velocity
We can estimate the velocity of ferromagnetic projectiles from the basic laws of mechanics (Figure 2.20). One non‐intuitive feature of projectile velocities is that they are broadly independent of the mass of the object. For example, if the densities of two objects are equal, then the translational force will scale with the mass, but acceleration scales with its inverse. Provided objects are the same shape (rather than size) and made of the same material, they will fly in equally fast with velocities of tens of meters per second‐ in under half a second for a given magnet!
Figure 2.20 Predicted projectile velocity for the objects and magnets in Figure 2.19 saturating at 1.6 and 0.5 T.
Torque
If the object is non‐spherical and has an axis at an angle θ with respect to B, there will be a twisting force or torque acting to align the object with the field. Torque, T, is a vector (Figure 2.21) with a magnitude
(2.14)
Figure 2.21 Torque Τ on a ferromagnetic object of length l. The force on either end is F = Τ/(0.5l) assuming a central axis of rotation.
Torque is measured in newton‐metres (N m). m is the object’s magnetic moment (= MV).
Torque on diamagnetic and paramagnetic objects
For |χm| << 1 the torque is [1]:
(2.15)
Perhaps at odds with “common sense” is the observation that the maximum torque is exerted at 45° (Figure 2.22a), but at this angle there is the greatest product of magnetization and interaction with B. The maximum torque for a long object becomes
(2.16)
Figure 2.22 Relative torque as a function of: (a) angle; (b) ratio of length‐diameter for a cylinder for unsaturated ferromagnetic objects.
As χ << 1, the torque on non‐ferromagnetic objects is very small.
Example 2.7 Torque on a weakly ferromagnetic magnetic implant
Suppose an implant of length 1 cm, diameter 0.2 cm is introduced into a 3 T scanner at 45° to the z‐axis, what is the torque and twisting force if the implant has χ = 0.01?
As χ is small, treat the implant as paramagnetic. Calculate the volume and use Equation 2.15
The torque is then
and the rotational force is
This is around one third of the force due to gravity.
MYTHBUSTER:
The maximum torque does not occur when an object is at a right angle to B0, but at 45°.
Torque on soft ferromagnetic objects
Of more significance is the torque on a soft ferromagnetic object. The torque on an unsaturated object is
(2.17)
Remarkably this is also independent of susceptibility, depending upon the square of B0 (so four times greater at 3 T compared with 1.5 T), with a maximum value within the magnet bore. Torque becomes larger for long needle‐like objects (
