2.22b). For a saturated object the torque becomes proportional to Bsat 2 and has a maximum value (Figure 2.23)
(2.18)
Figure 2.23 Predicted torque for ferromagnetic 0.1 kg objects of varying l/d ratios, with density 8000 kg m−3, susceptibility of 1000 and Bsat = 2 T on the axis of a 1.5 T shielded magnet. The bore entrance is at 0.8 m.
Usually an object will twist long before it reaches the bore, and hence the torque rapidly disappears, aligning the object for maximum projectile velocity. There is the possibility of serious injury if an implant is ferromagnetic.
MYTHBUSTER:
Torque on strongly ferromagnetic objects is independent of their magnetic susceptibility.
Example 2.8 Torque on a ferromagnetic cylinder
You can use Figure 2.22 and Equation 2.18 to calculate the torque on a ferromagnetic object which saturates at 1 T. Suppose a cylinder is at 30° to B0, and is 50 cm long, 10 cm diameter.
From the figure the multiplier for the angle is 0.43 and for the l/d ratio is 5. The volume is
The torque is then
The force exerted on the ends of the object is given by the torque divided by half the length = 2057 N.
Torque v translational force
A question of some significance, especially for implants, is which is stronger, the translational or the twisting force? Figure 2.24 shows the attractive and twisting force (from the torque) for ferromagnetic objects (mass 0.1 kg, density of 8000 kg m−3, χ = 1000) approaching a 3 T scanner. We assume that the orientation is such to produce either the maximum torque or translation (note: that these conditions are inconsistent with each other). What is quite surprising is that, in this simulation, the force exerted from twisting is greater than that from translation at most locations. This is bad news for MR safety as the first thing to occur will be the twisting to align with the field direction which is then the optimum orientation for projectiles. Also, for ferromagnetic implants, unless they are perfectly spherical (in which case the torque will be zero) or perfectly aligned with B0, the twisting forces will persist for the whole time spent in the bore of the magnet. In the figure the forces on a 5 cm long needle (l/d = 50) are shown, where the twisting can greatly exceed the projectile force. Although the attractive force on the needle appears low, remember that items tend to accelerate into the scanner with high velocities, so this situation is very dangerous.
Figure 2.24 Predicted maximum twisting force (solid lines) and translational (dashed lines) from ferromagnetic 0.1 kg cylinders of varying l/d ratios and a 5 cm needle with l/d = 50, with densities 8000 kg m−3, χ = 1000 and Bsat = 2 T on the axis of a 1.5 T shielded magnet. The lengths of the cylinders are 11.6, 7.5, 5, and 2.5 cm.
Forces on circuits
We saw in Chapter 1 that the tesla was defined in terms of a force upon a moving charge. A moving charge is an electrical current, so it is no surprise that magnetic fields exert a force on a current‐carrying conductor.
Force on a straight conductor
The force on a current‐carrying conductor at an angle θ to B0 is (Figure 2.25a):
(2.19)
Figure 2.25 The Lorentz force on a straight conductor with (a) B into the page; (b) Fleming’s left‐hand rule scheme to deduce the direction of the force.
This follows Fleming’s left‐hand rule: if your forefinger points along B0, your other fingers show the direction of current flow, then your thumb indicates the direction of the force (Figure 2.25b).
Example 2.9 Force on an electrical wire
What is the force on a 10 cm length of wire at an angle 45° to a B0 of 1.5 T and carrying 10 A?
If the wire weighs 10 g, then this force is ten times the gravitational force on the wire.
LORENTZ AND HYDRODYNAMIC FORCES
Moving charges are subject to an additional force, the Lorentz force. Charge moving within an external magnetic field produces an electric field by the hydrodynamic or Hall effect.
Lorentz force
The magnitude of the Lorentz force on a charge Q possessing velocity v is given as
(2.20)
The direction of the force can be determined by Fleming’s left‐hand rule.
Magneto‐hydrodynamic effect
A similar effect is the generation of an electric field E by the flow of charge within an external magnetic field (Figure
