Essentials of MRI Safety. Donald W. McRobbie
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 2.26). This is analogous to the Hall effect observed in semiconductors.
(2.21)
Figure 2.26 Magneto‐hydrodynamic and Hall effect.
In terms of induced voltage or electrical potential, V, where
(2.22)
and d is the distance between charged surfaces (as in a capacitor), we have an induced voltage
(2.23)
The effect is most commonly encountered in MRI as an artefact in ECG traces.
LAWS OF INDUCTION
The laws of induction follow from Maxwell’s third equation or Faraday’s law. If we consider a wire loop within a time‐varying B‐field the magnitude of the induced E‐field is [3]
(2.24)
This applies for both the electric field induced by the imaging gradients responsible for peripheral nerve stimulation (PNS), and the electric field induced by the RF B1‐field responsible for SAR and tissue (and implant) heating. The direction of E follows a left‐hand rule, as any magnetic field produced by the induced current in the wire opposes the rate of change of flux that induced it.
Faraday induction from the gradients
Biological tissues conduct electricity by means of water and electrolytes. Rather than considering electrical current in tissue (as in wires), we consider the current densityJ, a vector (Figure 2.27)
(2.25)
Figure 2.27 Ohm’s law in a circuit and a volume conductor.
σ is the tissue conductivity in siemens per meter (S m−1). Some representative values are shown in Table 2.3.
Table 2.3 Tissue conductivity at various frequencies. Electrical properties from https://itis.swiss/virtual‐population/tissue‐properties/database after [4].
| Tissue | Conductivity (S m−1) | ||
| 10 Hz | 1 kHz | 100 MHz | |
| Bone (cortical) | 0.02 | 0.02 | 0.064 |
| Brain (WM) | 0.028 | 0.063 | 0.32 |
| Fat | 0.038 | 0.042 | 0.068 |
| Heart muscle | 0.054 | 0.11 | 0.73 |
| Liver | 0.028 | 0.041 | 0.49 |
| Muscle | 0.20 | 0.32 | 0.71 |
In practice conductivity may be anisotropic, e.g. along a muscle fiber as opposed to across it; or, at radio frequencies, it may be complex with real and imaginary components. For now we shall assume the simplest situation: isotropic, non‐complex but frequency dependent. Human anatomy, with irregular shapes and differing tissue conductivities, will exhibit much more complex behavior, with E‐field lines and current loops being altered by tissue boundaries and electrostatic charges induced on these boundaries according to Gauss’s Law.
Induced fields from movement within the static fringe field gradient
Movement through the static fringe field gradient dB/dz exposes tissue to a changing magnetic flux, and hence induces an electric field and current density. Restricting this discussion to the z‐direction only
(2.26a)