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)
(2.26b)
The induced E and J are greatest for the highest level of dB/dz, i.e. close the scanner bore entrance, and scale with velocity. This mechanism is thought to be the cause for some of the acute sensory effects experienced around high field magnets (see Chapter 3).
Example 2.10 Movement in the fringe field gradient
A staff member moves towards the magnet at 1 ms−1 in a fringe field gradient of 5 Tm−1. What is the maximum induced electric field and current density around their head?
Use Equation 2.26 with r = 0.08 m and conductivity of 0.2 Sm −1
Lenz’s law
The eddy currents induced by movement generate a magnetic field which opposes the change in magnetic flux. This is Lenz’s law, a clarification upon Faraday’s law of induction. An example of this can be observed by introducing a sheet of non‐ferromagnetic metal such as aluminium or copper into the bore of the magnet. If you position the sheet vertically and transversely (normal to B0) and then allow it to drop towards the horizontal, the flux from B0 changes as the angle to B0
