Essentials of MRI Safety. Donald W. McRobbie
see Appendix 2) indicates that the field lines form circular paths around the wire.
Figure 2.5 Magnetic field lines from a long straight conductor carrying current I. The direction of the lines follows the right‐hand rule.
B field from loop conductors
The field on the z‐axis from a circular loop of radius a is directed along the z‐direction and is proportional to the current I. At the centre of the loop B is proportional to 1/a, so the field generated depends upon the radius of the coil. At long distances from the loop, it acts like a magnetic dipole. On‐axis the field only has a Bz component with a 1/z3 dependence. This is often cited to represent the dropping off of the B0 fringe field, but modern shielded MRI magnets do not exactly follow this behavior; they are not simple dipoles. Nevertheless, the 1/z 3 dependence serves as a useful approximation of the nature of the fringe field. The spatial gradient from a dipole varies with 1/z4 along the z‐axis. This is a very rapid decrease with distance from the iso‐centre and is intensely significant for projectile safety. The magnitude of Bz for a long straight wire, a loop and a dipole are plotted in Figure 2.6.
Figure 2.6 Relative magnitude of Bz along the z‐axis for a long straight wire, simple loop, magnetic dipole, solenoid, and simulated self‐shielded magnet with radii of 0.4 m. The iso‐centre is at z = 0.
B field from a solenoidal coil
The field generated at the centre of a solenoid of length d with windings of density N turns per metre ‐ this now looks more like a MR superconducting magnet‐ is
(2.1)
θ is the angle measured from the vertical at iso‐centre to the end of the solenoid (Figure 2.7). For very long solenoids θ tends to 90° (π/2) and the field is
(2.2)
Figure 2.7 Solenoid coil showing the angle θ. For a very long solenoid θ →90°.
This is where the definition of magnetic field strength or intensity H in A m−1 (from Chapter 1) comes in, as
(2.3)
B field from a shielded MRI magnet
The design of a MR magnet (at least for MR safety purposes) can be approximately simulated by two concentric solenoids: the inner one representing the main coil; the outer shield coil has current flowing in the opposite direction. This enables the fringe field to be significantly reduced in extent (Figure 2.8). It also results in a stronger spatial gradient of B0, dB/dz, close to the bore entrance, but weaker at greater distance. This poses a significantly increased hazard, as the projectile force on a ferromagnetic object may suddenly increase as you approach the bore entrance – and you only notice when it’s too late. Both B and dB/dz drop off with distance more rapidly than for a dipole (FIGURE 2.6).
Figure 2.8 B and dB/dt along the axis of a simulated shielded and unshielded 3 T MR magnet. Distances are from the iso‐centre, with the bore entrance at 0.8m. Simulated data for illustration only.
Spatial dependence of magnetic fields
Only the simplest coil geometries can be solved exactly with algebra. A generalized method of computing is given by the Biot‐Savart Law (see Appendix 1). Magnet and gradient coil designers use this to numerically compute the spatial responses of B0, Gx,y,z and B1 fields. It is also used in computer modeling of induced fields in tissue.
MAGNETIC MATERIALS
The previous section showed how a B‐field can be generated in free space or air. Now we consider how materials or physical media respond to an external magnetic field. At the atomic level the electrons in their shells orbiting the nucleus have intrinsic magnetic moments. In most atoms the magnetic moments from the electrons’ spin and orbital motion cancel. This is diamagnetism, the default “non‐magnetic” state. If the cancellation is incomplete then the material is paramagnetic. In ferromagnetic 1 materials, such as iron or steel, the electron spins become aligned in large groups or domains and their effect is significantly greater. Figure 2.9 shows the magnetic susceptibility spectrum covering a range of materials [1]. This is an enormous range extending over ten orders of magnitude (1010); each step in the chart represents a factor of ten.
Figure 2.9 Magnetic susceptibility spectrum.
When an object is placed in an external magnetic field, it becomes magnetized. Each of the types of material: dia‐, para‐, and ferromagnetic behave differently in the field, but because of Maxwell’s equations, the underlying physics is similar. In an external field, the magnetization of the material M (a vector) is
(2.4)
H is the magnetic field strength (A m−1) and χ is the magnetic susceptibility which is dimensionless2.