Electromagnetic induction
Maxwell’s third equation is also known as Faraday’s Law of Induction. We have met dB/dt already in Chapter 1, so clearly this equation is going to have significant implications for us. It states that a time‐varying magnetic field induces an electric field; also, that the electric field lines form complete loops unlike static electric fields (Figure 2.3). The induced electric field is sometimes called “conservative” as it involves no external static charges. Faraday’s Law is also responsible for the detection of the MR signal in an RF receive coil – so it’s important!
Figure 2.3 Electric fields induced by a time‐varying magnetic field form complete loops (unless there are static electrical charges present); dB/dt is into the page.
Electromagnetic waves
Maxwell’s fourth equation, or Ampere’s Law, tells us that magnetic fields can be generated both by electric currents and by time‐varying electric fields, allowing for the existence of electro‐magnetic waves – everything in the electromagnetic spectrum: gamma rays, X‐rays, ultraviolet, visible light, infrared, microwaves, and radiowaves (Figure 2.4). It has consequences for the more “wave‐like” behavior of the B1 excitation field at higher frequencies. It also results in field exposures from the gradients being higher than intuitively anticipated.
Figure 2.4 Electromagnetic wave: the magnetic and electric fields are orthogonal to each other and to the direction of propagation.
Generating magnetic fields
Maxwell’s equations teach us that a magnetic field (we shall drop the proper term “flux density”) is generated by an electrical current. In this section we consider the generation of magnetic fields from conductors and coils in various simple configurations. Further detail is given in Appendix 1.
B field from a long straight conductor
If we have a straight wire and pass a current I along it, then the magnetic field generated will have circular field lines (Figure 2.5). The direction of the field lines can be determined by the “right hand rule”, namely that if your right hand’s thumb represents the direction of current flow, then your cupped fingers will indicate the circular B field direction, denoted Bθ. The magnitude of the field at a radial distance r from the wire is proportional to 1/r. The subscript θ (from polar coordinates‐ 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
