Notes
1 1 In type 1 superconductors the magnetic field inside the material is zero due to the Meissner Effect.
2 2 This is not true in tissue. See Chapter 2, page 54.
3 3 Using complex notation where, the operator eiωt signifies circular motion.
2 Let’s get physical: fields and forces
BASIC LAWS OF MAGNETISM
The fundamental laws of magnetism were summarized by Scottish physicist James Clerk Maxwell in four equations. These equations are not for the faint‐hearted nor for the mathematically challenged, but if you aspire to be an expert in MRI safety, then you should have a good understanding of their consequences. By comparison, if you did not understand Newton’s laws of gravitation or Einstein’s theory of relativity you would not become a rocket scientist. Maxwell’s equations underpin everything in electromagnetism: the biological effects of EM fields, interactions with implants, electromagnetic modeling of field exposures and specific absorption rate (SAR), projectiles and magnet safety, magnetic shielding, fringe field gradients, and acoustic noise. A full understanding requires some knowledge of vector calculus and differential equations (see Appendix 2) but for now we will not need this. Those aspiring to be MR Safety Experts should read this chapter in conjunction with Appendix 1.
Understanding Maxwell’s Equations
Maxwell’s equations are given in Appendix 1. Here we describe their main consequences for MRI safety.
Electrical charge and electric fields
Gauss’s Law (Maxwell’s first equation) describes how electrical charges produce static electric fields E. Electric fields start at a positive charge and are directed towards their conclusion at negative charges (Figure 2.1). We are not going to use Gauss’s Law much, although it has relevance in minimizing unwanted electric fields in coil design, and at some tissue boundaries where charge may accumulate.
Figure 2.1 Electric field lines begin at a source of positive charge and terminate at a negative charge: (a) single point positive charge; (b) positive and negative point charges; (c) capacitor with a potential difference V between the plates.
Magnetic fields
Maxwell’s second equation states that the “divergence of B is zero.” This means that there is no magnetic equivalent of electrical charge – no “magnetic monopoles”. Magnetic sources are not like electrostatic ones, but exist as dipoles with a north and south pole (just like the Earth). Magnetic field lines have no beginning or end, but form complete loops from north pole to south (Figures 2.2, 1.23). The nature of the B0 fringe field depends upon this.
Figure 2.2 Magnetic field lines from a permanent bar magnet.
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