A familiarity with basic electromagnetic properties will assist the reader of this text. A detailed introduction to electromagnetism is beyond the scope of this appendix, but a handful of basic principles and concepts pertinent to MRI are presented, prima facie, in this appendix. The electricity and magnetism sections in introductory college physics texts for engineers and scientists are adequate background for the discussion below.
Maxwell’s Equations All electromagnetic fields are found to obey Maxwell’s equations
(a) ∇ · B =0 (b) ∇ · D = ρq (no magnetic charges)
(Gauss’s law) (c) ∇ × E + ∂B ∂t =0 (d) ∇ × H = J + ∂D ∂t
(Faraday’s law) (Amp`ere’s law) where
D = E B = μH
with dielectric constant and permeability μ. The electric charge density is ρq and the electric current density (current per unit area) is J.
In theory, all electromagnetic principles applied in this text can be directly derived from these equations but, in practice, it is much easier to work with particular forms and limits of some of which will be presented below. Integral Forms of Maxwell’s Equations Maxwell’s equations can also be written in integral form, instead of derivative form.
For constant and μ, we have (a) B · dA = 0 no magnetic charges (b) S E · dA = V d3xρq(x) = q Gauss’s law (c) E · d = − d dt B · dS Faraday’s law (d) B · d = μI + μ d dt E · dS Amp`ere’s law.
A key point of Maxwell’s equations is that time-varying magnetic fields give rise to electric fields, and vice versa.
This is the basis of the detection of the MRI signal. Another basic notion relevant to MR discussions is that currents (or current densities J) produce magnetic fields. In MRI, the static field is by definition time independent, the gradient field has mild time dependence, and the rf field is in a relatively low frequency band. Therefore, the time dependence in general can be factored out, and the spatial dependence determined by static methods. Faraday’s Law of Induction Faraday’s law of induction is
E · d = − d dt B · dS
Defining the magnetic flux through a surface to be
ΦB = B · dS
and considering the case where the surface is bounded by a wire loop, the electromotive force (emf) or voltage induced across the ends of the wire is given by
emf = E · d (A.6) When (A.5) and (A.6) are substituted into a more familiar form of Faraday’s law emf = −dΦB dt (A.7) is found.
Faraday’s law states that a time-varying magnetic flux through a loop of wire, or any circuit, generates a voltage whose amplitude is proportional to the (negative of the) time rate of change of the flux. The time-varying magnetic fields associated with the precessing magnetization in MRI induce voltages in the MRI rf receive coil, giving rise to the MRI signal . Electromagnetic Forces Individual electric charges immersed in an electric field or moving through a magnetic field experience electromagnetic forces. The electric force on a charged particle is proportional to, and along the direction of, the electric field. The force generated by a magnetic field on a charged particle is in the direction determined by the right-hand rule perpendicular to the plane defined by the particle’s velocity and the direction of the magnetic field. These relations are quantified by the respective Lorentz force and force density equations
F = qE + qv × B F = ρE + J × B (A.8)
While a charge at rest in a purely magnetic field experiences no force, a moving charge is deflected by a force perpendicular to its motion. Information about B and the charge-to-mass ratio can be determined by measuring such deflections. Magnetic Charges There is, at present, no physical evidence for the existence of individual magnetic charges (magnetic monopoles). However, the concept of a magnetic charge can be useful, especially when studying magnetic dipoles which can be modeled as two magnetic charges separated by a fixed distance. The forces on a magnetic charge qm would be given by
F = qmB − qmv × E
Dipoles in an Electromagnetic Field The electric dipole moment vector d for a system of two opposite charges ±q separated by a fixed distance
d is d = qdnˆ
where ˆn points from the negative to the positive charge along the line connecting the two charges. Analogously, the magnetic dipole moment vector μm corresponding to two magnetic charges ±qm in place of the electric charges is given by
μm = qmdnˆ
In considering the behavior of a magnetic dipole in a magnetic field (the situation of primary interest in MRI), it is useful to continue to keep in mind the analogous behavior of an electric dipole in an electric field. In a spatially uniform magnetic field, the forces on either charge in a dipole would be equal and opposite so there will be no net force on the dipole. However, there is a net torque on the dipole
N = μm × B Alternatively, if a dipole is placed in a spatially varying field, the force on the two charges are different, giving rise to a net force
F = ∇ (μm · B )
From a potential energy associated with an arbitrary orientation of a dipole in a magnetic field may be identified as
U = −μm
That is, the forces that apply torque to a dipole in a magnetic field, lead to a preferential (lower potential energy) orientation of the dipole parallel to the applied field. Magnetic Moment Defined by Current Loop A classical magnetic dipole moment μm may also be defined in terms of a current loop. The magnetic moment μm associated with a circular loop with radius a carrying a current I is given by
μm = IA = πa2 Inˆ
where ˆn is the normal to the coil. While a current loop and a magnetic charge dipole have fundamentally different internal fields, they have a similar interaction with an external magnetic field through their magnetic dipole moments. Formulas for Electromagnetic Energy In the absence of constraints, groups of electrical charges arrange themselves into a lowest energy state. One way of describing this is that the positive and negative charges move to positions where the electric field is minimized in some way over the volume of the system. Indeed, the electromagnetic energy in free space may be directly associated with the fields according to
U = 1 2μ0 V d3 x B 2 + 0 2 V d3 x E 2
The magnetic field energy in (A.16) may be recast in terms of familiar lumped circuit inductances, which is a form useful for a set of current carrying wires and is given by
U = 1 2 ⎛ ⎝ N i=1 LI2 i + N j=i N i=1 MijIiIj ⎞ ⎠
where Li is the self-inductance of each wire and their mutual inductances, Mij , are
Mij = μ0 4π di · dj 1 |xj − xi| (A.18)
Static Magnetic Field Calculations The Maxwell equation ∇ · B = 0 and the vector identity
∇ · (∇ × A) = 0 (A.19) imply that B can be written as the curl of a magnetic vector potential B = ∇ × A
The remaining freedom in choosing A (a ‘gauge’ freedom) allows an additional condition. If it is assumed that
∇ · A = 0 (A.21) then ∇× (∇× A) = ∇ (∇· A)−∇2A = −∇2A.
In the case of time-independent fields, (A.1d) can be rewritten as
∇2 A = −μ0J (A.22)
The solution of this equation can be shown to be
A(x) = μ0 4π volume d3 x J(x ) |x − x | (A.23) = μ0 4π loops i Ii di |x − xi| with the latter form appropriate for a current distribution J corresponding to a sum over a set of discrete current loops.
The Biot-Savart law for the magnetic field,
B (x) = μ0 4π volume d3 x J(x ) × (x − x ) |x − x | 3 (A.25) = μ0 4π
loops i Ii di × (x − xi) |x − xi| 3
These specifically relate time-independent electric currents to the magnetic fields they produce. Historically, such results were found by Biot and Savart from experiment. As the second formula indicates, they found that the differential magnetic field produced by a current element Iidi is perpendicular to both the direction of the current element and the direction from the current element to the observation point.
Taken from Magnetic Resonance Imaging: Physical Principles and Sequence Design Second Edition.
Published 16 May 2014