Biomedical Engineering Reference
In-Depth Information
Equation 4.13b states that the electromotive force (emf) around
a closed loop is equal to the rate of change of the magnetic flux
cutting the loop. This is known as Faraday's law. Equation 4.13c
indicates that the total magnetic field flux through a closed sur-
face is zero and so magnetic field lines always form closed loops.
Equation 4.13d shows the dependence of the magnetic field on
steady current (the first term on the right-hand side) and on the
displacement current, which is related to the rate of change of
electric field (the second term), which ensures conservation of
charge. Since the total current passing out through any closed sur-
face must be equal to the total change in charge within the surface,
where
E
0
,
B
0
are amplitudes and
k
is the wave vector that is
related to the angular frequency through
k
=
ω
=
2
π
λ
.
(4.20)
c
c
is the wave velocity and is determined by
1
.
00
c
=
(4.21)
µε
4.4 Interaction of Electric and
Magnetic Fields with tissues
=−
∂ρ
∂
∫
∫
JdS
.
dV
.
(4.14)
t
S
V
Interactions between electric and magnetic fields and media
in general can be explained in terms of conduction currents,
dielectric polarization, and magnetization. Tissues are essen-
tially nonmagnetic, and so only the first two mechanisms will
be outlined here. The electric field exerts a force on charges and
results in a drift of ions that is imposed on their random thermal
motion. This gives rise to a conduction current
J
c
= σ
E
where
σ (S m
−1
) is the electrical conductivity of the tissue. The electric
field also interacts with polar molecules within tissues causing
small displacements and reorientation from their equilibrium
positions in the absence of the field. The polarization density
P
is
related to the
E
field through the electric susceptibility χ:
The differential forms of Maxwell's equations are
E
=
ρ
ε
.
(4.15a)
0
×=−
∂
∂
B
t
E
(4.15b)
∇
.
B
= 0
(4.15c)
∂
∂
E
×=µ+ε
B
J
.
(4.15d)
0
0
t
Equation 4.15a indicates that charge density is a source of elec-
tric field and that electric field lines begin and end on charges.
Equation 4.15c shows that magnetic field lines are always closed
loops. From Equations 4.15b and 4.15d it can be seen that time-
varying electric and magnetic fields are intimately related but
that static electric and magnetic fields can be considered sepa-
rately since in this case time derivatives are zero. However, even
when the time derivatives are not zero but remain small, the
approximation that electric and magnetic fields are independent
remains valid. This is known as a quasi-static approximation
and in practical terms requires that the dimensions of the prob-
lem are small compared to the wavelength.
he
E
and
B
fields also satisfy the wave equation:
PE.
=εχ
(4.22)
χ accounts for both types of polarization and is related to the
relative permittivity ε
r
through
χ=ε−
1.
(4.23)
r
The permittivity of a medium ε indicates the ease with which it
is polarized:
ε=εε=+χε
(1
).
(4.24)
r
0
0
A quantity known as the complex permittivity ε∗ is used to
account for both drift of conduction charges and polarization:
2
∂
∂
E
2
E
−ε µ
=
0
(4.16)
00
2
t
(4.25)
ε= ε− ′′
2
(
j
ε
)
∂
∂
B
2
B
−ε µ
=
.
0
(4.17)
00
t
2
The real part of this complex quantity ε′ (= ε
r
) accounts for
polarization and is a measure of the energy stored in the
medium. The imaginary part ε″ accounts for conduction and is a
measure of energy loss from the E-field to the medium. It can be
expressed in terms of the conductivity σ and angular frequency
ω as
The simplest solution is to assume a sinusoidal wave of a single
frequency:
{
}
{
}
jt
ω
jt
ω
Er
(,)
t
=
e
Er
( )
e
or
Br
(,)
t
=
e
Br
( )
e
(4.18)
and in the case of a plane travelling wave:
′ε=
σ
ωε
.
(4.26)
Ee
j
0
−
kr
.
Be
j
0
−
kr
.
Er
()
=
or
Br
()
=
(4.19)
0
