50
Electromagnetic Fields in Biological Systems
in each of these figures. It can be seen from these figures that for all cases the transmitted
amplitude is the highest at the leading surface of the spherical head and that it decreases
initially with increasing distances into the model. The pulse amplitude approaches a zero
or a minimum near the center of the sphere, then increases with increasing radial distance,
and finally reaches a maximum at the trailing surface. However, the peak amplitude at the
trailing surface is slightly lower than that at the leading surface. Hence, the transmitted
pulse amplitude is asymmetric about the center of the spherical model.
Similar to planar tissue models, the transmitted pulse has both positive and negative components. In fact, the positive portion of the transmitted EMP is almost equal
to the negative portion at any given location inside the spherical head. The widths of
the positive and negative pulses are about the same as the width of the incident pulse.
Although not shown, the principal difference between a 1-μs and a 10-μs pulse is that
the transmitted EMP for the former is approximately 10 times higher than for the latter.
Therefore, the transmitted EMP amplitude is inversely proportional to the pulse width,
which is the principal distinguishing characteristic for the two Gaussian pulses, thereby
illustrating the dependence of the transmitted EMP on pulse widths.
To appreciate the magnitude of the transmitted EMPs in both humans and animals,
it is possible to consider a typical EMP with a peak electric field strength of 50 kV/m.
Figure 1.26 indicates that a peak transmitted pulse amplitude is only 4.75 V/m at the
leading surface of a sphere of radius 10 cm and the transmitted amplitude one quarter of
the way into the sphere is 2.5 V/m. These are relatively small values.
1.12.4 Constant Conductivity Spherical Model for
Electromagnetic Pulse Propagation
The imaginary part of the complex dielectric permittivity is related to the conductivity
as ε″ = σ/ωε 0 , where ε 0 is the permittivity of vacuum. Since ε′ is small compared to ε″ for
brain matter at lower frequencies, ε′ may be neglected. This is similar to neglecting the
displacement current. Furthermore, if we assume a constant conductivity for the brain,
Equation 1.60 becomes
H(r,ω) = jω[3ε 0 /σ x − r/(2c) (cos θ x − sin θ cos θ z)]
(1.61)
On substituting Equation 1.61 into Equation 1.59 and then taking the inverse Fourier
transform, the induced electric field can be calculated:
e t (r,t) = [(3ε 0 /σ) x − r/(2c)(cos θ x − sin θ cos θ z)](de 0 /dt)
(1.62)
Note that this result is in a closed form and can be evaluated exactly. It is seen that for
the case of frequency-independent σ the induced pulse inside a spherical model of the
brain is a function of the time rate of change of the incident EMP. After substituting the
explicit expression of Equation 1.57 for e 0 in Equation 1.62 and performing the indicated
differentiation, we obtain
e t (r,t) = −e 0 (0) (t/2τ 2 ) exp [−t 2 /(2τ 2 )] [(3ε 0 /σ) x − r/(2c) (cos θ x − sin θ cos θ z)] (1.63)
Electromagnetic Fields in Biological Systems
in each of these figures. It can be seen from these figures that for all cases the transmitted
amplitude is the highest at the leading surface of the spherical head and that it decreases
initially with increasing distances into the model. The pulse amplitude approaches a zero
or a minimum near the center of the sphere, then increases with increasing radial distance,
and finally reaches a maximum at the trailing surface. However, the peak amplitude at the
trailing surface is slightly lower than that at the leading surface. Hence, the transmitted
pulse amplitude is asymmetric about the center of the spherical model.
Similar to planar tissue models, the transmitted pulse has both positive and negative components. In fact, the positive portion of the transmitted EMP is almost equal
to the negative portion at any given location inside the spherical head. The widths of
the positive and negative pulses are about the same as the width of the incident pulse.
Although not shown, the principal difference between a 1-μs and a 10-μs pulse is that
the transmitted EMP for the former is approximately 10 times higher than for the latter.
Therefore, the transmitted EMP amplitude is inversely proportional to the pulse width,
which is the principal distinguishing characteristic for the two Gaussian pulses, thereby
illustrating the dependence of the transmitted EMP on pulse widths.
To appreciate the magnitude of the transmitted EMPs in both humans and animals,
it is possible to consider a typical EMP with a peak electric field strength of 50 kV/m.
Figure 1.26 indicates that a peak transmitted pulse amplitude is only 4.75 V/m at the
leading surface of a sphere of radius 10 cm and the transmitted amplitude one quarter of
the way into the sphere is 2.5 V/m. These are relatively small values.
1.12.4 Constant Conductivity Spherical Model for
Electromagnetic Pulse Propagation
The imaginary part of the complex dielectric permittivity is related to the conductivity
as ε″ = σ/ωε 0 , where ε 0 is the permittivity of vacuum. Since ε′ is small compared to ε″ for
brain matter at lower frequencies, ε′ may be neglected. This is similar to neglecting the
displacement current. Furthermore, if we assume a constant conductivity for the brain,
Equation 1.60 becomes
H(r,ω) = jω[3ε 0 /σ x − r/(2c) (cos θ x − sin θ cos θ z)]
(1.61)
On substituting Equation 1.61 into Equation 1.59 and then taking the inverse Fourier
transform, the induced electric field can be calculated:
e t (r,t) = [(3ε 0 /σ) x − r/(2c)(cos θ x − sin θ cos θ z)](de 0 /dt)
(1.62)
Note that this result is in a closed form and can be evaluated exactly. It is seen that for
the case of frequency-independent σ the induced pulse inside a spherical model of the
brain is a function of the time rate of change of the incident EMP. After substituting the
explicit expression of Equation 1.57 for e 0 in Equation 1.62 and performing the indicated
differentiation, we obtain
e t (r,t) = −e 0 (0) (t/2τ 2 ) exp [−t 2 /(2τ 2 )] [(3ε 0 /σ) x − r/(2c) (cos θ x − sin θ cos θ z)] (1.63)