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OCR for page 157
Appendix B
Calculations of Induced Electric Fields
INTRODUCTION
THIS APPENDW PROVIDES calculations on the relative strengths of induced
electric fields in various biota exposed to 76-Hz electric and magnetic fields.
THEORETICAL MODEL
A number of investigations have used tissue-equivalent spheroidal models
as an index of induced field. The spheroidal mode! is attractive because sim-
ple expressions can be obtained for all body sizes. The wavelength at 76 Hz
is very large, compared with the longest dimension of the body, so the quasi-
static field theory can be appropriately applied to calculate the induced electric
field in the body (Michaelson and Lin 1987~. For uniform external electric
and magnetic fields, the magnitude of the induced electric field inside a homo-
geneous dielectric tissue sphere resulting from the applied electric field is
Ee = 3/cEo
(Bar)
and the peak magnitude of the induced electric field resulting from the applied
magnetic field is
Em = ~a/BO
157
(B-2)
OCR for page 158
~ 58 EVALUATION OF ELF ECOLOGICAL MONITORING PROGRAM
where ~ is the dielectric permittivity, a is the radius, f is the frequency, Eo is
the imposed or applied electric-f~eld strength, and Be is the magnetic flux
density. The uniform external electric field gives rise to a constant induced
electric field inside the dielectric sphere that has the same direction but is
reduced by 3/e from the applied electric field for the biologic object and is
independent of body size. The magnetically induced electric field produces an
internal electric field that varies directly with the radius of the spherical body
and is proportional to the source frequency.
For some species, a prolate spheroidal mode! approximates more closely
their elongated bodies. The magnitude of the electric field induced inside a
homogeneous dielectric spheroid with semimajor axis a and semiminor axis
b by a uniform applied electric field oriented along the semimajor axis is
Bee = Eo/C~
and for an electric field oriented along the semi-minor axis of the body is
Ehe = -Eo/C2.
(B-3)
(B-4)
Similarly, the peak magnitude of the electric field induced by a uniform mag-
netic field oriented along the semimajor axis is
Eem = to
and for a magnetic field oriented along the semiminor axis of the body is
Ehm = Prado.
(B-S)
(B-6)
Because Cat and C2 are constants, the induced fields are uniform. However,
they are dependent on the orientation of the applied electric and magnetic
fields with respect to the major axis of the body. in particular, because a >
b, the higher induced field is associated with an applied magnetic field ori-
ented along the minor axis of the body.
For both spheroidal models, the electrically induced current is in the
direction of the applied field and is uniform. The magnetically induced cur-
rent is a circulating current with an amplitude of zero at the center of the body
and increases with distance from the center. In all cases, the electrically
induced field is uniform, but the magnetically induced field increases with
increasing size of the subject, such as the average radius or longest dimension
of the body.
OCR for page 159
CALCULATIONS OF INDUCED ELECTRIC FIELDS
NUMERICAL CALCULATIONS
159
To provide an index of induced electric fields in biota and a guide to the
extrapolation of data from the ecological monitoring program to other experi-
mental subjects, the committee has made numerical calculations of induced
electric field as a function of size (! mg to 500 g), using spheres to approxi-
mate the shapes of insects, birds, and small vertebrates (see Equations B-! and
B-2 above and data given In Polk and Postow 19861. In addition, an elongated
prolate spheroid (see Equations B-3 through B-6 above and data given in
Nelson 1991) is used to mode! upland hardwood-tree stands whose average
diameter is 15-25 cm and average height is 10-20 m. The exposure parame-
ters considered are applied electric fields of 10-5,000 mV/m and applied
magnetic fields of 1.0-50 mG. Results are shown in Figures B-! through B-3.
60
50
40
._
IL
.O
~ x
a, ~
IL.I °
·C) E
a, ~ 20
1 0
o
m-SOOg ~
it/
1
0 10 20 30 40 50
So (mG)
Applied Magnetic Field
FIGURE B-l Magnetically induced electric field in insects and birds and
other vertebrates.
OCR for page 160
60 EVALUATION OF ELF ECOLOGICAL MONITORING PROGRAM
INSECTS AND BIRDS AND
OTHER SMALL VERTEBRATES
it is noted that electrically induced fields are the same for all body sizes
and are proportional to the strength of the applied electric fields. For the
parameters considered, the values arelower than 16 x 10~5 mV/m and are less
than one-millionth of the applied electric fields.
As shown in Figure B-l, the magnetically induced electric field varies
with both body size and magnetic field. For a lOO-mg insect, the maximal
induced electric field can vary from 6.9 x lo-s mV/m at 1.0 mG to 3.4 x
i0~3 mV/m at 50 mG. Likewise, for a 100-g bird or other vertebrate, the
highest induced electric field varies from 6.9 x 10~4 mV/m at I.0 mG to 3.4
x 10~2mV/mat50mG.
30
a, 20
._
c' _
E._
_ _
~ >
Q. E
us
~0
c
o
1
a-25 m
~ED m
0 10 20 30 40 50
So (mG)
Applied Magnetic Field
FIGURE B 2 Vertical magnetic-field-induced electric field in hardwood-tree
stand.
OCR for page 161
CALCULATIONS OF INDUCED ELECTRIC FIELDS
HARDWOOD-TREE STANDS
161
The induced electric field resulting from an applied electric field oriented
vertically along the height (major axis) of a hardwood stand is independent of
stand size. Because the vertical electric field is tangential to the major dimen-
sion of the tree stand, the induced and applied electric fields are the same (10-
5,000 mV/m).
Although the induced electric field resulting from an applied electric field
oriented horizontally along the width (minor axis) of the tree stand is also
independent of stand size, the values are drastically reduced and vary from I.6
x lo-2 to S. 19 mV/m for applied fields of 10-5,000 mV/m.
The results of a vertically oriented magnetic field are shown in Figure B-
2. Induced electric fields depend both on the width of the tree stand and on
the magnitude of the applied magnetic field. For a lO-cm width, the induced
electric fields resulting from applied magnetic fields of 10 and 50 mG are I.2
al ~ O
._
IL
-
E
c, -
._
~ E
c' ~
a., x
N
lo
~ ~ 5
Cal ~
o
FIGURE B 3
tree stand.
/
~--1
~20 call
0 10 20 30 40 50
Bo (mG)
Applied Magnetic Field
Horizontal magnetic-field-induced electric field in hardwood
OCR for page 162
62 EVALUATION OF ELF ECOLOGICAL MONITORING PROGRAM
x lo-2 and 6.0 x lo-2 mV/m, respectively. For these magnetic-field magni-
tudes, the induced electric fields in a 25-cm tree stand are 3.0 x lo-2 and
0.15 mV/m, respectively.
Tf the applied magnetic field is oriented horizontally along the minor axis
of the tree stand, the induced electric field will be proportional to the height
of the tree stand and the strength of the magnetic field. The electric fields
induced by magnetic fields of t.0 and 50 mG in a tree 10 m tall are 0.24 and
tI.9 mV/m, respectively, and in a 25-m tree are 0.6 and 29.8 mV/m, respec-
tively (see Figure Bob.
SUMMARY
in summary, because of small size, the calculated 76-Hz electric fields
induced in insects, birds, and small vertebrates by electric fields of up to
5,000 mV/m and magnetic fields of up to 50 mG are fairly low. In contrast,
electric fields induced by the same electric and magnetic fields in large
hardwood-tree stands could be substantial. Calculations based on these simple
models suggest that the electric field induced in a 25-m tree by a vertically
oriented electric field could be as high as 5,000 mV/m and that induced by a
horizontally oriented magnetic field could be as high as 29.S mV/m. It is
emphasized that because of shielding and other phenomena, the applied or
impinging electric field would decrease in strength with distance from the
antenna wire and as a function of the landscape. However, magnetic-field
strength would remain unattenuated by its environment and would decrease in
strength only with distance from the antenna wire because magnetic permeabil-
ity remains unchanged. Therefore, at greater distances from the antenna, the
electric field induced in tree stands by a horizontal magnetic field could be-
come a dominant factor.
Representative terms from entire chapter:
induced electric
