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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)
~ 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.
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.
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.
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
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.