Fields Induced in Objects by Incident B Fields in Free Space

Figure 2.29 shows the calculated internal E in a simple two-dimensional model of a con-ducting prolate spheroid placed in a 60 Hz sinusoidally time-varying and spatially uni-form B field in free space. The B field in this case is directed out of the paper (the B field is parallel to the minor [short] axis of the spheroid, which is also directed out of the paper). As explained in Section 1.4, the time-varying B field produces E fields that encircle the B field.

A close-up view of the E fields in the spheroid is shown in Figure 2.30. When the B field is parallel to the major (long) axis of the spheroid, the induced E field pattern is as shown in Figure 2.31, which shows one cross section of the spheroid, with the B field directed out of the paper. A view with a finer grid is shown in Figure 2.32. The pattern is similar in each cross section of the spheroid perpendicular to the major axis. As in the other orientation of the object with respect to the B field, the induced E field encircles the B field. Although the rectangular mathematical cells of the model do not represent the smooth boundaries of a prolate spheroid very well, the pattern of induced E fields is approximately the same as that for an actual prolate spheroid.

Figure 2.33 shows the results of three-dimensional analytical calculations for the E fields induced in a prolate spheroid by a sinusoidally time-varying, spatially uniform B field.

When the B field is parallel to the major axis of the spheroid, as shown in Figure 2.33(a), the induced E field in any cross section perpendicular to the major axis is given by ωBr/2, where r is the distance from the spheroid major axis to the point at which the E field is evaluated, and ω is the radian frequency of the B field. This shows that the higher the

B The direction of B is out of the paper

Figure 2.29

Calculated E fields in a cross section of a two-dimensional model of a prolate spheroid in free space exposed to a uniform 60 Hz B field perpendicular to the major axis of the spheroid (i.e., B is directed out of the paper).

The fields were calculated on a finer grid and displayed on a coarser grid to show more clearly the overall field pattern.

radian frequency ω, the more electric field that is produced. Also, the farther from the axis, the greater the electric field. The maximum E field induced by a B field of 1 mT at a frequency of 60 Hz (ω = 2π × 60) is 25.8 mV/m, located at the outer surface of the spheroid.

The induced E encircles the applied B

Figure 2.31

Calculated E fields in a cross section of a two-dimensional model of a prolate spheroid in free space exposed to a uniform 60 Hz B field perpendicular to the minor axis of the spheroid (i.e., B is directed out of the paper).

The fields were calculated on a finer grid and displayed on a coarser grid to show more clearly the overall field pattern. The E encircles the applied B and is stronger near the outside of the object.

The induced E encircles the applied B

Figure 2.30

The E field pattern in the left half of the spheroid in Figure 2.29, shown as calculated on the finer grid.

When the B field is parallel to the minor axis of the spheroid (Figure 2.33(b)), the maximum induced E field, located at the outer surface of the spheroid, is almost twice that value. The difference can be explained in terms of the cross-sectional area intercepted by the B field in each case. In Figure 2.33(a), the cross-sectional area intercepted by B is considerably less than that intercepted by the B field in Figure 2.33(b). This is an illustration of a general behavior: the E fields induced in a body by a spatially uniform B field are generally greater when the cross-sectional area intercepted by B is greater, and are found near the outer periphery of the body.

The E fields induced in a coarse two-dimensional model of an animal in free space exposed to a 60 Hz, spatially uniform B field are shown in Figure 2.34. Again, the E fields tend to circle around the applied B field, which is directed out of the paper. They are gen-erally larger in the air surrounding the conducting tissue of the model than they are in the tissue itself. The E fields also tend to be small near the center of the system and larger around the outside, as in the spheroidal models. Figure 2.35 shows the E fields inside the model only (plotted to a different scale), where the circulating pattern is more obvious.

It is interesting to note that the E fields tend to circulate around the center of the trunk, but also to a lesser extent around the center of the head and the center of the legs. The cir-culation around the center of the left leg is shown more clearly in Figure 2.36, which shows just the left leg of Figure 2.34, but still attached to the whole animal. Figure 2.37 shows the E fields in a leg that has been detached from the rest of the animal but exposed to the same B field, plotted to the same scale as in Figure 2.36. Although the fields at the very top of the detached leg are different from those of the attached leg, the fields at the bottom of the detached leg differ from those in the attached leg by less than one-half of 1%. This

The induced E encircles the applied B

Figure 2.32

A close-up view of the fields in Figure 2.31, showing the fields calculated on the finer grid.

comparison indicates that the fields tend to circulate around the center of the attached leg as though it were a separate entity, and only in the region where the leg is attached are the fields significantly different from those of a detached leg. This effect becomes more pro-nounced as the leg becomes longer and thinner, and less propro-nounced as the leg becomes shorter and fatter (and blends more into the body as a whole).

Figure 2.38 shows a close-up of just the E fields in the attached head and neck of the model of Figure 2.34. The pattern very clearly shows how the fields are circulating around the center of the head, almost as if it were detached. Only the fields near the neck are significantly different from those of a detached head. This effect becomes more pronounced as the area of the appendage becomes a greater portion of the entire area.

POWEr LInES AnD PEOPLE

Whether or not living under or near power lines causes cancer (particularly leukemia and some types of brain tumors) continues to be a question for international debate and research. The late 1980s brought a number of epidemiological studies from the United States and Europe that showed statistical links between childhood leuke-mia and proximity to power lines. Later European studies showed weaker or non-existent links, particularly when normalizing for the distance from roadways (car exhaust is a known carcinogen). At power line frequencies (continued on next page)

2a

B_inc

Binc

E_int = ωBr/2 E_int = 25.8 mV/m max

E_int = 51 mV/m max E_int

E_int 2b

(a) (b)

The maximum E_int is larger when the cross-sectional area intercepted by the applied B is larger

Figure 2.33

Comparison of the internal fields in two spheroids, (a) with the incident B parallel to the long axis of the spher-oid, and (b) with the incident B perpendicular to the long axis of the spheroid. In both cases the incident B is 1 mT, the conductivity is 0.067 S/m, the frequency is 60 Hz, a = 0.875 m, and b = 0.138 m. The internal E fields were calculated in three dimensions using a long-wavelength approximation to Maxwell’s equations. The E fields are stronger when the cross-sectional area that B passes through is larger (case b).

(60 Hz in the United States and 50 Hz elsewhere) the electric and magnetic fields are decoupled. They do not generate each other and can be evaluated independently.

The magnetic field from a power line encircles the line (according to the right-hand rule described in Section 1.3). If a person is standing near the power line, this can be evaluated as a frontally incident magnetic field. This field will generate circulating electric fields within the body. However, biological effects from the magnetic field have generally not been implicated in the debate, and attention has focused on the electric field and the associated currents within the body.

The electric field goes from the power line to the ground and is vertically polar-ized with respect to a standing person under the power line. This field enters the body through the head and shoulders, passes through the torso, and exits through the feet and legs, as shown in Figure 2.39. If the person is isolated from the ground (wearing tennis shoes and standing on a dry surface, for instance), the field tends to exit uniformly from the legs, as illustrated in Figure 2.39(a), but if the person is grounded (wearing leather-soled shoes and standing in wet grass), the current passes out mainly through the bottoms of the feet, as shown in Figure 2.39(b).

Using the finite-difference time-domain method described in Chapter 5, the peak current density in the body is found to be in the ankle and knee, as indicated in Figure 2.39(c). This is expected, because both of these regions are relatively bony (bone does not conduct much) with very little conductive material in their cross sec-tion. The current passing down through these regions is concentrated in the rela-tively small regions of surrounding muscle and fat, giving large current density. A less anticipated result was that there are regions of large current density in the torso also. This is because the lungs are also not very conductive, and the current must flow through the outer region of the torso, mainly in the muscle regions. The muscles of the back, directly behind the lungs, have several regions with large current density because of this concentration of the current.

Epidemiological and bioeffect research continues today. The bioeffect research often uses laboratory animals instead of humans. The current density in these ani-mals is obviously going to be very different than in a human because of their differ-ence in size and orientation. For a 1 µT, 60 Hz magnetic field, for instance, a human will have a calculated average current density of 1.3 to 1.9 µA/m², while a rat will have 0.3 µA/m² and a mouse 0.12 µA/m². Under these conditions, the maximum cal-culated current density for the human is 8 µA/m², the rat 1.3 µA/m², and the mouse 0.4 µA/m². This research relies on dosimetry (Chapter 5) to determine the relative doses for different exposure conditions and different animals.