Fields Induced in Objects by Incident E Fields in Free Space

Figure 2.2 shows the calculated internal E in a section of a simple two-dimensional model of a dielectric prolate spheroid placed in the uniform E (similar to that shown in Figure 1.3) that existed between two metallic plates before the object was inserted. (A prolate spheroid has the shape of an egg.) When the spacing between the plates is very large compared to the length of the body, this configuration produces approximately the same field pattern that occurs when the body is placed in a uniform E field in free space. In this numeri-cal numeri-calculation (a finite-difference frequency-domain solution of Maxwell’s equations), we

E_inc

The internal and scattered E and B fields due to (a) an incident E alone and (b) an incident B alone. The inter-nal and scattered fields are generally not the same in (a) and (b).

did not make the spacing between the plates larger because that requires more computer memory, but the characteristics of the field patterns are approximately the same as those for a prolate spheroid placed in a uniform E in free space.

In Figure 2.2, the relative permittivity (see Section 1.6) of the prolate spheroid is εr = 2, which is a comparatively small value, since the relative permittivity of free space is 1.

We used this comparatively small value of relative permittivity for illustrative purposes, because larger values make the internal fields too small to be seen. The presence of the dielectric object perturbs the originally uniform E field in a way that can be thought of as the dielectric pulling in the E field vectors. In smooth prolate spheroids, the magnitude of the E field inside is uniform in space, but in this model, in which the outer boundary is stair-stepped (a consequence of the numerical calculation method), E is only approximately uniform because of the stair stepping. The E vectors in Figure 2.2 were calculated on a finer grid, but displayed on a coarser grid to show more clearly the overall field pattern.

The dielectric spheroid

“pulls in” the originally uniform E-field lines

A spheroid of _εr = 2 in free space between two metallic plates

Figure 2.2

Calculated E fields in a two-dimensional model of a prolate spheroid between two metallic plates, approximat-ing a spheroid placed in a uniform E field. The E fields were calculated on a finer grid but displayed on a coarser grid to show more clearly the overall field pattern. (The fields on the finer grid are shown in Figure 2.3.) The electric field tends to be pulled in toward the spheroid, where its normal component is reduced at the bound-ary. This type of model indicates roughly what would happen to a human with the electric field oriented from head to toe.

Figure 2.3 shows an enlarged view of the field pattern around the top portion of the spheroid as calculated on the finer grid. Note that the E field is weaker inside the object than outside. For quasi-static E fields, the E field is virtually always weaker in materials with higher permittivity. The electric flux density D, on the other hand, is controlled only by the source of the field (the distribution of charges on the metal plates in a capacitor, for instance). For the models shown here, D is uniform throughout. This means that the sources (charge distribution) are uniform, thus creating a uniform set of D flux lines. The effect that these flux lines have in a region on a charge (i.e., the force on the charge) is measured by E. The dielectric material introduces electric dipoles that rotate in a direction to partially cancel the incident E field, and hence reduce the force (and E) a charge would experience.

The fact that E is weaker inside the dielectric than in the surrounding air can be math-ematically described in terms of the boundary conditions. As explained in connection with Equation 1.17 and Figure 1.32, the boundary conditions on E require that the normal component of E must be discontinuous at the boundary between two dielectrics by the ratio of the permittivities. In this case, if ε1 is the permittivity of air and ε2 is the permittiv-ity of the spheroid, then Equation 1.17 requires that E_n2 = (ε1/ε2)E_n1 = (1/2)E_n1 at the boundary.

From Figure 2.3, you can see that at the top of the spheroid, the E fields are approximately normal to the boundary, and that the normal component in the dielectric is about half that of the normal component in air. These fields are not the fields right at the boundary, but they are close enough to the boundary that they approximately satisfy the relationship required by the boundary conditions.

Figures 2.4 and 2.5 show the E field patterns for a prolate spheroid with its long axis perpendicular to the originally uniform E field in which it is placed, essentially turned 90° with respect to the orientation of Figure 2.2. The behavior is similar, with the E fields weaker in the dielectric than in the surrounding air above and below the object.

The normal E

The field pattern around the top portion of the spheroid in Figure 2.2 as calculated on the finer grid. From this picture, it is clearer that the normal components of the field are reduced at the boundary, but the tangential components remain the same.

If the permittivity of the dielectric were increased to a much larger value, the bound-ary conditions would require the normal E in the dielectric at the boundbound-ary to be much smaller than the normal E in the air at the boundary. This illustrates the following impor-tant characteristic of E fields in this low-frequency region of the spectrum: the E fields inside dielectric objects with relatively high permittivities are usually much smaller than the E fields in the surrounding air. This behavior will be illustrated and explained more fully in subsequent sections.

Behavior similar to that in Fig. 2.3

Figure 2.4

Calculated E fields in a model similar to that used in Figure 2.2, but with the long axis of the spheroid perpen-dicular to the originally uniform E in which it was placed. This type of model indicates roughly what would happen to a human with the electric field oriented from shoulder to shoulder. The fields are shown in more detail in Figure 2.5.

Figure 2.5

The field pattern near the left half of the spheroid in Figure 2.4 as calculated on the finer grid.

Three-dimensional quasi-static solutions of Maxwell’s equations in spheroidal coordi-nates for a prolate spheroidal model give more specific results about the internal fields in spheroidal objects, as illustrated in Figure 2.6, in this case for objects with more realistic relative permittivities near those of tissues. In both cases shown, the internal fields are six or seven orders of magnitude smaller than the incident field in which the objects were placed. This is attributable to the high permittivity and conductivity of the objects, which are approximate averages of all the tissues in the human body. Although the relative per-mittivity is of the order of 10⁶, at this low frequency the conductivity probably dominates in determining the internal fields.

The ratio of the internal to incident E field for the case shown in (a) is almost fifteen times greater than that for the case shown in (b) even though the object is the same in each case.

This difference can be explained in terms of the boundary conditions. The incident field in (b) is mostly normal to the dielectric-air interface over a much larger portion of the sur-face of the body than it is in (a). Because the boundary condition as stated in Equation 1.17 requires the internal normal field to be smaller than the external normal field by the ratio of the permittivities, the internal field is smaller in (b) because the boundary condition forces it to be that way over a larger portion of the surface at the boundary.

2a E_inc

Einc

E_int = 260 µV/m E_int/E_inc = 2.6 (10^–7)

E_int = 18 µV/m E_int/E_inc= 0.18 (10^–7) 2b

(a) (b)

The internal E fields are uniform in both (a) and (b)

Figure 2.6

Comparison of the internal fields in two spheroids, (a) with the incident E parallel to the long axis of the spher-oid, and (b) with the incident E perpendicular to the long axis of the spheroid. In both cases the incident E is 1 kV/m, 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. Data for cross sections of similar two-dimensional models are shown in Figures 2.2 to 2.5.