5Symmetries, Images, and Dielectrics

5.1 Symmetries

When there are physical symmetries in a geometry description, the same symmetries must be present in L, q, and V. These symmetries can be used to reduce the number of variables required for a given resolution model.

Consider the 12-cell parallel plate capacitor shown in Figure 5.1.

All 12 cells are rectangular — not necessarily square but all aligned the same. The top and bottom electrodes are in the (z = −h/2) and (z = +h/2) planes. They are at V = −0.5 and V = +0.5 V.

c5-fig-0001

FIGURE 5.1 Twelve-cell parallel plate capacitor.

Because of the physical and voltage symmetries of the geometry description, there is no need to include the extra equation to guarantee charge neutrality.

From the physical symmetries of the geometry, we know that

(5.1)images

and also that

(5.2)images

The first two equations describing this structure are

(5.3)images

Substituting all the symmetry–based equalities yields

(5.4)images

In other words, this structure is accurately described by only two ...

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