Chapter 5 Field decompositions and the EM potentials 5.1 Spatial symmetry decompositions Spatial symmetry can often be exploited to solve electromagnetics problems. For analytic solutions, symmetry can be used to reduce the number of boundary conditions that must be applied. For computer solutions the storage requirements can be reduced. Typical symmetries include rotation about a point or axis, and reflection through a plane, along an axis, or through a point. We shall consider the common case of reflection through a plane. Reflections through the origin and through an axis will be treated in the exercises Note that spatial symmetry decompositions may be applied even if the ields possess no spatial symmetry. As long as the boundaries and material media are symmetric, the sources and fields may be decomposed into constituents that individually mimic the symmetry of the environment 5.1.1 Planar field symmetry Consider a region of space consisting of linear, isotropic, time-invariant media having laterial parameters E(r), u(r), and o(r). The electromagnetic fields(E, H)within this region are related to their impressed sources (, Jm)and their secondary sources J '=gE through Maxwells curl equations der dEy dHr der dE dey aEr dHz m Hy dEx at +oEr+J ahr dh. dey +oe+ aH aHr ae ②2001Chapter 5 Field decompositions and the EM potentials 5.1 Spatial symmetry decompositions Spatial symmetry can often be exploited to solve electromagnetics problems. For analytic solutions, symmetry can be used to reduce the number of boundary conditions that must be applied. For computer solutions the storage requirements can be reduced. Typical symmetries include rotation about a point or axis, and reflection through a plane, along an axis, or through a point. We shall consider the common case of reflection through a plane. Reflections through the origin and through an axis will be treated in the exercises. Note that spatial symmetry decompositions may be applied even if the sources and fields possess no spatial symmetry. As long as the boundaries and material media are symmetric, the sources and fields may be decomposed into constituents that individually mimic the symmetry of the environment. 5.1.1 Planar field symmetry Consider a region of space consisting of linear, isotropic, time-invariant media having material parameters (r), µ(r), and σ(r). The electromagnetic fields (E, H) within this region are related to their impressed sources (Ji , Ji m) and their secondary sources Js = σE through Maxwell’s curl equations: ∂Ez ∂y − ∂Ey ∂z = −µ ∂ Hx ∂t − J i mx , (5.1) ∂Ex ∂z − ∂Ez ∂x = −µ ∂ Hy ∂t − J i my , (5.2) ∂Ey ∂x − ∂Ex ∂y = −µ ∂ Hz ∂t − J i mz, (5.3) ∂ Hz ∂y − ∂ Hy ∂z = ∂Ex ∂t + σ Ex + J i x , (5.4) ∂ Hx ∂z − ∂ Hz ∂x = ∂Ey ∂t + σ Ey + J i y , (5.5) ∂ Hy ∂x − ∂ Hx ∂y = ∂Ez ∂t + σ Ez + J i z . (5.6)