where each of the quantities P, L, M, Q may be dyadics in the usual sense, or dyadic operators containing space or time derivatives or integrals, or some nonlinear operations on the fields. We may write these expressions as a single matrix equation E H where the6×6 matrix [C]= P M Q This most general relationship between fields is the property of a bianisotropic material We may wonder why d is not related to (E, B, h),e to (D, B),etc. The reason that since the field pairs(e, B)and(D, H)convert identically under a lorentz transfor- mation, a constitutive relation that maps fields as in(2. 18)is form invariant, as are the Maxwell-Minkowski equations. That is, although the constitutive parameters may vary numerically between observers moving at different velocities, the form of the relationship Many authors choose to relate(, B)to (E, H), often because the expressions are simpler and can be more easily applied to specific problems. For instance, in a linear isotropic material (as shown below) D is directly proportional to E and B is directly proportional to H. To provide the appropriate expression for the constitutive relations re need only remap(2.18). This gives s·H B=sE+乒·H, 2] where the new constitutive parameters E, 4, S, i can be easily found from the original constitutive parameters P, L, M, Q. We do note, however, that in the form(2. 19 )-(2.20) the Lorentz invariance of the constitutive equations is not obvious In the following paragraphs we shall characterize some of the most common materials according to these classifications. With this approach effects such as temporal or spatial dispersion are not part of the classification process, but arise from the nature of the constitutive parameters. Hence we shall not dwell on the particulars of the constitutive parameters, but shall concentrate on the form of the constitutive relations. Constitutive relations for fields in free space. In a vacuum the fields are related by the simple constitutive equation H B (223) The quantities uo and Eo are, respectively, the free-space permeability and permittivity constants. It is convenient to use three numerical quantities to describe the electromag- netic properties of free space--Ao, E0, and the speed of light c- and interrelate them through the equation C=1/(u0∈0)/2 @2001 by CRC Press LLCwhere each of the quantities P¯,L¯ , M¯ , Q¯ may be dyadics in the usual sense, or dyadic operators containing space or time derivatives or integrals, or some nonlinear operations on the fields. We may write these expressions as a single matrix equation  cD H  = [C¯ ]  E cB  (2.18) where the 6 × 6 matrix [C¯ ] =  P¯ L¯ M¯ Q¯  . This most general relationship between fields is the property of a bianisotropic material. We may wonder why D is not related to (E,B, H), E to (D,B), etc. The reason is that since the field pairs (E,B) and (D, H) convert identically under a Lorentz transfor￾mation, a constitutive relation that maps fields as in (2.18) is form invariant, as are the Maxwell–Minkowski equations. That is, although the constitutive parameters may vary numerically between observers moving at different velocities, the form of the relationship given by (2.18) is maintained. Many authors choose to relate (D,B) to (E, H), often because the expressions are simpler and can be more easily applied to specific problems. For instance, in a linear, isotropic material (as shown below) D is directly proportional to E and B is directly proportional to H. To provide the appropriate expression for the constitutive relations, we need only remap (2.18). This gives D =
¯ · E + ξ¯ · H, (2.19) B = ζ¯ · E + µ¯ · H, (2.20) or  D B  = C¯ E H  E H  , (2.21) where the new constitutive parameters
¯, ξ¯, ζ¯, µ¯ can be easily found from the original constitutive parameters P¯,L¯ , M¯ , Q¯ . We do note, however, that in the form (2.19)–(2.20) the Lorentz invariance of the constitutive equations is not obvious. In the following paragraphs we shall characterize some of the most common materials according to these classifications. With this approach effects such as temporal or spatial dispersion are not part of the classification process, but arise from the nature of the constitutive parameters. Hence we shall not dwell on the particulars of the constitutive parameters, but shall concentrate on the form of the constitutive relations. Constitutive relations for fields in free space. In a vacuum the fields are related by the simple constitutive equations D = 0E, (2.22) H = 1 µ0 B. (2.23) The quantities µ0 and 0 are, respectively, the free-space permeability and permittivity constants. It is convenient to use three numerical quantities to describe the electromag￾netic properties of free space — µ0, 0, and the speed of light c — and interrelate them through the equation c = 1/(µ0 0) 1/2