Once Maxwells equations are in definite form, standard methods for partial differential a nutshell, the system(2. 1)-(2.2)of hyperbolic differential equations is well-posed if and only if we specify e and H throughout a volume region V at some time instant and alse specify, at all subsequent times 1. the tangential component of E over all of the boundary surface S,or 2. the tangential component of H over all of S, or 3. the tangential component of e over part of S, and the tangential component of H over the remainder of s Proof of all three of the conditions of well-posedness is quite tedious, but a simplified uniqueness proof is often given in textbooks on electromagnetics. The procedure used by Stratton [187 is reproduced below. The interested reader should refer to Hansen [81 for a discussion of the existence of solutions to Maxwells equations 2.2.1 Uniqueness of solutions to Maxwell equations Consider a simply connected region of space V bounded by a surface S, where both V and S contain only ordinary points. The fields within V are associated with a current distribution J, which may be internal to V(entirely or in part). By the initial conditions that imply the auxiliary Maxwells equations, we know there is a time, say t=0, prior to which the current is zero for all time, and thus by causality the fields throughout v are identically zero for all times t<0. We next assume that the fields are specified hroughout V at some time to >0, and seek conditions under which they are determine uniquely for all t >to Let the field set(E1, D, Bl, Hi) be a solution to Maxwells equations(2.1)-(2.2) associated with the current J(along with an appropriate set of constitutive relations) and let(E,, D,, B,, H,) be a second solution associated with j. To determine the con- ditions for uniqueness of the fields, we look for a situation that results in El= e2 B1= B2, and so on. The electromagnetic fields must obey aB V×E1 V×H1=J+ V×E Subtracting d(B1-B2) V×(H1-H2)= d(D (2.10) hence defining Eo =El-e2, Bo= B1-B2, and so on, we have Eo·(V×Ho)=Eo H·(×E0)=-、aB (212) @2001 by CRC Press LLCOnce Maxwell’s equations are in definite form, standard methods for partial differential equations can be used to determine whether the electromagnetic model is well-posed. In a nutshell, the system (2.1)–(2.2) of hyperbolic differential equations is well-posed if and only if we specify E and H throughout a volume region V at some time instant and also specify, at all subsequent times, 1. the tangential component of E over all of the boundary surface S, or 2. the tangential component of H over all of S, or 3. the tangential component of E over part of S, and the tangential component of H over the remainder of S. Proof of all three of the conditions of well-posedness is quite tedious, but a simplified uniqueness proof is often given in textbooks on electromagnetics. The procedure used by Stratton [187] is reproduced below. The interested reader should refer to Hansen [81] for a discussion of the existence of solutions to Maxwell’s equations. 2.2.1 Uniqueness of solutions to Maxwell’sequations Consider a simply connected region of space V bounded by a surface S, where both V and S contain only ordinary points. The fields within V are associated with a current distribution J, which may be internal to V (entirely or in part). By the initial conditions that imply the auxiliary Maxwell’s equations, we know there is a time, say t = 0, prior to which the current is zero for all time, and thus by causality the fields throughout V are identically zero for all times t < 0. We next assume that the fields are specified throughout V at some time t0 > 0, and seek conditions under which they are determined uniquely for all t > t0. Let the field set (E1, D1,B1, H1) be a solution to Maxwell’s equations (2.1)–(2.2) associated with the current J (along with an appropriate set of constitutive relations), and let (E2, D2,B2, H2) be a second solution associated with J. To determine the con￾ditions for uniqueness of the fields, we look for a situation that results in E1 = E2, B1 = B2, and so on. The electromagnetic fields must obey ∇ × E1 = −∂B1 ∂t , ∇ × H1 = J + ∂D1 ∂t , ∇ × E2 = −∂B2 ∂t , ∇ × H2 = J + ∂D2 ∂t . Subtracting, we have ∇ × (E1 − E2) = −∂(B1 − B2) ∂t , (2.9) ∇ × (H1 − H2) = ∂(D1 − D2) ∂t , (2.10) hence defining E0 = E1 − E2, B0 = B1 − B2, and so on, we have E0 · (∇ × H0) = E0 · ∂D0 ∂t , (2.11) H0 · (∇ × E0) = −H0 · ∂B0 ∂t . (2.12)