by(B 49). This requires that V. B be constant with time, say V. B(r, t)= CB(r) The constant CB must be specified as part of the postulate of Maxwells theory, and the choice we make is subject to experimental validation. We postulate that CB(r=0, which leads us to(2. 4). Note that if we can identify a time prior to which B(r, t)=0 then CB(r) must vanish. For this reason, CB(r)=0 and(2.4)are often called the"initial conditions"for Faraday's law [159. Next we take the divergence of (2.2) to find that v·(×H=V·J+-(V·D Using(2.5)and(B49), we obtain and thus p-V. D must be some temporal constant Cp(r). Again, we must postulate the value of C as part of the Maxwell theory. We choose Cp(r)=0 and thus obtain Gauss's law(2.3). If we can identify a time prior to which both D and p are everywhere ual to zero, then CD(r)must vanish. Hence CD(r)=0 and (2.3) may be regarded as "initial conditions"for Ampere's law. Combining the two sets of initial conditions we find that the curl equations imply the divergence equations as long as we can find a time prior to which all of the fields e, D, B, h and the sources j and p are equal to zero ( since all the fields are related through the curl equations, and the charge and current are elated through the continuity equation). Conversely, the empirical evidence supporting the two divergence equations implies that such a time should exist Throughout this book we shall refer to the two curl equations as the "fundamental Maxwell equations, and to the two divergence equations as the "auxiliary"equations The fundamental equations describe the relationships between the fields while, as we have seen, the auxiliary equations provide a sort of initial condition. This does not imply that the auxiliary equations are of lesser importance; indeed, they are required to establish uniqueness of the fields, to derive the wave equations for the fields, and to properly describe static fields. Field vector terminology. Various terms are used for the field vectors, sometimes harkening back to the descriptions used by Maxwell himself, and often based on the physical nature of the fields. We are attracted to Sommerfeld,'s separation of the fields into entities of intensity(E, B)and entities of quantity(D, H). In this system E is called the electric field strength, B the magnetic field strength, D the electric excitation, and H the magnetic excitation [ 185. Maxwell separated the fields into a set(e, h) of vectors that appear within line integrals to give work-related quantities, and a set (B, D) of vectors that appear within surface integrals to give flux-related quantities: we shall see this clearly when considering the integral forms of Maxwells equations. By this system authors such as Jones 97] and Ramo, Whinnery, and Van Duzer [153 call e the electric intensity, H the magnetic intensity, b the magnetic fiur density, and d the electric fur Maxwell himself designated names for each of the vector quantities. In his classic paper A Dynamical Theory of the Electromagnetic Field, [178 Maxwell referred to the quantity we now designate e as the electromotive force, the quantity d as the elec- tric displacement (with a time rate of change given by his now famous "displacement current"), the quantity H as the magnetic force, and the quantity B as the magnetic @2001 by CRC Press LLCby (B.49). This requires that ∇ · B be constant with time, say ∇ · B(r, t) = CB(r). The constant CB must be specified as part of the postulate of Maxwell’s theory, and the choice we make is subject to experimental validation. We postulate that CB(r) = 0, which leads us to (2.4). Note that if we can identify a time prior to which B(r, t) ≡ 0, then CB(r) must vanish. For this reason, CB(r) = 0 and (2.4) are often called the “initial conditions” for Faraday’s law [159]. Next we take the divergence of (2.2) to find that ∇ · (∇ × H) =∇· J + ∂ ∂t (∇ · D). Using (2.5) and (B.49), we obtain ∂ ∂t (ρ −∇· D) = 0 and thus ρ −∇· D must be some temporal constant CD(r). Again, we must postulate the value of CD as part of the Maxwell theory. We choose CD(r) = 0 and thus obtain Gauss’s law (2.3). If we can identify a time prior to which both D and ρ are everywhere equal to zero, then CD(r) must vanish. Hence CD(r) = 0 and (2.3) may be regarded as “initial conditions” for Ampere’s law. Combining the two sets of initial conditions, we find that the curl equations imply the divergence equations as long as we can find a time prior to which all of the fields E, D,B, H and the sources J and ρ are equal to zero (since all the fields are related through the curl equations, and the charge and current are related through the continuity equation). Conversely, the empirical evidence supporting the two divergence equations implies that such a time should exist. Throughout this book we shall refer to the two curl equations as the “fundamental” Maxwell equations, and to the two divergence equations as the “auxiliary” equations. The fundamental equations describe the relationships between the fields while, as we have seen, the auxiliary equations provide a sort of initial condition. This does not imply that the auxiliary equations are of lesser importance; indeed, they are required to establish uniqueness of the fields, to derive the wave equations for the fields, and to properly describe static fields. Field vector terminology. Various terms are used for the field vectors, sometimes harkening back to the descriptions used by Maxwell himself, and often based on the physical nature of the fields. We are attracted to Sommerfeld’s separation of the fields into entities of intensity (E,B) and entities of quantity (D, H). In this system E is called the electric field strength, B the magnetic field strength, D the electric excitation, and H the magnetic excitation [185]. Maxwell separated the fields into a set (E, H) of vectors that appear within line integrals to give work-related quantities, and a set (B, D) of vectors that appear within surface integrals to give flux-related quantities; we shall see this clearly when considering the integral forms of Maxwell’s equations. By this system, authors such as Jones [97] and Ramo, Whinnery, and Van Duzer [153] call E the electric intensity, H the magnetic intensity, B the magnetic flux density, and D the electric flux density. Maxwell himself designated names for each of the vector quantities. In his classic paper “A Dynamical Theory of the Electromagnetic Field,” [178] Maxwell referred to the quantity we now designate E as the electromotive force, the quantity D as the elec￾tric displacement (with a time rate of change given by his now famous “displacement current”), the quantity H as the magnetic force, and the quantity B as the magnetic