Then Ji cos 01= J2 cos 02 by (3. 39), while o2J1 sin 01=01J2 sin e2 by(3.40). Hence It is interesting to consider the case of current incident from a conducting material onto material. If insulator, then J2n =J2=0: by (3.39)we ha Jin =0. But(3.40)does not require Jit =0; with 02=0 the right-hand side of (3.40) is indeterminate and thus JI, may be nonzero. In other words, when current moving through a conductor approaches an insulating surface, it bends and Hows tangential to the surface. This concept is useful in explaining how wires guide current Interestingly, (3.42 )shows that when o2<on we have 02-0: current passing fror onducting region into a slightly-conducting region does so normally. 3.2.3 Uniqueness of the electrostatic field In§ ic field is ur V when the tangential component of E is specified over the surrounding surface. Unfortunately this condition is not appropriate in the electrostatic case. We should remember that an additional requirement for uniqueness of solution to Maxwells equations is that the field be specified throughout V at some time to. For a static field this would completely determine e without need for the surface field! Let us determine conditions for uniqueness beginning with the static field equations Consider a region V surrounded by a surface S. Static charge may be located entirely or partially within V, or entirely outside V, and produces a field within V. The region may also contain any arrangement of conductors or other materials. Suppose 1,E1) and (D2, E) represent solutions to the static field equations within V with source p(r) We wish to find conditions that guarantee both El=E2 and DI=D Since V. DI= p and V. D2 p, the difference field Do= D2 - DI obeys the homogeneous equation Consider the quantity V·(DoΦ0)=Φo(VDo)+Do·(vΦo) here Eo=E2-E1=-Vφo=-V(Φ2-中1).We divergence theorem and ( 3.43)to obtain do[D·的dS=/Do·(vφo)dV D0·EodV 3.44 Now suppose that o=0 everywhere on S, or that f. Do=0 everywhere on S, or that po=0 over part of S and f Do =0 elsewhere on S. Then dv Since V is arbitrary, either Do =0 or Eo =0. Assuming E and D are linked by the constitutive relations, we have e= e and D,= D2 Hence the fields within V are unique provided that either the normal component of D, or some combination of the two, is specified over S. We often use a multiply connected surface to exclude conductors. By (3. 33)we see that specification of the ②2001 by CRC Press LLCThen J1 cos θ1 = J2 cos θ2 by (3.39), while σ2 J1 sin θ1 = σ1 J2 sin θ2 by (3.40). Hence σ2 tan θ1 = σ1 tan θ2. (3.42) It is interesting to consider the case of current incident from a conducting material onto an insulating material. If region 2 is an insulator, then J2n = J2t = 0; by (3.39)we have J1n = 0. But (3.40)does not require J1t = 0; with σ2 = 0 the right-hand side of (3.40) is indeterminate and thus J1t may be nonzero. In other words, when current moving through a conductor approaches an insulating surface, it bends and flows tangential to the surface. This concept is useful in explaining how wires guide current. Interestingly, (3.42)shows that when σ2 σ1 we have θ2 → 0; current passing from a conducting region into a slightly-conducting region does so normally. 3.2.3 Uniqueness of the electrostatic field In § 2.2.1 we found that the electromagnetic field is unique within a region V when the tangential component of E is specified over the surrounding surface. Unfortunately, this condition is not appropriate in the electrostatic case. We should remember that an additional requirement for uniqueness of solution to Maxwell’s equations is that the field be specified throughout V at some time t0. For a static field this would completely determine E without need for the surface field! Let us determine conditions for uniqueness beginning with the static field equations. Consider a region V surrounded by a surface S. Static charge may be located entirely or partially within V, or entirely outside V, and produces a field within V. The region may also contain any arrangement of conductors or other materials. Suppose (D1,E1) and (D2,E2) represent solutions to the static field equations within V with source ρ(r). We wish to find conditions that guarantee both E1 = E2 and D1 = D2. Since ∇ · D1 = ρ and ∇ · D2 = ρ, the difference field D0 = D2 − D1 obeys the homogeneous equation ∇ · D0 = 0. (3.43) Consider the quantity ∇ · (D00) = 0(∇ · D0) + D0 · (∇0) where E0 = E2 − E1 = −∇0 = −∇(2 − 1). We integrate over V and use the divergence theorem and (3.43)to obtain S 0 [D0 · nˆ] d S = V D0 · (∇0) dV = − V D0 · E0 dV. (3.44) Now suppose that 0 = 0 everywhere on S, or that nˆ · D0 = 0 everywhere on S, or that 0 = 0 over part of S and nˆ · D0 = 0 elsewhere on S. Then V D0 · E0 dV = 0. (3.45) Since V is arbitrary, either D0 = 0 or E0 = 0. Assuming E and D are linked by the constitutive relations, we have E1 = E2 and D1 = D2. Hence the fields within V are unique provided that either , the normal component of D, or some combination of the two, is specified over S. We often use a multiply￾connected surface to exclude conductors. By (3.33)we see that specification of the