At the boundaries of isotropic media, the two formulations again lead to substantially dif- ferent force densities. The contribution of the B-field to the surface force is negligible as, for non-magnetic media(u=1), the B-field is continuous at the boundary; also, the discontinuity of aP/at, if any, is finite. In contrast, the perpendicular component of the E-field at the bound ary, E1, has a sharp discontinuity, which results in a Dirac S-function behavior for the E-field gradient VEi(r). When evaluating the integral of (P. V)E across the boundary between the free space and a medium of dielectric constant E, one finds a force density(per unit interfacial rea)F2(r)=,P1(ea1-Ebi); here Pi is the normal component of the medium's polarization density at the surface, Eal is the perpendicular E-field in the free space region just outside the medium, and Ebi is the perpendicular E-field within the medium just beneath the surface, Ref. [ 5]. Note that Pi, being identical to the bound surface charge density ob, may be derived from the discontinuity in the magnitude of E l at the surface, namely, P1=Ob=Eo(Eal-ebi In the first formulation based on Eq (Al), the surface charge density ob derived from-(V.P) is equal to Pi, as mentioned before. This Ob, however, must be multiplied by the effective E-field at the boundary to yield the surface force density(per unit area)FI(r)=ObE. The tangential component El of the E-field is continuous at the boundary and, therefore, defined unambiguously. The perpendicular component, however, can be shown to be exactly equal to the average e⊥ at the boundary, namely,E⊥=i(Eal+Eb⊥) We give two examples from electrostatics to demonstrate the difference between the two for- mulations of the Lorentz law as concerns the E-field contribution to the force within isotropic media. In the first example, shown on the left-hand side of Fig. Al, the medium is heteroge- neous with a dielectric constant E(x). Let the D-field be constant and oriented along the x-axis, that is, D(x)=DoX=Eo Ex(x)+P(x)i=EE(x)E(r)i ( Clearly VD=0 and VxE=0,as required by Maxwells electrostatic equations. )From the first formulation, Eq (Al), we find F1x(x=0)=-(O)E0+E1(O)=-(D0/2e)+1/e(O)2(O),x=0(A4) Fiuk(x)=-Ex(x)dP(x)dx=(1/2Eo)d(Do-P(x)1/dx, 0<x<L(A5) F1xa(x=L)=(D0/20)+1/e(L)P(L) x=L (A6) The second formulation, Eq (A2), yields (x=0)=Px(0)Ex(0)-Eo]=-(D0/2eo)1-1/e(O)P(O) x=0(A7) F2r k(x)=P(x)dEx (x)/ dx=-(1/2Eo)dP (x)/dx, 0<x<L(A8) F2(x=L)=(D0/2e0)1-1/e(L)P(L) x=L(A9) Clearly the two formulations predict different distributions for the force density, both at the surface and in the bulk. However, the total force -obtained by adding the surface contributions to the integral of the force density within the bulk- turns out to be exactly the same in the two In the second example, shown on the right-hand-side of Fig. Al, the medium is homogeneous (dielectric constant=E), and the D-field profile is D(r)=(Doro/r)e In the first formulation, based on Eq (Al), the bulk force is zero, and the surface force density is Fma(6=61)=±(1/2)(1-1/e2)Db/r)26 6,61(A10) In the second formulation, Eq (A2)yields 2ma(r=)=±(1/2)(1-1/e)(Doo/r36 e=0,61(A11) (;6 /ee)(1-1/e)(D/r3)f 6<6<61(A12) #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April 2006 (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESS 3673At the boundaries of isotropic media, the two formulations again lead to substantially dif￾ferent force densities. The contribution of the B-field to the surface force is negligible as, for non-magnetic media (μ = 1), the B-field is continuous at the boundary; also, the discontinuity of ∂P/∂t, if any, is finite. In contrast, the perpendicular component of the E-field at the bound￾ary, E⊥, has a sharp discontinuity, which results in a Dirac δ-function behavior for the E-field gradient ∇E⊥(r). When evaluating the integral of (P · ∇)E across the boundary between the free space and a medium of dielectric constant ε, one finds a force density (per unit interfacial area) F2(r) = 1 2P⊥(Ea⊥ −Eb⊥); here P⊥ is the normal component of the medium’s polarization density at the surface, Ea⊥ is the perpendicular E-field in the free space region just outside the medium, and Eb⊥ is the perpendicular E-field within the medium just beneath the surface, Ref. [5]. Note that P⊥, being identical to the bound surface charge density σ b, may be derived from the discontinuity in the magnitude of E⊥ at the surface, namely, P⊥ = σb = ε0(Ea⊥ −Eb⊥). In the first formulation based on Eq.(A1), the surface charge density σ b derived from −(∇·P) is equal to P⊥, as mentioned before. This σb, however, must be multiplied by the effective E-field at the boundary to yield the surface force density (per unit area) F 1(r) = σbE. The tangential component E of the E-field is continuous at the boundary and, therefore, defined unambiguously. The perpendicular component, however, can be shown to be exactly equal to the average E⊥at the boundary, namely, E⊥ = 1 2 (Ea⊥ +Eb⊥). We give two examples from electrostatics to demonstrate the difference between the two for￾mulations of the Lorentz law as concerns the E-field contribution to the force within isotropic media. In the first example, shown on the left-hand side of Fig. A1, the medium is heteroge￾neous with a dielectric constant ε(x). Let the D-field be constant and oriented along the x-axis, that is, D(x) = D0xˆ = [ε0Ex(x)+Px(x)]xˆ = ε0ε(x)Ex(x)xˆ. (Clearly ∇·D = 0 and ∇×E = 0, as required by Maxwell’s electrostatic equations.) From the first formulation, Eq.(A1), we find: Fsur f ace 1x (x = 0) = −1 2 Px(0)[E0 +Ex(0)] = −(D0/2ε0)[1+1/ε(0)]Px(0); x = 0 (A4) Fbulk 1x (x) = −Ex(x)dPx(x)/dx = (1/2ε0)d[D0 −Px(x)]2 /dx; 0 < x < L (A5) Fsur f ace 1x (x = L)=(D0/2ε0)[1+1/ε(L)]Px(L); x = L (A6) The second formulation, Eq.(A2), yields: Fsur f ace 2x (x = 0) = 1 2 Px(0)[Ex(0)−E0] = −(D0/2ε0)[1−1/ε(0)]Px(0); x = 0 (A7) Fbulk 2x (x) = Px(x)dEx(x)/dx = −(1/2ε0)dP2 x (x)/dx; 0 < x < L (A8) Fsur f ace 2x (x = L)=(D0/2ε0)[1−1/ε(L)]Px(L); x = L (A9) Clearly the two formulations predict different distributions for the force density, both at the surface and in the bulk. However, the total force - obtained by adding the surface contributions to the integral of the force density within the bulk- turns out to be exactly the same in the two formulations (Ftotal = 0). In the second example, shown on the right-hand-side of Fig. A1, the medium is homogeneous (dielectric constant=ε), and the D-field profile is D(r)=(D0r0/r)θˆ. In the first formulation, based on Eq.(A1), the bulk force is zero, and the surface force density is Fsur f ace 1 (r,θ = θ0,1) = ±(1/2ε0)(1−1/ε2 )(D0r0/r) 2 θˆ; θ = θ0,θ1 (A10) In the second formulation, Eq.(A2) yields: Fsur f ace 2 (r,θ = θ0,1) = ±(1/2ε0)(1−1/ε) 2 (D0r0/r) 2θˆ; θ = θ0,θ1 (A11) Fbulk 2 (r,θ) = −(1/ε0ε)(1−1/ε)(D2 0r 2 0/r 3 )rˆ; θ0 < θ < θ1 (A12) #67575 - $15.00 USD Received 15 February 2006; revised 7 April 2006; accepted 10 April 2006 (C) 2006 OSA 17 April 2006 / Vol. 14, No. 8 / OPTICS EXPRESS 3673