832 Kemp, Grzegorczyk, and Kong for various choices of integration radius yielding zero for all R a and F=y21190. 10-8[N/m]for all R> a, which is in agreement with the value reported by [14] 3. LORENTZ FORCE ON BOUND CURRENTS AND CHARGES The Lorentz force can be applied directly to bound currents and charges in a lossless medium [14]. The bulk force density in (N/m1 computed throughout the medium by 1 Re{-iP×B D”}, here the electric polarization Pe =(Ep -EbE and the magnetic polarization Pm =-(up -ubH are given in terms of the background constitutive parameters (ub, Eb) and the particle constitutive parameters (up, Ep). The surface force density in [N/m R where the bound electric surface charge density is Pe = n.(El Eoeb [12 the bound magnetic surface charge density is Pm =n 1-Ho)ub [14], and the unit vector n is an outward pointing normal to the surface. The fields(Eo, Ho) and(E1, H1) are the total fields just inside the particle and outside the particle, respectively, and the fields in(11) are given by Eaug =(E1+ Eo)/2 and Haug=(H1+Ho)/2 The distributed Lorentz force is applied to the problem of Fig. 1 Because TE polarized waves are incident upon a dielectric particle the bound charges at the surface ero and the total force F obtained by integrating the bulk Lorentz force density foulk over the cross section of the cylinder. The numerical integration is performed by summing the contribution from M discrete area elements. The area elements△A=△x△ y are taken to be identical so that the numerical integration is F=/1()△A∑n(圆xm,(2 here the dielectric polarization Pe[m] and magnetic field Hm] are evaluated at each point indexed by m in the cross section of the cylinder. The y-directed force is plotted in Fig 3 versus the number of integration points. The integral converges much slower than the832 Kemp, Grzegorczyk, and Kong for various choices of integration radius yielding zero for all R<a and F¯ = ˆy2.1190 · 10−18 [N/m] for all R>a, which is in agreement with the value reported by [14]. 3. LORENTZ FORCE ON BOUND CURRENTS AND CHARGES The Lorentz force can be applied directly to bound currents and charges in a lossless medium [14]. The bulk force density in [N/m3] is computed throughout the medium by ¯fbulk = 1 2 Re{−iωP¯e × B¯∗ − iωP¯m × D¯ ∗}, (10) where the electric polarization P¯e = (p − b)E¯ and the magnetic polarization P¯m = −(µp − µb)H¯ are given in terms of the background constitutive parameters (µb, b) and the particle constitutive parameters (µp, p). The surface force density in [N/m2] is given by ¯fsurf = 1 2 Re{ρeE¯∗ avg + ρmH¯ ∗ avg}, (11) where the bound electric surface charge density is ρe = ˆn · (E¯1 − E¯0)b [12], the bound magnetic surface charge density is ρm = ˆn · (H¯1 − H¯0)µb [14], and the unit vector ˆn is an outward pointing normal to the surface. The fields (E¯0, H¯0) and (E¯1, H¯1) are the total fields just inside the particle and outside the particle, respectively, and the fields in (11) are given by E¯avg = (E¯1 + E¯0)/2 and H¯avg = (H¯1 + H¯0)/2. The distributed Lorentz force is applied to the problem of Fig. 1. Because TE polarized waves are incident upon a dielectric particle, the bound charges at the surface are zero and the total force F¯ is obtained by integrating the bulk Lorentz force density ¯fbulk over the cross section of the cylinder. The numerical integration is performed by summing the contribution from M discrete area elements. The area elements ∆A = ∆x∆y are taken to be identical so that the numerical integration is F¯ =  S dA1 2 Re{ ¯fbulk} ≈ ∆A  M m=1 1 2 Re{−iωP¯e[m] × µ0H¯ ∗[m]}, (12) where the dielectric polarization P¯e[m] and magnetic field H¯ [m] are evaluated at each point indexed by m in the cross section of the cylinder. The ˆy-directed force is plotted in Fig. 3versus the number of integration points. The integral converges much slower than the