Lorentz force on dielectric and magnetic particles 829 time-average force on a material body can be calculated from a single divergence integral. The proof of this fact is shown by derivation of the momentum conservation theorem with the only assumption that all fields have a force provides the fundamental relationship between lence electromagnetic fields and the mechanical force on charges and currents[16]. The time average Lorentz force is given in terms of the electric field strength E and magnetic flux density b by 了={E+了×B where p and represent the electric charge and current, respectively, Ref represents the real part of a complex quantity, and()denotes the complex conjugate. The Maxwell Equations p=V·D =V×H+iD relate the sources p and to the electric flux density D and magnetic field strength H. Substitution yields F=Re(v DE+(VxH)XB-Dx( B).(3) After applying the remaining two Maxwell equations B B=V×E the force can by expressed as 了=2{(,DE+(×E)xD+(,B)B+(xB)×B() The momentum conservation theorem for time harmonic fields is 了=2{V)} where f is the time average force density in N/m3, and the Maxwell 6 ()=5(D.E*+B*·H)1-DE-B In(7), DE*and B* H are dyadic products and is the(3×3) identity matrix. By integration over a volume V enclosed by a surface S andLorentz force on dielectric and magnetic particles 829 time-average force on a material body can be calculated from a single divergence integral. The proof of this fact is shown by derivation of the momentum conservation theorem with the only assumption that all fields have e−iωt dependence. The Lorentz force provides the fundamental relationship between electromagnetic fields and the mechanical force on charges and currents [16]. The time average Lorentz force is given in terms of the electric field strength E¯ and magnetic flux density B¯ by ¯f = 1 2 Re ρE¯∗ + J¯× B¯∗ , (1) where ρ and J¯ represent the electric charge and current, respectively, Re{} represents the real part of a complex quantity, and (∗) denotes the complex conjugate. The Maxwell Equations ρ = ∇ · D¯ J¯ = ∇ × H¯ + iωD¯ (2) relate the sources ρ and J¯ to the electric flux density D¯ and magnetic field strength H¯ . Substitution yields ¯f = 1 2 Re (∇ · D¯)E¯∗ + (∇ × H¯ ) × B¯∗ − D¯ × (iωB¯) ∗ . (3) After applying the remaining two Maxwell equations 0 = ∇ · B¯ iωB¯ = ∇ × E, ¯ (4) the force can by expressed as ¯f = 1 2 Re (∇·D¯)E¯∗+(∇×E¯∗)×D¯ +(∇·B¯∗)H¯ +(∇×H¯ )×B¯∗ . (5) The momentum conservation theorem for time harmonic fields is reduced to ¯f = −1 2 Re ∇ · T ¯¯(¯r) , (6) where ¯f is the time average force density in N/m3, and the Maxwell stress tensor is [16] T ¯¯(¯r) = 1 2  D¯ · E¯∗ + B¯∗ · H¯  ¯¯I − D¯E¯∗ − B¯∗H. ¯ (7) In (7), D¯E¯∗ and B¯∗H¯ are dyadic products and ¯¯I is the (3×3) identity matrix. By integration over a volume V enclosed by a surface S and