Abstract

We examine the force of the electromagnetic radiation on linear, isotropic, homogeneous media specified in terms of their permittivity ε and permeability μ. A formula is proposed for the electromagnetic Lorentz force on the magnetization M, which is treated here as an Amperian current loop. Using the proposed formula, we demonstrate conservation of momentum in several cases that are amenable to rigorous analysis based on the classical Maxwell equations, the Lorentz law of force, and the constitutive relations. Our analysis yields precise expressions for the density of the electromagnetic and mechanical momenta inside the media that are specified by their (ε,μ) parameters. An interesting consequence of this analysis is the identification of an “intrinsic” mechanical momentum density, ½E × M/c 2, analogous to the electromagnetic (or Abraham) momentum density, ½E × H/c 2. (Here E and H are the magnitudes of the electric and magnetic fields, respectively, and c is the speed of light in vacuum.) This intrinsic mechanical momentum, associated with the magnetization M in the presence of an electric field E, is apparently the same “hidden” momentum that was predicted by W. Shockley and R. P. James nearly four decades ago.

© 2007 Optical Society of America

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References

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  1. J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973).
    [CrossRef]
  2. D. F. Nelson, "Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy," Phys. Rev. A 44, 3985 (1991).
    [CrossRef] [PubMed]
  3. I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum. tensor," Phys. Reports 52, 133-201 (1979).
    [CrossRef]
  4. R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
    [CrossRef]
  5. R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004).
    [CrossRef]
  6. R. Loudon, S. M. Barnett, and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005).
    [CrossRef]
  7. M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004).
    [CrossRef] [PubMed]
  8. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Radiation pressure on a dielectric wedge," Opt. Express 13, 2064-2074 (2005).
    [CrossRef] [PubMed]
  9. M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005).
    [CrossRef] [PubMed]
  10. M. Mansuripur, "Angular momentum of circularly polarized light in dielectric media," Opt. Express 13, 5315-5324 (2005).
    [CrossRef] [PubMed]
  11. M. Mansuripur, "Radiation pressure and the distribution of electromagnetic force in dielectric media," SPIE Proc. 5930, 154-160 (2005).
    [CrossRef]
  12. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Equivalence of total force (and torque) for two formulations of the Lorentz law," SPIE Proc. 6326, 63260G (2006).
    [CrossRef]
  13. M. Mansuripur, "Radiation Pressure on Submerged Mirrors: Implications for the Momentum of Light in Dielectric Media," Opt. Express 15, 2677-2682 (2007).
    [CrossRef] [PubMed]
  14. B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
    [CrossRef]
  15. S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
    [CrossRef]
  16. B. Kemp, T. Grzegorczyk, and J. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005).
    [CrossRef] [PubMed]
  17. B. A. Kemp, J. A. Kong, and T. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
    [CrossRef]
  18. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
    [CrossRef]
  19. A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).
  20. W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
    [CrossRef]
  21. M. Mansuripur, "Momentum of the electromagnetic field in transparent dielectric media," SPIE Proc. 6644, 664413 (2007).
    [CrossRef]
  22. W. Shockley and R. P. James, "Try simplest cases discovery of hidden momentum forces on magnetic currents," Phys. Rev. Lett. 18, 876-879 (1967).
    [CrossRef]
  23. P. Penfield and H. A. Haus, Electrodynamics of Moving Media, (MIT Press, Cambridge, 1967).
  24. R. P.  Feynman, R. B.  Leighton, and M. Sands, The Feynman Lectures on Physics, (Addison-Wesley, Reading, Massachusetts 1964) Vol. 2, Chap. 27.

2007 (3)

M. Mansuripur, "Radiation Pressure on Submerged Mirrors: Implications for the Momentum of Light in Dielectric Media," Opt. Express 15, 2677-2682 (2007).
[CrossRef] [PubMed]

B. A. Kemp, J. A. Kong, and T. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

M. Mansuripur, "Momentum of the electromagnetic field in transparent dielectric media," SPIE Proc. 6644, 664413 (2007).
[CrossRef]

2006 (2)

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Equivalence of total force (and torque) for two formulations of the Lorentz law," SPIE Proc. 6326, 63260G (2006).
[CrossRef]

2005 (6)

2004 (2)

2002 (1)

R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

1999 (1)

A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).

1991 (1)

D. F. Nelson, "Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy," Phys. Rev. A 44, 3985 (1991).
[CrossRef] [PubMed]

1990 (1)

L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
[CrossRef]

1979 (1)

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum. tensor," Phys. Reports 52, 133-201 (1979).
[CrossRef]

1973 (1)

J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973).
[CrossRef]

1968 (1)

W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
[CrossRef]

1967 (1)

W. Shockley and R. P. James, "Try simplest cases discovery of hidden momentum forces on magnetic currents," Phys. Rev. Lett. 18, 876-879 (1967).
[CrossRef]

1962 (1)

B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
[CrossRef]

Barnett, S. M.

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

R. Loudon, S. M. Barnett, and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Baxter, C.

R. Loudon, S. M. Barnett, and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Brevik, I.

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum. tensor," Phys. Reports 52, 133-201 (1979).
[CrossRef]

Gordon, J. P.

J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973).
[CrossRef]

Grzegorczyk, T.

B. A. Kemp, J. A. Kong, and T. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

B. Kemp, T. Grzegorczyk, and J. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005).
[CrossRef] [PubMed]

James, R. P.

W. Shockley and R. P. James, "Try simplest cases discovery of hidden momentum forces on magnetic currents," Phys. Rev. Lett. 18, 876-879 (1967).
[CrossRef]

Kemp, B.

Kemp, B. A.

B. A. Kemp, J. A. Kong, and T. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Kong, J.

Kong, J. A.

B. A. Kemp, J. A. Kong, and T. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Loudon, R.

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

R. Loudon, S. M. Barnett, and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005).
[CrossRef]

R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004).
[CrossRef]

R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

Mansuripur, M.

Moloney, J. V.

M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Equivalence of total force (and torque) for two formulations of the Lorentz law," SPIE Proc. 6326, 63260G (2006).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Radiation pressure on a dielectric wedge," Opt. Express 13, 2064-2074 (2005).
[CrossRef] [PubMed]

Nelson, D. F.

D. F. Nelson, "Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy," Phys. Rev. A 44, 3985 (1991).
[CrossRef] [PubMed]

Shockley, W.

W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
[CrossRef]

W. Shockley and R. P. James, "Try simplest cases discovery of hidden momentum forces on magnetic currents," Phys. Rev. Lett. 18, 876-879 (1967).
[CrossRef]

Tellegen, B. D. H.

B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
[CrossRef]

Vaidman, L.

L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).

Zakharian, A. R.

M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Equivalence of total force (and torque) for two formulations of the Lorentz law," SPIE Proc. 6326, 63260G (2006).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Radiation pressure on a dielectric wedge," Opt. Express 13, 2064-2074 (2005).
[CrossRef] [PubMed]

Am. J. Phys. (2)

B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962).
[CrossRef]

L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990).
[CrossRef]

Fortschr. Phys. (1)

R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004).
[CrossRef]

IEEE Anten. Prop. Soc. Symp. (1)

A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).

J. Mod. Opt. (1)

R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys. (1)

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006).
[CrossRef]

Opt. Express (6)

Phys. Reports (1)

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum. tensor," Phys. Reports 52, 133-201 (1979).
[CrossRef]

Phys. Rev. A (4)

J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973).
[CrossRef]

D. F. Nelson, "Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy," Phys. Rev. A 44, 3985 (1991).
[CrossRef] [PubMed]

B. A. Kemp, J. A. Kong, and T. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

R. Loudon, S. M. Barnett, and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

W. Shockley and R. P. James, "Try simplest cases discovery of hidden momentum forces on magnetic currents," Phys. Rev. Lett. 18, 876-879 (1967).
[CrossRef]

W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968).
[CrossRef]

SPIE Proc. (3)

M. Mansuripur, "Momentum of the electromagnetic field in transparent dielectric media," SPIE Proc. 6644, 664413 (2007).
[CrossRef]

M. Mansuripur, "Radiation pressure and the distribution of electromagnetic force in dielectric media," SPIE Proc. 5930, 154-160 (2005).
[CrossRef]

M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Equivalence of total force (and torque) for two formulations of the Lorentz law," SPIE Proc. 6326, 63260G (2006).
[CrossRef]

Other (2)

P. Penfield and H. A. Haus, Electrodynamics of Moving Media, (MIT Press, Cambridge, 1967).

R. P.  Feynman, R. B.  Leighton, and M. Sands, The Feynman Lectures on Physics, (Addison-Wesley, Reading, Massachusetts 1964) Vol. 2, Chap. 27.

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Figures (5)

Fig.1.
Fig.1.

The local magnetization M = Mx + My ŷ + Mz of the material is subject to various local B-field components: Bx (brown), By (blue), and Bz (red). A circulating current I around a loop of area δ 2 (green squares) produces a magnetic dipole m = 2 n̂ along the perpendicular unit vector . The magnetization density is M = Nm, where N is the number density of the loops.

Fig. 2.
Fig. 2.

A linearly polarized plane-wave having E-field amplitude E o and H-field amplitude H o is normally incident at the interface between the free space and a LIH slab of permittivity ε and permeability μ. The Fresnel reflection coefficient at the surface is ρ. Inside the slab, the transmitted beam has field amplitudes Ex and Hy .

Fig. 3.
Fig. 3.

Variant of the Einstein box experiment featuring a short pulse of light and a transparent slab of length L and mass M o. In the free-space region outside the slab, the pulse, having energy E = mc 2 and momentum p = mc , travels with speed c. Inside the slab, the pulse travels with the group velocity Vg . The entrance and exit facets of the slab are anti-reflection coated to ensure the passage of the entire pulse through the slab. In one experiment, the pulse travels entirely in the free-space region outside the slab, while in another, the pulse spends a fraction of its time inside the slab. Since no external forces are at work, the center of mass of the system (consisting of the light pulse and the slab) must be displaced equally in the two experiments.

Fig. 4.
Fig. 4.

Collimated beam of light propagating along z and having a finite diameter along x. In (a) the beam is p-polarized, that is, its field components are (Ex , Ez , Hy ). In (b) the polarization state is s, corresponding to the field components (Ey , Hx , Hz ). The lateral electromagnetic force at the beam’s sidewalls, denoted by Fsw is oriented in opposite directions on opposite sidewalls.

Fig. 5.
Fig. 5.

Linearly polarized plane-wave incident at oblique angle θ at the interface between the free space and a homogeneous slab of permittivity μ and permeability [l. The incident beam is p-polarized in (a) and s-polarized in (b). In both cases the foot-print of the beam along x is assumed to be unity, making the cross-sections of the incident and transmitted beams equal to cosθ and cosθ́, respectively. The transmitted beam’s lower sidewall has a segment of length sinθ́, which is subject to the sidewall force density Fsw .

Equations (80)

Equations on this page are rendered with MathJax. Learn more.

F a = M x ( B x y ) y ̂ + M x ( B x z ) z ̂ M x ( B y y ) x ̂ M x ( B z z ) x ̂ ,
F b = M y ( B y x ) x ̂ + M y ( B y z ) z ̂ M y ( B x x ) y ̂ M y ( B z z ) y ̂ ,
F c = M z ( B z x ) x ̂ + M z ( B z y ) y ̂ M z ( B x x ) z ̂ M z ( B y y ) z ̂ ,
F m x y z t = M × ( × B ) + ( M ) B ( B ) M .
F m x y z t = μ 0 [ M × ( × H ) + ( M ) H ] .
F m x y z t = ½ μ o ( μ 1 ) ( H . H ) .
ρ = ( 1 ε μ ) ( 1 + ε μ ) .
< S z > = 1 2 Z o 1 E o 2 ( 1 ρ 2 ) = 2 Z o 1 E o 2 Re ( ε μ ) 1 + ε μ 2
F = ( P t ) × B + μ o M × ( × H ) .
F z = μ o ε o ( ε 1 ) ( E x t ) ( μ H y ) μ o ε o ε ( E x t ) [ ( μ 1 ) H y ] .
< F z > = ½ μ o ε o Re [ i 2 πf ( ε 1 ) E x μ * H y * + i 2 πf ε E x ( μ * 1 ) H y * ] 0 exp [ 4 πf ( z c ) Im εμ ] dz
= ¼ ε o ( 1 + ρ ) E o 2 Im [ ( ε 1 ) μ * ε * μ * ε ( μ * 1 ) ε * μ * Im ( εμ ) ]
= ε o E o 2 ( 1 + ε μ ) [ 1 + ε μ ] 2 .
< F z > = ½ ε o E o 2 ( 1 + ρ 2 ) = ε o E o 2 ( 1 + ε μ ) 1 + ε μ 2 .
<F z > = ¼ ε o Z o E x H y μ ε [ 1 + ( ε μ ) ] .
p = <F z > z ̂ V = ¼ ( μ + ε ) E x H y z ̂ c 2 = ¼ ε o E × B + ¼ μ o D × H .
p photon = <F z > <S z > ( hf ) = ½ ( ε μ + ε μ ) ( hf c ) .
< F z > = ¼ ε o E o 2 ( 1 ε μ ) ( 1 μ ε ) .
p M = M o ( Δ z Δ t ) z ̂ = m ( c V g ) z ̂ .
p E = m V g z ̂ = ( E c ) ( V g c ) z ̂ .
F z ( z , t ) = μ o ε o [ ( ε 1 ) μ ( μ 1 ) ε ] H y E x t
= μ o ε o ( ε μ ) Z o 1 ε μ E x E x t
= ε o [ 1 ( ε μ ) ] E x E x z
= ½ ε o [ 1 ( ε μ ) ] E x 2 z .
F z ( t ) = ± ½ ε o [ ( ε μ ) 1 ] E x 2 ( z = z o , t ) .
<F z > = ± ¼ ε o [ ( ε μ ) 1 ] E x 2 .
p mech = ¼ μ o ε o ( ε μ ) E x H y z ̂ = ¼ ( ε μ ) E × H c 2 .
p = ½ ( E × H c 2 ) + ¼ ( E × M c 2 ) + ¼ ( μ o P × H ) .
p mech = ¼ ( ε + μ 2 ) E × H c 2 .
F = ( P t ) × B + μ o M × ( × H )
= ( ε 1 ) ε 1 ( D t ) × ( μ o μ H ) + μ o ( μ 1 ) H × ( × H )
= μ o [ 1 ( μ ε ) ] [ ( H y x ) z ̂ ( H y z ) x ̂ ] × H y y ̂
= ½ μ o [ ( μ ε ) 1 ] [ ( H y 2 x ) x ̂ + ( H y 2 z ) z ̂ ] .
F sw ( p ) = 0 F x d x x ̂ = ½ μ o [ ε) 1 ] 0 ( H y 2 x ) d x x ̂ = ½ μ o [ 1 ( μ ε ) ] H y 2 ( x = 0 , y , z , t ) x ̂ .
F sw ( p ) = ± ¼ μ o [ 1 ( μ ε ) H y 2 x ̂ = ± ¼ ε 0 [ ( ε μ ) 1 ] E x 2 x ̂ . ]
F ( extra ) = ( P ) E = ε o ( ε 1 ) [ ( E x E x x + E z E x z ) x ̂ + ( E x E z x + E z E z z z ̂ ) ] .
F x ( extra ) = ε 0 ( ε 1 ) ( E x E x x + E z E z x μ o μ E z H y t )
= ε 0 ( ε 1 ) [ 1 2 E x 2 x + 1 2 E z 2 x μ o μ ( E z H y ) t + μ o μ H y E z t ]
= ε 0 ( ε 1 ) [ 1 2 ( E x 2 + E z 2 ) x + μ o μ ( S x t ) + 1 2 ( μ o ε o ) ( μ ε ) H y 2 x ] .
< 0 F x ( extra ) d x > = 1 2 ε 0 ( ε 1 ) E x 2 .
< F sw ( p ) > = ± 1 4 ε o [ ( ε μ ) 2 ε + 1 ] E x 2 x ̂ .
F = ( P t ) × B + μ o M × ( × H ) + μ o [ ( M x H x x + M z H x z ) x ̂ + ( M z H z x + M z H z z ) z ̂ ] .
F x = ε o μ o μ ( ε 1 ) ( E y H z ) t + 1 2 ε o ( ε 1 ) E y 2 x + 1 2 μ o ( μ 1 ) ( H x 2 + H z 2 ) x .
< F sw ( s ) > = ± 1 4 ε o [ ( ε μ ) 2 ε + 1 ] E y 2 x ̂ .
E i = E o ( cos θ x ̂ sin θ z ̂ ) exp { i 2 πf [ ( x sin θ + z cos θ ) c t ] }
H i = Z o 1 E o y ̂ exp { i 2 πf [ ( x sin θ + z cos θ ) c t ] }
E r = ρ p E o ( cos θ x ̂ + sin θ z ̂ ) exp { i 2 πf [ ( x sin θ + z cos θ ) c t ] }
H r = Z o 1 ρ p E o y ̂ exp { i 2 πf [ ( x sin θ + z cos θ ) c t ] }
E t = E o [ ( 1 + ρ p ) cos θ x ̂ ε 1 ( 1 ρ p ) sin θ z ̂ ] exp { i 2 πf [ x sin θ + z με sin 2 θ c t ] }
H t = Z o 1 ( 1 ρ p ) E o y ̂ exp { i 2 πf [ x sin θ + z με sin 2 θ c t ] } .
ρ p = ( με sin 2 θ ε cos θ ) ( με sin 2 θ + ε cos θ ) .
F ( bulk ) = ε o E o 2 cos 2 θ με sin 2 θ + ε cos θ 2 { 2 Re [ με sin 2 θ ] sin θ x ̂ + ( ε 2 + με sin 2 θ sin 2 θ ) z ̂ } ,
F ( surface ) = ε o E o 2 sin θ cos 2 θ με sin 2 θ + ε cos θ 2 { 2 Re [ ( ε * 1 ) με sin 2 θ ] x ̂ ( ε 2 1 ) sin θ z ̂ } ,
F ( total ) = ε o E o 2 cos 2 θ με sin 2 θ + ε cos θ 2 { 2 Re [ ε * με sin 2 θ ] sin θ x ̂ + ( ε 2 cos 2 θ + με sin 2 θ ) z ̂ } .
E t = E x 2 + E z 2 = 2 με E o cos θ ( με sin 2 θ + ε cos θ ) .
F ( bulk ) = 1 4 ε o E t 2 { 2 sin θ ' cos θ ' x ̂ + [ 1 + ( ε μ ) 2 sin 2 θ ' ] z ̂ } ,
F ( surface ) = 1 4 ε o E t 2 { 2 ( ε 1 ) sin θ ' cos θ ' x ̂ ( ε 2 1 ) sin 2 θ ' z ̂ } ,
F ( total ) = 1 4 ε o E t 2 { 2 ε sin θ ' cos θ ' x ̂ + [ 1 + ( ε μ ) ( 1 + ε 2 ) sin 2 θ ' ] z ̂ } .
F ( flux ) = 1 4 ε o E t 2 [ 1 + ( ε μ ) ] cos θ ' ( sin θ ' x ̂ + cos θ ' z ̂ ) .
Δ F = F ( bulk ) F ( flux ) = 1 4 ε o E t 2 [ 1 ( ε μ ) ] sin θ ' ( cos θ ' x ̂ sin θ ' z ̂ ) ,
F ( sidewall ) = 1 4 ε o E t 2 [ 2 ε ( ε μ ) 1 ] sin θ ' ( cos θ ' x ̂ sin θ ' z ̂ ) ,
F ( surface ) = 1 4 ε o E t 2 ( ε 1 ) 2 sin θ ' z ̂ ,
E i = E o y ̂ exp { i 2 πf [ ( x sin θ + z cos θ ) c t ] } ,
H i = Z o 1 E o ( cos θ x ̂ + sin θ z ̂ ) exp { i 2 πf [ ( x sin θ + z cos θ ) c t ] } ,
E r = ρ s E o y ̂ exp { i 2 πf [ ( x sin θ z cos θ ) c t ] } ,
H r = Z o 1 ρ s E o ( cos θ x ̂ + sin θ z ̂ ) exp { i 2 πf [ ( x sin θ z cos θ ) c t ] } ,
E t = E o ( 1 ρ s ) y ̂ exp { i 2 πf [ ( x sin θ + z με sin 2 θ ) c t ] } ,
H t = Z o 1 E o [ ( 1 + ρ s ) cos θ x ̂ + μ 1 ( 1 ρ s ) sin θ z ̂ ] exp { i 2 πf [ ( x sin θ + z με sin 2 θ ) c t ] } .
ρ s = ( με sin 2 θ μ cos θ ) ( με sin 2 θ + μ cos θ ) .
F ( bulk ) = ε o E o 2 cos 2 θ { 2 Re [ μ * με sin 2 θ ] sin θ x ̂ + [ μ 2 με sin 2 θ + ( 1 2 Re ( μ ) ) sin 2 θ ] z ̂ με sin 2 θ + μ cos θ 2 ,
F ( surface ) = ε o E o 2 [ 2 Re ( μ ) μ 2 1 ] sin 2 θ cos 2 θ z ̂ με sin 2 θ + μ cos θ 2 ,
F ( total ) = ε o E o 2 cos 2 θ { 2 Re [ μ * με sin 2 θ ] sin θ x ̂ + ( μ 2 cos 2 θ + με sin 2 θ ) z ̂ } με sin 2 θ + μ cos θ 2 .
F ( total ) = ¼ ε 0 E t 2 { 2 ε sin θ ' cos θ ' x ̂ + [ 1 + ( ε μ ) cos 2 θ ' ( εμ ) sin 2 θ ' ] z ̂ } .
F 1 = ¼ ε o E t 2 [ 1 + ( ε μ ) ] cos θ ' ( sin θ ' x ̂ + cos θ ' z ̂ ) ,
F 2 = ¼ ε o E t 2 [ 2 ε ( ε μ ) 1 ] sin θ ( cos θ x ̂ sin θ z ̂ ) ,
F 3 = ¼ ε o E t 2 [ 2 ε εμ ( ε μ ) ] sin 2 θ ' z ̂ .
M z x t = H z ( x , z = 0 , t ) H z ( x , z = 0 + , t ) = Z O 1 E t ( 1 μ 1 ) sin θ exp { i 2 πf [ ( x c ) sin θ t ] } .
Δ H z = H z ( x , z = 0 + , t ) ½ [ H z ( x , z = 0 , t ) + H z ( x , z = 0 + , t ) ] = ½ M z x t .
F total ( x , y , z , t ) = ( P ) E + ( P t ) × B + μ o M × ( × H ) + μ o ( M ) H + ( E × M c 2 ) t .
F total ( x , y , z , t ) = ( P ) E + μ o ( M ) H + μ o ( P t ) × H μ o ε o ( M t ) × E .

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