Abstract

We derive an expression for the Maxwell stress tensor in a magnetic dielectric medium specified by its permittivity ε and permeability µ. The derivation proceeds from the generalized form of the Lorentz law, which specifies the force exerted by the electromagnetic E and H fields on the polarization P and magnetization M of a ponderable medium. Our stress tensor differs from the well-known tensors of Abraham and Minkowski, which have been at the center of a century-old controversy surrounding the momentum of the electromagnetic field in transparent materials.

© 2008 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.
  2. R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, "Momentum of an electro-magnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
    [CrossRef]
  3. M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13502-13518 (2007).
    [CrossRef] [PubMed]
  4. T. B.  Hansen and A. D.  Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications, IEEE Press, New York (1999).
    [CrossRef]
  5. 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]
  6. A. D.  Yaghjian, "Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material," IEEE Trans. Anten. Prop. 55, 1495-1505 (2007).
    [CrossRef]

2007 (4)

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, "Momentum of an electro-magnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[CrossRef]

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]

A. D.  Yaghjian, "Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material," IEEE Trans. Anten. Prop. 55, 1495-1505 (2007).
[CrossRef]

M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13502-13518 (2007).
[CrossRef] [PubMed]

IEEE Trans. Anten. Prop. (1)

A. D.  Yaghjian, "Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material," IEEE Trans. Anten. Prop. 55, 1495-1505 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

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]

Rev. Mod. Phys. (1)

R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, "Momentum of an electro-magnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007).
[CrossRef]

Other (2)

T. B.  Hansen and A. D.  Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications, IEEE Press, New York (1999).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.

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Figures (3)

Fig. 1.
Fig. 1.

A small cube of dimensions Δx×Δy×Δz within a magnetic dielectric is separated from the surrounding medium by a fictitious vacuum-filled gap; the medium is specified by its (ε, µ) parameters. Assuming the gap is sufficiently narrow (compared to the wavelength of the electromagnetic field), its presence should not affect the distribution of the fields throughout the medium. Within the gap, however, the various components of the electromagnetic field are determined by the standard boundary conditions derived from Maxwell’s equations. In general, the tangential components of E and H remain continuous across the gap, while, in the perpendicular direction, the components of D and B retain continuity.

Fig. 2.
Fig. 2.

A collimated beam of light having a finite diameter along the x-axis, propagates along z in a homogeneous medium specified by its (ε,µ) parameters. The beam has transverse magnetic (TM) or p-polarization, that is, its electromagnetic field components are (Ex , Ez , Hy ). A narrow gap opened in the central region of the beam reveals the existence of a force on the adjacent layers of dipoles. Continuity of D yields the E-field within the gap as ε E o . The E-field acting on the negative charges of the upper layer of the dipoles (as well as that acting on the positive charges of the lower dipoles) is ½(ε+1)E o , whereas the field acting on the positive charges of the upper dipoles (or negative charges of the lower dipoles) is E o . These boundary dipole layers, therefore, experience an E-field gradient proportional to ½(ε-1)E o . The net force of the E-field gradient exerted on the upper boundary layer is downward, while that on the lower boundary layer is upward. The two forces, being equal in magnitude, cancel each other out, but each force must be taken into account when considering the total force on the upper or lower halves of the medium. In addition to forces at the boundary layers, the sidewalls of the beam exert a force on the medium as well; the density of this force (per unit area of the sidewall) is denoted by F sw .

Fig. 3.
Fig. 3.

A collimated beam of finite diameter in the x-direction (and infinite diameter along y) propagates in a medium specified by its (ε, µ) parameters. The propagation direction makes an angle θ with the z-axis in the xz-plane. The beam’s foot-print on the x-axis has unit length, making the beam width equal to cosθ, as shown. The beam is transverse magnetic (TM) or p-polarized, that is, its electromagnetic field components are (Ex , Ez , Hy ). A narrow gap, opened parallel to the x-axis, reveals the force exerted on the boundary layer electric dipoles due to Ez discontinuity. The effective E-field gradient acting on the boundary dipole layer is proportional to ½(ε-1)E osinθ z ^ . There is also an imbalance between the forces acting at the beam’s upper and lower sidewalls, due to the extra length sinθ of the lower wall. When the force on the boundary dipole layer as well as the imbalance of the sidewall forces are taken into account, the stress tensor component Txz +Tzz yields the rate of flow of momentum along the propagation direction.

Equations (43)

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F ( r , t ) = ( P · ) E + ( M · ) H + ( P t ) × μ o H ( M t ) × ε o E .
· D = ρ free ,
× H = J free + D t ,
× E = B t ,
· B = 0 .
D = ε o E + P = ε o ( 1 + χ e ) E = ε o ε E ,
B = μ o H + M = μ o ( 1 + χ m ) H = μ o μ H .
( P · ) E + ( · P ) E = ( P x E ) x + ( P y E ) y + ( P z E ) z ,
· P = · ( D ε o E ) = ρ free ε o · E = ε o · E .
( M · ) H + ( · M ) H = ( M x H ) x + ( M y H ) y + ( M z H ) z .
· M = · ( B μ o H ) = . B μ o · H = μ o · H .
( P t ) × μ o H = ( D t ε o E t ) × μ o H = μ o ( × H ) × H ε o μ o ( E t ) × H ,
( M t ) × ε o E = ( B t μ o H t ) × ε o E = ε o ( × E ) × E ε o μ o ( H t ) × E .
F ( r , t ) = ( x ) ( P x E + M x H ) + ( y ) ( P y E + M y H ) + ( z ) ( P z E + M z H )
+ ε o [ ( · E ) E + ( × E ) × E ] + μ o [ ( · H ) H + ( × H ) × H ] ε o μ o ( E × H ) t .
F ( r , t ) + ( t ) ( E × H c 2 ) = ( x ) { P x E + M x H + ε o [ 1 2 ( E x 2 E y 2 E z 2 ) x ̂ + E x E y y ̂ + E x E z z ̂ ]
+ μ o [ 1 2 ( H x 2 H y 2 H z 2 ) x ̂ + H x H y y ̂ + H x H z z ̂ ] }
+ ( y ) { P y E + M y H + ε o [ E x E y x ̂ + 1 2 + ( E y 2 E x 2 E z 2 ) y ̂ + E y E z z ̂ ]
+ μ o [ H x H z x ̂ + 1 2 ( H y 2 H x 2 H z 2 ) y ̂ + H y H z z ̂ ] }
+ ( z ) { P z E + M z H + ε o [ E x E z x ̂ + E y E z y ̂ + 1 2 + ( E z 2 E x 2 E y 2 ) z ̂ ]
+ μ o [ H x H z x ̂ + H y H z y ̂ + 1 2 ( H z 2 H x 2 H y 2 ) z ̂ ] } .
F ( r , t ) + ( t ) ( E × H c 2 ) = ( x ) { [ 1 2 ( ε o 1 D x 2 ε o E y 2 ε o E z 2 ) x ̂ + D x E y y ̂ + D x E z z ̂ ]
+ [ 1 2 ( μ o 1 B x 2 μ o H y 2 μ o H z 2 ) x ̂ + B x H y y ̂ + B x H z z ̂ ] }
+ y { [ E x D z x ̂ + 1 2 ( ε o 1 D y 2 ε o E x 2 ε o E z 2 ) y ̂ + D y E z z ̂ ] + [ H x B y x ̂ + 1 2 ( μ o 1 B y 2 μ o H x 2 μ o H z 2 ) y ̂ + B y H z z ̂ ] }
+ z { [ E x D z x ̂ + E y D z y ̂ + 1 2 ( ε o 1 D z 2 ε o E x 2 ε o E y 2 ) z ̂ ] + [ H x B z x ̂ + H y B z y ̂ + 1 2 ( μ o 1 B z 2 μ o H x 2 μ o H y 2 ) z ̂ ] } .
T xx = 1 2 ( ε o E y 2 + ε o E z 2 ε o 1 D x 2 ) + 1 2 ( μ o H y 2 + μ o H z 2 μ o 1 B x 2 ) ,
T yx = D x E y B x H y ,
T zx = D x E z B x H z ,
T xy = D x E y B x H y ,
T yy = 1 2 ( ε o E x 2 + ε o E z 2 ε o 1 D y 2 ) + 1 2 ( μ o H x 2 + μ o H z 2 μ o 1 B y 2 ) ,
T zy = D y E z B y H z ,
T xz = E x D z H x B z ,
T yz = E y D z H y B z ,
T zz = 1 2 ( ε o E x 2 + ε o E y 2 ε o 1 D z 2 ) + 1 2 ( μ o H x 2 + μ o H y 2 μ o 1 B z 2 ) .
· T + F ( r , t ) + G ( r , t ) t = 0 .
T zz = 1 2 ( ε o E x 2 + μ o H y 2 ) + 1 2 ε o [ 1 + ( ε μ ) ] E o 2 cos 2 ( ω t ) .
S z = E x H y = Z o 1 ε μ E o 2 cos 2 ( ω t ) .
< T xx > = < 1 2 ε o 1 D x 2 + 1 2 μ o H y 2 > = 1 4 ε o [ ( ε μ ) ε 2 ] E o 2 .
< T xz x ̂ + T zz z ̂ > = - < E x D z > x ̂ + 1 2 < ε o E x 2 ε o 1 D z 2 + μ o H y 2 > z ̂
= 1 2 ε o E o 2 { ε cos θ sin θ x ̂ + 1 2 [ cos 2 θ ε 2 sin 2 θ + ( ε μ ) ] z ̂ } .
d < p > d t = 1 4 ε o [ 1 + ( ε μ ) ] E o 2 cos θ ( sin θ x ̂ + cos θ z ̂ ) .
< F sw > = 1 4 ε o [ ( ε μ ) 2 ε + 1 ] E o 2 sin θ ( cos θ x ̂ + sin θ z ̂ ) .
< F > = 1 4 ε o ( ε 1 ) 2 E o 2 sin 2 θ z ̂ .

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