## Electromagnetic stress tensor in ponderable media

Optics Express, Vol. 16, Issue 8, pp. 5193-5198 (2008)

http://dx.doi.org/10.1364/OE.16.005193

Acrobat PDF (113 KB)

### 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

**fields on the polarization**

*H***and magnetization**

*P***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.**

*M*© 2008 Optical Society of America

## 1. Introduction

**(**

*S***,**

*r**t*)=

**×**

*E***specifies the rate of flow of energy at a given location in space**

*H***and time**

*r**t*. The integral of

**(**

*S***,**

*r**t*) over a closed surface is equal to the rate of increase of stored energy, ∂

**/∂**

*ε**t*, within the volume enclosed by the surface, plus the rate of loss (or minus the rate of gain) of energy throughout that volume [1]. Similarly, the stress tensor

**(**

*T***,**

*r**t*) specifies the rate of flow of momentum (i.e., momentum crossing unit area per unit time) at a given point in space and time.

**and magnetization density**

*P***describe the electromagnetic properties of the material (and where the macroscopic version of Maxwell’s equations, incorporating**

*M***and**

*P***, are applicable), the form of the stress tensor has been the subject of debate and controversy for the past century [2**

*M*2. R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. **79**, 1197–1216 (2007). [CrossRef]

## 2. Stress tensor in ponderable medium

3. M. Mansuripur, “Radiation pressure and the linear momentum of the electro-magnetic field in magnetic media,” Opt. Express **15**, 13502–13518 (2007). [CrossRef] [PubMed]

*µ*and

*ε*parameters:

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]

*et al*, in their analysis of momentum in left-handed media [5

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]

3. M. Mansuripur, “Radiation pressure and the linear momentum of the electro-magnetic field in magnetic media,” Opt. Express **15**, 13502–13518 (2007). [CrossRef] [PubMed]

**=**

*F**q*(

**+**

*E***×**

*V***), but it soon became apparent that magnetic dipoles cannot be treated as simple Amperian current loops; conservation of momentum demanded certain modifications of the original Lorentz law. In particular, a new term had to be introduced to account for the force experienced by magnetic dipoles. Rather than attempting to justify Eq. (1) on the basis of the original Lorentz law, we believe that one should simply accept it as a law of nature, on par with Maxwell’s equations. Not only are these five equations consistent among themselves, they also comply with the laws of energy and momentum conservation.**

*B***is readily set to zero in accordance with Maxwell’s 4**

*B*^{th}equation.

**in terms of**

*P***and**

*D***, then invoking Maxwell’s 2**

*E*^{nd}equation. Similarly, the fourth term is rewritten by substituting for

**in terms of**

*M***and**

*B***, then invoking Maxwell’s 3**

*H*^{rd}equation. We find

*x*×Δ

*y*×Δ

*z*cube depicted in Fig. 1, normalize the resultant by the cube’s volume, and consider the limit when (Δ

*x*, Δ

*y*, Δ

*z*)→0. The left-hand side of Eq. (8) thus remains intact, but several changes occur on the right-hand side. For instance, in the first term, integration over

*x*yields the argument of ∂/∂

*x*, evaluated in the gaps on the left- and right-hand sides of the cube, then subtracted from each other. In these gaps,

*P*=0,

_{x}*M*=0,

_{x}*ε*

_{o}*E*=

_{x}*D*, and

_{x}*µ*

_{o}*H*=

_{x}*B*, while the remaining components of

_{x}**and**

*E***retain the values that they have in the adjacent material environment. (These gap fields are found by invoking standard boundary conditions, namely, the continuity of tangential**

*H***and**

*E***, as well as perpendicular**

*H***and**

*D***components.) Similar arguments apply to the second and third terms on the right-hand side of Eq. (8), provided that, in the case of the 2**

*B*^{nd}(3

^{rd}) term, the initial integration is carried over

*y*(

*z*). When the integrals are fully evaluated and the result is normalized by the volume of the cube, we find, in the limit of a vanishing cube,

**×**

*E***/**

*H**c*

^{2}as the electromagnetic momentum density

**(**

*G***,**

*r**t*), and yields the following stress tensor

*T*(i.e., rate of flow of momentum per unit area per unit time) within the medium:

_{ij}**,**

*E***,**

*D***and**

*H***fields, the stress tensor of Eq. (10) differs from both Abraham and Minkowski tensors. A similar (although by no means identical) tensor has been derived by Yaghjian [6**

*B*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]

**Example 1.**A plane electromagnetic wave propagates along the

*z*-axis inside a medium specified by its (

*ε*,

*µ*) parameters. The linearly polarized plane-wave has

*E*-field amplitude

*E*

_{o}

*x̂*and

*H*-field amplitude

*ω*, the rate of flow of momentum (per unit area per unit time) along the

*z*-axis will be given by Eq. (10i) as follows:

**=**

*S***×**

*E***. Therefore,**

*H**N*photons cross the

*xy*-plane at

*z*=0 during the time interval [0,

*τ*]. Since each photon has energy

*hf*, we have <

*S*>

_{z}*τ*=

*Nhf*(angled brackets denote time-averaging). The total momentum crossing the same plane during the same time interval will therefore be

3. M. Mansuripur, “Radiation pressure and the linear momentum of the electro-magnetic field in magnetic media,” Opt. Express **15**, 13502–13518 (2007). [CrossRef] [PubMed]

*hf*/

*c*. In general, the photon momentum consists of an electromagnetic part and a mechanical part. In a non-dispersive medium where the group velocity of light equals its phase velocity, the electromagnetic momentum of a single photon is

**Example 2**. With reference to Fig. 2, consider a collimated, monochromatic beam of light propagating along the

*z*-axis within a linear, isotropic, and homogeneous medium specified by its (

*ε*,

*µ*) parameters. The beam, which has a finite-diameter along the

*x*-axis and an infinite diameter along

*y*, is linearly polarized, having

*E*-field amplitude

*E*

_{o}

**and**

*x̂**H*-field amplitude

*E*component of the field, which is an odd function of

_{z}*x*and goes to zero at the center. At the central

*yz*-plane, the time-averaged rate of flow of

*x*-momentum along the

*x*-axis is given by Eq. (10a), as follows:

*z*-axis and a second force,

*, exerted on the medium by the upper sidewall of the beam. (The system being symmetric with respect to the*

**F**_{sw}*yz*-plane, identical forces, albeit in opposite directions, act on the lower half of the medium.) The force on the dipoles immediately above the

*z*-axis is best understood if one introduces a gap in the middle of the beam as indicated in Fig. 2. The continuity of

*D*_{⊥}at this interface reveals the

*E*-field within the gap as being equal to

*ε*

*E*

_{o}

**. The average**

*x̂**E*-field at the interface is thus ½(

*ε*+1)

*E*

_{o}, and the field gradient sensed by the interfacial dipole layer is proportional to ½(

*ε*-1)

*E*

_{o}. The dipole density being

**=**

*P**ε*

_{o}(

*ε*-1)

*E*

_{o}

**, we find a force density at the interface given by <**

*x̂**F*>=¼

_{x}*ε*

_{o}(

*ε*-1)

^{2}

*E*

_{o}

^{2}. Adding this force density to <

*T*> of Eq. (14) yields <

_{xx}*F*

_{x}^{(sw)}>=¼

*ε*

_{o}[(

*ε*/

*µ*)-2

*ε*+1]

*E*

_{o}

^{2}, which is consistent with the sidewall force density of finite-diameter beams found in [3

**15**, 13502–13518 (2007). [CrossRef] [PubMed]

**Example 3**. Figure 3 shows a collimated, monochromatic beam of finite width propagating in a homogeneous medium specified by its (

*ε*,

*µ*) parameters. The propagation direction makes an angle

*θ*with the

*z*-axis in the

*xz*-plane. The stress tensor of Eq. (10) gives the following time-averaged rate of flow of momentum (per unit area per unit time) across the

*xy*-plane:

*θ*should be

*x*-axis, reveals the force exerted on the boundary electric dipoles due to the

*E*discontinuity. The strength of this dipole layer is

_{z}**=**

*P**ε*

_{o}(

*ε*-1)

*E*

_{o}sin

*θ*z ^ , and the effective

*E*-field gradient acting on it is proportional to ½(

*ε*-1)

*E*

_{o}sin

*θ*z ^ . The effective force on the dipole layer (per unit area) is thus given by

## 3. Concluding remarks

**,**

*E***,**

*D***, and**

*H***. When the forces acting at the boundaries of a specified region of the medium are properly taken into account, we have shown that the stress tensor, in accordance with Eq. (11), yields the correct rate of flow of momentum, including both electromagnetic and mechanical momenta.**

*B*## References and links

1. | J. D. Jackson, |

2. | R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. |

3. | M. Mansuripur, “Radiation pressure and the linear momentum of the electro-magnetic field in magnetic media,” Opt. Express |

4. | T. B. Hansen and A. D. Yaghjian, |

5. | B. A. Kemp, J. A. Kong, and T. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A |

6. | A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Anten. Prop. |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 27, 2007

Revised Manuscript: March 25, 2008

Manuscript Accepted: March 27, 2008

Published: April 1, 2008

**Citation**

Masud Mansuripur, "Electromagnetic stress tensor in ponderable media," Opt. Express **16**, 5193-5198 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5193

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### References

- J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.
- 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]
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13502-13518 (2007). [CrossRef] [PubMed]
- T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications, IEEE Press, New York (1999). [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]

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