## Radiation pressure and the linear momentum of the electromagnetic field in magnetic media

Optics Express, Vol. 15, Issue 21, pp. 13502-13518 (2007)

http://dx.doi.org/10.1364/OE.15.013502

Acrobat PDF (232 KB)

### 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, ½

*×*

**E***/*

**M***c*

^{2}, analogous to the electromagnetic (or Abraham) momentum density, ½

*×*

**E***/*

**H***c*

^{2}. (Here

*and*

**E***are the magnitudes of the electric and magnetic fields, respectively, and*

**H***c*is the speed of light in vacuum.) This intrinsic mechanical momentum, associated with the magnetization

*in the presence of an electric field*

**M***, is apparently the same “hidden” momentum that was predicted by W. Shockley and R. P. James nearly four decades ago.*

**E**© 2007 Optical Society of America

## 1. Introduction

1. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

7. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

*ε*and permeability

*μ*.

*of a material medium as an Amperian current loop [14*

**M**14. B. D. H. Tellegen, “Magnetic-Dipole Models,” Am. J. Phys. **30**, 650 (1962). [CrossRef]

*. Our belief in the validity of this expression stems from our analysis of radiation pressure on semi-infinite slabs, the results of which turn out to be in complete agreement with the momentum conservation law.*

**M***, the Lorentz force of the electromagnetic field on induced electrical charges and currents [4–7*

**M**4. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. **49**, 821–838 (2002). [CrossRef]

**= (**

*F**∂*/

**P***∂t*)×

**, where**

*B***is the polarization density of the medium, and**

*P**=*

**B***μ*

_{o}(

*+*

**H***) is the magnetic induction. As for the induced (bound) charge density*

**M***ρ*= -

_{b}**∇**∙

*, the corresponding force density may be written as*

**P****= -(**

*F***∇**∙

*)*

**P****. For the LIH media discussed in the present paper,**

*E***∇**∙

*=*

**P***ε*

_{o}(

*ε*- 1)

**∇**∙

*vanishes everywhere within the bulk of the medium, thus confining the force of the*

**E***E*-field to surfaces and interfaces (where the induced charge density can be non-zero). An alternative formula for the

*E*-field’s contribution to the Lorentz force density,

*=(*

**F***∙*

**P****∇**)

**, has been used extensively in the literature [1–6**

*E*1. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

*total*force (and torque) on rigid bodies [12

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, Optical Trapping and Optical Micro-manipulation III, K. Dholakia and G. C. Spalding, Eds. (2006). [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]

*= -(*

**F****∇**∙

*)*

**P****can differ substantially from that obtained using**

*E***= (**

*F**∙*

**P****∇**)

**. In what follows, whenever the total force exerted by the**

*E**E*-field happens to be non-zero, we will present two sets of results, one for each formulation.

*ε*,

*μ*), to determine the radiation pressure at the entrance facet as well as the momentum density of the light inside the medium. The “Einstein box” Gedanken experiment [5

5. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1134–1140 (2004). [CrossRef]

7. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

*ε*,

*μ*) parameters; the direction of the force, which also depends on the aforementioned parameters, can be either expansive or compressive. In Sec. 6 we derive expressions for the optical force density at the sidewalls of finite-diameter beams inside a transparent medium (i.e., one for which both

*ε*and

*μ*are real-valued and have the same sign).

## 2. Lorentz force of the electromagnetic field on the magnetization of a medium

**(**

*M**x*,

*y*,

*z*,

*t*) =

*M*_{x}

*x̂*+

**M**_{y}

*ŷ*+

*M*_{z}

*ẑ*of a material at a given point in space and time is subject to the Lorentz force of the local magnetic field. The magnetic induction

**exerts a force on electric currents, and since**

*B**is ultimately rooted in Amperian current loops on the atomic scale [14*

**M**14. B. D. H. Tellegen, “Magnetic-Dipole Models,” Am. J. Phys. **30**, 650 (1962). [CrossRef]

*B*-field on

**as the sum of contributions from all the various atomic currents that make up the**

*M**M*-field. A current circulating around a small loop of area

*δ*

_{2}produces a magnetic dipole moment

**=**

*m**Iδ*

^{2}

**, where**

*n̂**is a unit vector perpendicular to the loop’s surface. Denoting the number density of the loops in the medium by*

**n̂***N*, we will have

*=*

**M***. Equivalently, one may assign a magnetic dipole moment*

**Nm***=*

**m**

**M**δ^{3}to each cubic region of volume

*δ*

^{3}; the three loop currents depicted in Fig. 1 will then be

*M*,

_{x}δ*M*, and

_{y}δ*M*, respectively.

_{z}δ*(green arrows) as well as the relevant components of*

**M***(brown, blue, and red arrows). According to the Lorentz law, the electromagnetic force on each leg of each loop is produced by the action of the local*

**B***B*-field. The various components of the force density (i.e., force per unit volume) for the loops of Fig. 1 may thus be written as follows:

**∇**∙

*= 0. As for the remaining terms, we note that the defining relation*

**B***=*

**B***μ*

_{o}(

**+**

*H***) indicates that a certain fraction of the**

*M**B*-field is produced by the local magnetization

*. If we exclude this part of*

**M***from exerting a force on its own progenitor, we are left with*

**B***μ*

_{o}

*as the effective field that exerts a force on the current loops. We thus have*

**H***of magnetic (or magnetizable) materials. When integrated over the volume of interest, Eq. (3a) should yield the total force exerted by the*

**M***H*-field on the magnetic dipoles of the material. In LIH materials where

*=*

**M***χ*and

**H****=**

*B**μ*

_{o}(1+

*χ*)

*=*

**H***μ*

_{o}

*μ*, the vector identity

**H***×(*

**A****∇**×

*) + (*

**A***∙*

**A****∇**)

*= ½*

**A****∇**(

*∙*

**A***) further simplifies Eq. (3a) as follows:*

**A***,*

**B***, and*

**H***beyond the defining relation*

**M***=*

**B***μ*

_{o}(

*+*

**H***) and the Maxwell equation*

**M****∇**∙

*= 0. In this formulation there is no natural way to introduce the magnetic charge density, which is usually defined as*

**B***ρ*= -

_{m}1**∇**∙

*and considered analogous to the bound electric charge density*

**M***ρ*= -

_{b}**∇**∙

*[16*

**P**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]

*E*-field in accordance with the Lorentz law

*=*

**F***ρ*, in our formulation there is no corresponding interaction between the magnetic charge density

_{b}**E***ρ*and the magnetic fields. Note, however, that when the field components whose derivatives appear in Eq. (3a) happen to be discontinuous at the boundaries and interfaces between adjacent media, one must be careful to account for the forces experienced by the magnetic dipoles at such boundaries.

_{m}*is analogous to that of the Lorentz force experienced by the polarization*

**M***, namely,*

**P***=(*

**F**_{e}*∙*

**P****∇**)

*+(∂*

**E***/∂*

**P***t*)×

*, which can be equivalently written as*

**B***=(*

**F**_{e}*∙*

**P****∇**)

*+*

**E***×(*

**P****∇**×

*) + ∂(*

**E***×*

**P***)/*

**B***∂t*. In LIH media, where

*=*

**P***ε*

_{o}(

*ε*- 1)

*, the force density experienced by*

**E***will be*

**P***(*

**F**e*x*,

*y*,

*z*,

*t*) = ½

*ε*

_{o}(

*ε*- 1)

**∇**(

*∙*

**E***)+ ∂(*

**E***×*

**P***)/*

**B***∂t*.

## 3. Radiation pressure and momentum in magnetic media

*ε*and

*μ*, respectively. In general,

*ε*and

*μ*are complex functions of the frequency

*f*; only when transparent materials are considered shall we assume that both

*ε*and

*μ*are real-valued. By convention, the real parts of

*ε*and

*μ*could be positive or negative, but their imaginary parts are always greater than or equal to zero. In the following discussion both

*z*→ ∞. In particular, when both

*ε*and

*μ*are real and negative (i.e., the case of “negative-index” materials), one may resort to a limiting argument to show that

*πf*[(

*z*/

*c*) -

*t*]} and field amplitudes

*E*

_{o}and

*H*

_{o}=

*E*

_{o}/

*Z*

_{o}, where

*Z*

_{o}=

*ε*and

*μ*from the free space region on the left-hand side. The Fresnel reflection coefficient at the entrance facet of the slab is readily found to be

*πf*[

*z*/

*c*) -

*t*]}, while the field amplitudes are

*E*= (1 +

_{x}*ρ*)

*E*

_{o}and

*H*= (1 +

_{y}*ρ*)

*E*

_{o}/

*Z*

_{o}. Using the time-averaged Poynting vector component along

*z*, namely, <

*S*> = ½Re(

_{z}*E*

_{x}H_{y}^{*}), it is easy to verify that the rate of flow of optical energy in the incident beam minus that in the reflected beam is exactly equal to the rate of flow of energy into the slab, namely,

*B*-field on the bound current density

*=*

**J**_{b}*∂*/

**P***∂t*, as well as that on the magnetization

*= (*

**M***μ*- 1)

*. According to Eq. (3a), the force density experienced by the magnetic dipoles of the slab in the system of Fig. 2 is*

**H***μ*

_{o}

*×(*

**M****∇**×

*); the second term vanishes for this geometry. [Note that the contribution of ∂(*

**H***×*

**E***/*

**M***c*

^{2})/∂

*t*may be ignored for now, as this term’s time-average is zero under steady-state conditions. Also, the contribution of bound charges to the Lorentz force,

*ρ*, is zero in the present example, as

_{b}**E***ρ*= 0 both inside the slab and at its entrance facet.] The total force density is thus given by

_{b}^{nd}equation,

**∇**×

*= ∂*

**H***/∂*

**D***t*, simplifies the above expression to

*E*- and

*H*-fields, and assuming for the moment that

*ε*and

*μ*are complex-valued, the time-averaged force density, integrated over the thickness of the slab (i.e.,

*z*ranging from 0 to ∞), is evaluated as follows:

*F*in Eq. (7) is force per unit volume, whereas

_{z}*F*in Eq. (8) represents force per unit surface area of the slab. The last line of Eq. (8) is precisely the result one would expect from a consideration of the rates of flow of incident and reflected momenta in the free space, namely,

_{z}*ε*and

*μ*, is an extremely important factor in support of the Lorentz force formula of Eq. (6) and, by extension, of Eq. (3). Similar equalities obtained in Secs. 7 and 8, in cases of oblique incidence with p- and s-polarized light, lend further credence to Eq. (3) as the correct expression for the Lorentz force exerted by the electromagnetic field on the material’s magnetization

*under steady-state conditions.*

**M***μ*, it turns out to be valid even when these parameters are real-valued; the reason being that the term Im(

*z*) in the denominator in the second line of Eq. (8), cancels out in the end; thus the fact that Im(

*√*/

*μ*) → 0 for a transparent medium does not affect the final result. In a transparent medium where

*ε*and

*μ*are real-valued (both positive or both negative), <

*F*> of Eq. (8) should be interpreted as the time-averaged rate of momentum flow at any cross-section of the beam inside the slab. In terms of the field amplitudes |

_{z}*E*| and |

_{x}*H*| within the transparent medium, Eq. (8) may be rewritten as

_{y}*ε*/

*μ*and

*ε*and

*μ*are both positive or both negative. In a transparent, dispersionless medium where the speed of light is

*V*=

*c*/

*ε*and

*μ*are necessarily positive, as negative-index materials cannot be free from dispersion), the momentum per unit volume becomes

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]

*μ*= 1, the two terms on the right-hand side of Eq. (11) correspond, respectively, to the Abraham and Minkowski momentum densities, yielding a net density that is half-way between the two, as has been argued in our previous papers [7

7. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

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]

*ε*and

*μ*, Eqs. (5) and (9) yield the rates of flow of optical energy and momentum into a transparent slab. For an incident pulse of duration

*τ*and cross-sectional area

*A*, the number of photons entering the slab will be <

*S*>

_{z}*Aτ*/(

*hf*), where

*h*is Planck’s constant; therefore, the momentum of each photon inside the (transparent) material will be

*ε*and permeability

*μ*is applicable to both positive- and negative-index media (in both cases

*ε*/

*μ*> 0). The value of

*P*

_{photon}in Eq. (12) is always greater than or equal to the free-space momentum

*hf*/

*c*. For a proof, note that ½ (

*ε*/

*μ*+

*ε*/

*μ*) ≥ 2

^{2}≥ 0, an obviously valid inequality.

21. M. Mansuripur, “Momentum of the electromagnetic field in transparent dielectric media,” SPIE Proc.6644, Optical Trapping and Optical Micro-manipulation IV, K. Dholakia and G. C. Spalding, Eds. (2007). [CrossRef]

*ε*and

*μ*(both real and positive, of course, as negative-index materials cannot be free from dispersion), without changing the final conclusions. In Ref. [21

21. M. Mansuripur, “Momentum of the electromagnetic field in transparent dielectric media,” SPIE Proc.6644, Optical Trapping and Optical Micro-manipulation IV, K. Dholakia and G. C. Spalding, Eds. (2007). [CrossRef]

*ε*| and permeability √|

*μ*|. A straightforward calculation similar to that in Ref. [7

**12**, 5375–5401 (2004). [CrossRef] [PubMed]

*ε*/

*μ*> 1 or

*μ*/

*ε*> 1. (When

*ε*=

*μ*, the slab’s Fresnel reflection coefficient is zero and, therefore, there is no need for AR coating.) The negative force on the AR coating layer given by Eq. (13) accounts for the excess photon momentum when it enters from the free space into a transparent slab.

## 4. Division of momentum into electromagnetic and mechanical parts

5. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1134–1140 (2004). [CrossRef]

*E*=

*mc*

^{2}and (free-space) momentum

*=*

**p***mc*travels either outside or inside a transparent slab of length

**ẑ***L*and mass

*M*

_{o}. The entrance and exit facets of the slab are anti-reflection coated to ensure the passage of the entire pulse through the slab, with no reflection losses whatsoever. The pulse crosses the slab in a time interval Δ

*t*=

*L*/

*V*, where the group velocity

_{g}*V*is a function of the optical frequency

_{g}*f*and the (frequency-dependent) material parameters

*ε*and

*μ*.

*z*-axis at the constant velocity

*V*=

_{CM}*mc*/(

*m*+

*M*

_{o}). The displacement of the center of mass during a time interval

*τ*is, therefore,

*mcτ*/(

*m*+

*M*

_{o}). If the pulse goes through the slab, however, its velocity, while inside the slab, will drop down to the group velocity

*V*. The emergent pulse will thus stay behind the pulse that has traveled in the free-space by a distance [(

_{g}*c*/

*V*) - 1]

_{g}*L*, as shown in Fig. 3. Therefore, for the system’s center of mass to be in the same place in both experiments, it is necessary for the slab in the latter case to have shifted to the right by Δ

*z*= [(

*c*/

*V*) - 1]

_{g}*Lm*/

*M*

_{o}. This displacement, which occurs during the time interval Δ

*t*=

*L*/

*V*when the pulse is inside the slab, requires the slab’s mechanical momentum during the passage of the pulse to be

_{g}*=*

**p***mc*, we conclude that the pulse’s electromagnetic momentum inside the slab must be

**ẑ***V*/

_{g}*c*relative to its free-space value [5

5. R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. **52**, 1134–1140 (2004). [CrossRef]

9. M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express **13**, 2245–2250 (2005). [CrossRef] [PubMed]

*V*=

_{g}*c*/

*n*,

*n*being the refractive index, the electromagnetic momentum of the pulse will be

*= (*

**P**_{E}*E*/

*nc*)

*ẑ*, which is commonly referred to as the Abraham momentum. The difference between the free-space momentum of the pulse and its electromagnetic (or Abraham) momentum is thus transferred to the slab in the form of mechanical momentum,

*, causing the slab’s eventual displacement in a manner consistent with the demands of the Einstein box experiment.*

**P**_{M}## 5. Mechanical momentum in dispersionless magnetic media

*×*

**E***/*

**M***c*

^{2}, thus suggesting the existence of an

*intrinsic*mechanical momentum associated with the magnetization of the medium. As it turns out, this seemingly implausible momentum was predicted nearly forty years ago by Shockley and James [22

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]

*×*

**E***/*

**M***c*

^{2}, is based on purely classical arguments, relying on Maxwell’s equations, standard constitutive relations, the expression of the Lorentz force on

*given in Eq. (3), and the conclusions reached from the “Einstein box” Gedanken experiment of Sec. 4. The relevant physics in which the inherent momentum is rooted, however, is quantum electrodynamics, as argued by Shockley [20*

**M**20. W. Shockley, “Hidden linear momentum related to the or *α* ∙ * E* term for a Dirac-electron wave packet in an electric field,” Phys. Rev. Lett.

**20**, 343–346 (1968). [CrossRef]

*ε*and

*μ*are real and positive, as lack of dispersion excludes negative-index media from such considerations. However, a treatment similar to that of Ref. [9

9. M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express **13**, 2245–2250 (2005). [CrossRef] [PubMed]

*H*=

_{y}*Z*

_{o}

^{-1}

*E*and ∂

_{x}*E*/∂

_{x}*t*+ (

*c*/

*E*/∂

_{x}_{z}= 0 have been used in the above derivation, the latter being a direct consequence of dispersionless propagation along

*z*at the constant velocity

*c*/

*z*-axis, from the mid-point of the pulse, say,

*z*=

*z*

_{o}to

*z*= +√ (leading edge) or

*z*= -0√ (trailing edge) yields

*c*/

*μ*

_{o}(

*+*

**H***) for*

**M***and (*

**B***ε*

_{o}

*+*

**E***) for*

**P***. The momentum density of a plane-wave inside a transparent, dispersionless medium may thus be written in the following equivalent form:*

**D***×*

**E***/*

**M***c*

^{2}, which appears to be some sort of mechanical momentum inherent in the magnetization of the material. In other words, just as the electromagnetic (or Abraham) momentum density inside the medium is ½

*×*

**E***/*

**H***c*

^{2}, there appears to reside within the medium an “intrinsic” mechanical momentum density of ½

*×*

**E***/*

**M***c*

^{2}as well. It is the sum of this intrinsic mechanical momentum and the momentum imparted to the medium by the leading edge of the pulse, given by Eq. (19), that produces the total mechanical momentum of Eq. (21). It thus appears that the missing force density at the pulse edges must be

*F*(

_{z}*z*,

*t*) = ∂(

*E*/

_{x}M_{y}*c*

^{2})/∂

*t*, which was mentioned toward the end of Sec. 2. When this new term is added to Eq. (16), the resulting mechanical momentum density in Eq. (19) will coincide with that given by Eq. (21). We emphasize that this additional force density does not modify the steady-state analysis of Sec. 3, as the time-averaged value of the new term would be zero everywhere. For the same reason, the new term will be ignored in the sections that follow, as the discussion in these sections is confined to steady-state situations.

*×*

**E***/*

**M***c*

^{2}mechanical momentum density. This momentum density in turn requires the addition of a new term, ∂(

*×*

**E***/*

**M***c*

^{2})/∂

*t*, to the classical formula for the Lorentz force density on magnetization

*, namely, Eq. (3). The physical mechanism responsible for the additional force is believed to be some sort of energy flux inside the Amperian current loops of magnetic dipoles, a quantum mechanical phenomenon that lies outside the domain of classical electrodynamics.*

**M**## 6. Lateral pressure at the sidewalls of a finite-diameter beam

*x*-axis, as shown in Fig. 4. When the beam is linearly polarized with its

*H*-field along the

*y*-axis (i.e., the case of transverse magnetic, TM, or p-polarization), the force density may be written as

^{nd}equation,

**∇**×

*=∂*

**H***/∂*

**D***t*, has been used in going from the second to the third line. Integrating the lateral force component

*F*from

_{x}*x*= 0 to ∞ yields

*H*

_{y}^{2}we obtain at the beam center, and using the relation |

*E*| =

_{x}*Z*

_{o}√

*μ*/

*ε*|

*H*|, we obtain

_{y}*μ*, and compressive otherwise. Upon setting

*μ*= 1, the above formula reduces to the result obtained for non-magnetic dielectrics in Ref [7

**12**, 5375–5401 (2004). [CrossRef] [PubMed]

*∙ ∇)*

**P***in addition to those already present in Eq. (22). The additional force density will be*

**E**^{nd}and 3

^{rd}equations we have ∂

*E*/∂

_{x}_{z}= ∂

*E*/∂

_{z}*x*-

*μ*

_{o}

*μ*∂

*H*/∂

_{y}*t*and ∂

*H*/∂

_{y}*x*=

*ε*

_{o}

*ε*∂

*E*/∂

_{z}*t*. We also know that ∂(

*E*)/∂

_{z}H_{y}*t*=

*E*/∂

_{z}∂H_{y}*t*+

*H*∂

_{y}*E*/∂

_{z}*t*and that

*E*= -

_{z}H_{y}*S*, where

_{x}*S*is the

_{x}*x*-component of the Poynting vector. Substitution in Eq. (25) then yields

*x*from 0 to ∞, averaging over time, and setting

*E*(

_{z}*x*= 0,

*y*,

*z*,

*t*) = 0, <

*E*

_{x}^{2}(

*x*= 0,

*y*,

*z*,

*t*) = ½|

*E*|

_{x}^{2}, and <

*H*

_{y}^{2}(

*x*= 0,

*y*,

*z*,

*t*) = ½|

*H*|

_{y}^{2}, we find

*ε*/

*μ*)-2

*ε*+ 1] in Eq. (28) is positive or negative, this sidewall force will be expansive or compressive. Equations (24) and (28) provide alternative expressions for radiation pressure at the sidewalls of a p-polarized beam. The former applies when the

*E*-field contribution to the Lorentz force is expressed as

*= -(*

**F****∇**∙

*)*

**P***; the latter when it is (*

**E***∙*

**P****∇**)

*. Both expressions apply to negative-index as well as positive-index media.*

**E***E*,

_{y}*H*,

_{x}*H*), and the force density in accordance with Eq. (3a) is written

_{z}*E*/∂

_{y}*x*= -∂

*B*/∂

_{z}*t*, ∂

*E*/∂

_{y}*z*=∂

*B*/∂

_{x}*t*, and ∂

*H*/∂

*z*-∂

*H*/∂

_{z}*x*= ∂

*D*/∂

_{y}*t*, the

*x*-component of the force density in Eq. (29) is found to be

*x*= 0 to ∞, recognizing that the

*x*-component of the Poynting vector,

*S*=

_{x}*E*, time-averages to zero (i.e., no net lateral energy flux), and using the fact that at the beam center

_{y}H_{z}*H*(

_{z}*x*=0,

*y*,

*z*,

*t*) vanishes, while

*E*(

_{y}*x*= 0,

*y*,

*z*,

*t*) = ½|

*E*|, and <

_{y}*H*(

_{x}*x*= 0,

*y*,

*z*,

*t*) = ½|

*H*|

_{x}^{2}= ½(

*ε*

_{o}/

*μ*

_{o})(

*ε*/

*μ*)|

*E*|

_{y}^{2}, we find

*E*-field contribution to the force is expressed as -(

**∇**∙

*)*

**P***or (*

**E***∙*

**P****∇**)

*, for s-light the lateral pressure given by Eq. (31) does not distinguish between the two alternatives, as the*

**E***E*-field in the latter case makes no contribution to the force.

## 7. Oblique incidence on a magnetic slab – case of p-polarization

*ε*and permeability

*μ*, as shown in Fig. 5(a). The various field components are listed below, with the subscripts

*i*,

*r*,

*t*referring to the incident, reflected, and transmitted beams:

*ρ*for a p-polarized plane-wave is

_{p}**12**, 5375–5401 (2004). [CrossRef] [PubMed]

*ε*

_{o}|

*E*

_{o}|

^{2}cos

*θ*[(1-|

*ρ*|)sin

_{p}*θ*+ (1 + |

**x̂***ρ*|)cos

_{p}*θ*], which is the combined rate of change of the incident and reflected momenta. (The extra cosθ factor is the ratio of the incident beam’s cross-sectional area to its footprint at the slab’s surface.) These formulas are valid for all values of

**ẑ***ε*and

*μ*, whether complex or real. For a transparent medium having

*εμ*≥ sin

^{2}

*θ*, the forces may be written in terms of the refracted angle

*θ*́, where sin

*θ*= √

*με*sin

*θ*́, and the

*E*-field magnitude inside the medium, namely,

**F**^{(bulk)}= ¼

*ε*

_{o}|

*E*|

_{t}^{2}[1+(

*ε*/

*μ*)]

*, which is consistent with Eq. (10). However, when the same beam propagates at an angle*

**ẑ***θ*́ relative to the surface normal, its cross-sectional area shrinks to cos

*θ*́, and its momentum flux becomes

*θ*́ factor in Eq. (38) is the excess length of the lower sidewall in Fig. 5(a). A more detailed discussion of this point is given in Ref. [7

**12**, 5375–5401 (2004). [CrossRef] [PubMed]

*∙*

**P****∇**)

*instead of -(*

**E****∇**∙

*)*

**P***, then the force density on the sidewalls will be given by Eq. (28) rather than Eq. (24). Also, the contribution of surface charges to the total force will differ from that given by Eq. (34b). However, the sum of all the forces on the slab turns out to be the same, irrespective of which formula is used to compute the forces. Listed below are expressions for the modified forces in the system of Fig. 5(a), when the*

**E***E*-field contribution to the Lorentz force is written as (

*∙*

**P****∇**)

*.*

**E**## 8. Oblique incidence on a magnetic slab – case of s-polarization

*i*,

*r*,

*t*refer to incident, reflected, and transmitted beams):

*ρ*for an s-polarized plane-wave is

_{s}*, and on the magnetization*

**J**_{b}*. The*

**M***E*-field contribution to the Lorentz force, whether expressed as -(

**∇**∙

*)*

**P***or (*

**E***∙*

**P****∇**)

*, turns out to be zero for s-light. However, both terms in Eq. (3a) will be needed to account for the force experienced by*

**E***. Unlike the case of p-light discussed in Sec. 7, there are no bound electric charges (or electric dipoles) at the entrance facet of the slab. Due to the discontinuity of*

**M***H*at the entrance facet, however, the magnetic dipoles, via the term

_{z}*μ*

_{o}(

*∙*

**M****∇**)

*, contribute a surface force. The expression of the Lorentz force density for s-light, given in Eq. (29), must be time-averaged and integrated over the (infinite) thickness of the slab to yield the force per unit surface area as follows:*

**H***με*≥ sin

^{2}

*θ*, the total force (per unit surface area) given by Eq. (42c) may be written in terms of the it-field magnitude inside the medium, |

*E*|= |

_{t}*E*|, and the refracted angle

_{y}*θ*́, where sin

*θ*́=√

*με*sin

*θ*́, as follows:

**F**_{1}is the rate of flow of momentum along the transmitted beam in accordance with Eq. (10); here the beam propagates at angle

*θ*́ and has cross-sectional area cos

*θ*́, exactly as in the case of p-light discussed in the preceding section. In fact,

**F**_{1}of Eq. (44a) is the same as

**F**^{(flux)}of Eq. (37). The second force,

**F**_{2}, is the excess force exerted on the beam’s lower sidewall; see Eq. (31) and the corresponding discussion in Ref. [7

**12**, 5375–5401 (2004). [CrossRef] [PubMed]

*θ*́ factor in Eq. (44b) is the excess length of the lower sidewall in Fig. 5(b). The third force,

**F**_{3}, is the surface force of Eq. (42b); just as the force density (

*∙*

**P****∇**)

*in the case of p-polarized light gave rise to*

**E**

**F**^{(surface)}of Eq. (39b), so is

*μ*

_{o}(

*∙*

**M****∇**)

*, in the present case, producing the force*

**H**

**F**_{3}at the surface. The only non-zero component of (

*∙*

**M****∇**)

*associated with the discontinuities at the entrance facet is*

**H***M*(∂

_{z}*H*/∂

_{z}*). Now, the*

**z***B*continuity allows one to determine

_{z}*M*from the discontinuity of

_{z}*H*, namely,

_{z}*z*at the entrance facet are acted upon by

*μ*

_{o}

*H*(

_{z}*x*,

*z*=0

^{+},

*t*) at one pole and by ½

*μ*

_{o}[

*H*(

_{z}*x*,

*z*= 0

^{-},

*t*) +

*H*(

_{z}*x*,

*z*= 0+,

*t*)] at the other pole. The net

*H*-field acting on this dipole layer is thus given by

*F*

_{z}^{(surface)}½ = ½Re(

*μ*

_{o}

*M*Δ

_{z}*H*

_{z}^{*}), may now be shown to be given by Eq. (44c). The bottom line is that the total force of Eq. (43), exerted on a transparent slab by an s-polarized beam at oblique incidence, is compatible with the rate of flow of momentum inside the slab as given by Eq. (10). The proof of this statement, however, required an analysis of the forces exerted on the medium within the triangular region immediately beneath the entrance facet of the slab of Fig. 5(b). In the above discussion, we showed that

**F**_{2}and

**F**_{3}of Eq. (44) are associated with the forces exerted on this triangular region, leaving the remaining term

**F**_{1}to account for the momentum flux along the beam’s propagation direction.

## 9. Concluding remarks

*×*

**E***/*

**M***c*

^{2})/∂

*t*, describes the force of the electromagnetic field on a material’s magnetization density

*. To demonstrate the validity of this formula, we compared the total force exerted on a semi-infinite slab illuminated by a plane-wave (both at normal incidence and at oblique incidence) with the rates of flow of the incident and reflected momenta. Complete agreement between the two methods of calculation was obtained in every case. Along the way, we obtained expressions for (i) the momentum density of electromagnetic waves inside linear, isotropic, homogeneous materials, Eq. (11); (ii) the photon momentum inside transparent magnetic media, Eq. (12); and (iii) the lateral force experienced by a host medium at the sidewalls of a finite-diameter beam, Eqs. (24), (28) and (31). Also, relying on an “Einstein box” Gedanken experiment, we concluded that the “intrinsic” mechanical momentum density ½*

**M***×*

**E***/*

**M***c*

^{2}of a magnetic medium arises from the interaction between

*and the electric component of the electromagnetic field.*

**M***and*

**P***, the total force exerted by the electromagnetic field is the sum of the forces experienced by*

**M***and*

**P***. The density of the*

**M***E*-field’s force on bound electric charges can be written either as -(

**∇**∙

*)*

**P***or as (*

**E***∙*

**P***)*

**∇***; the two formulations yield identical results for the total force (and total torque) on a given solid object, provided that the forces acting at the boundaries of the object are properly taken into account [12*

**E**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, Optical Trapping and Optical Micro-manipulation III, K. Dholakia and G. C. Spalding, Eds. (2006). [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]

*E*-field on the bound electrical charges is written as (

*∙*

**P****∇**)

*, the complete expression of the total force density will be*

**E****∇**∙

*)*

**P***. Now, using Maxwell’s equations to combine its various terms, Eq. (47) can be streamlined into the following “generalized Lorentz law” for isotropic media:*

**E***and*

**P***to the overall force density. It also indicates that, as far as electromagnetic force is concerned, the relevant fields are*

**M***and*

**E***(rather than*

**H***and*

**B***). This is noteworthy, considering that the same fields (*

**D***and*

**E***) also appear in the expression of the Poynting vector*

**H***.*

**S**20. W. Shockley, “Hidden linear momentum related to the or *α* ∙ * E* term for a Dirac-electron wave packet in an electric field,” Phys. Rev. Lett.

**20**, 343–346 (1968). [CrossRef]

*=*

**P***ε*

_{o}

*and magnetization*

**E***=*

**M***in the presence of the*

**H***and*

**E***fields, then the optical momentum density of ½*

**H***×*

**E***/*

**H***c*

^{2}in vacuum could be said to have arisen by equal contributions from this

*and*

**P***in the form of ¼/*

**M***μ*

_{o}

*×*

**P***and ¼*

**H***×*

**E***/*

**M***c*

^{2}.

## Acknowledgements

## References and links

1. | J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A |

2. | D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A |

3. | I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum. tensor,” Physics Reports |

4. | R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. |

5. | R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. |

6. | R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A |

7. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express |

8. | M. Mansuripur, A. R. Zakharian, and J. V. Moloney, “Radiation pressure on a dielectric wedge,” Opt. Express |

9. | M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express |

10. | M. Mansuripur, “Angular momentum of circularly polarized light in dielectric media,” Opt. Express |

11. | M. Mansuripur, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” SPIE Proc.5930, Optical Trapping and Optical Micromanipulation II, K. Dholakia and G. C. Spalding, Eds. (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, Optical Trapping and Optical Micro-manipulation III, K. Dholakia and G. C. Spalding, Eds. (2006). [CrossRef] |

13. | M. Mansuripur, “Radiation Pressure on Submerged Mirrors: Implications for the Momentum of Light in Dielectric Media,” Opt. Express |

14. | B. D. H. Tellegen, “Magnetic-Dipole Models,” Am. J. Phys. |

15. | S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B: At. Mol. Opt. Phys. |

16. | B. Kemp, T. Grzegorczyk, and J. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express |

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

18. | L. Vaidman, “Torque and force on a magnetic dipole,” Am. J. Phys. |

19. | A. D. Yaghjian, “Electromagnetic forces on point dipoles,” IEEE Anten. Prop. Soc. Symp. |

20. | W. Shockley, “Hidden linear momentum related to the or 20, 343–346 (1968). [CrossRef] |

21. | M. Mansuripur, “Momentum of the electromagnetic field in transparent dielectric media,” SPIE Proc.6644, Optical Trapping and Optical Micro-manipulation IV, K. Dholakia and G. C. Spalding, Eds. (2007). [CrossRef] |

22. | W. Shockley and R. P. James, “Try simplest cases discovery of hidden momentum forces on magnetic currents,” Phys. Rev. Lett. |

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,” Vol. II, Chap. 27, Addison-Wesley, Reading, Massachusetts (1964). |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 27, 2007

Revised Manuscript: September 26, 2007

Manuscript Accepted: September 27, 2007

Published: October 1, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Masud Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express **15**, 13502-13518 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-21-13502

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### References

- 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]
- I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum. tensor," Phys. Reports 52, 133-201 (1979). [CrossRef]
- R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002). [CrossRef]
- R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004). [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]
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004). [CrossRef] [PubMed]
- M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Radiation pressure on a dielectric wedge," Opt. Express 13, 2064-2074 (2005). [CrossRef] [PubMed]
- M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005). [CrossRef] [PubMed]
- M. Mansuripur, "Angular momentum of circularly polarized light in dielectric media," Opt. Express 13, 5315-5324 (2005). [CrossRef] [PubMed]
- 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]
- 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. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962). [CrossRef]
- 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]
- 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]
- 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]
- L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990). [CrossRef]
- A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).
- 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]
- M. Mansuripur, "Momentum of the electromagnetic field in transparent dielectric media," SPIE Proc. 6644, 664413 (2007). [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]
- 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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