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In this paper I compare the Abraham and the Minkowski forms for the momentum pertaining to an electromagnetic wave inside a dielectric or a magnetic material. The discussion is based on a careful treatment of the surface charges and currents and of the forces acting on them. While in the dielectric case the Abraham momentum is certainly more appealing from the physical point of view, for a magnetic material it suggests an interpretation in terms of magnetic charges and related magnetic currents. The Minkowski momentum for magnetic non conducting materials, on the contrary, has a natural interpretation in terms of an amperian model, in which the dynamics is determined by the Lorentz force acting on bulk and surface electric currents.

At the beginning of the twentieth century considerable attention has been devoted to the determination of the form of the mechanical momentum to be assigned to an electromagnetic (e.m.) wave propagating inside a given material. Two contrasting solutions to this problem were proposed at the time, the Minkowski espression [

and the Abraham one [

where ^{1} with arguments alternatively in favour of one or the other proposal. The reason for this variety of conclusions is due to the fact that this is a question that finally can only be settled empirically, but experiments are difficult and often do not match the material’s idealizations needed to derive the theoretical results.

In Ref. [^{2}. Using an expansion in powers of the susceptibility

It is the purpose of the present paper to use and extend the methodology presented in [

In Section 2 we make some general considerations about the interaction of an e.m. wave with a material body.

In Sections 3, 4 and 5 we discuss the case of a dielectric material, both in the Abraham and the Minkowski setup.

Finally, in Section 6, we extend the analysis to the case of magnetic non conducting materials.

We consider an extended, in general deformable, body,

The equations of motion for this system, together with the action-reaction law, imply

where

If we choose the external forces so that every part of

must be satisfied. In Equations (5) and (6)

Integrating Equation (5) over the body volume gives

while, integrating Equation (6) over the body surface, gives

so that we have

where

Since the material body is in mechanical equilibrium, we have

Given an expression for

It is important to notice that, following the procedure just described, we do not need to restrict in any way the entity of the body stresses, provided we use, in the formulas we get, the dielectric and magnetic polarizabilities pertaining to the stressed body in equilibrium. In the following, in order not to make the presentation too heavy, we will consider the simplified situation in which the stressed deformed body is uniform with respect to its elecric and magnetic properties. The most general, non uniform case, can be easily treated along the same lines.

In the following sections we will examine the consequences of assuming for

Although it would be possible to schematize the transition of dielectric or magnetic polarizabilities from the material body to the vacuum in a continuous way, we will find it convenient to consider the electric and magnetic properties of the material body as homogeneous, with a discontinuity at the body surface.

A uniform linear3 dielectric

which defines the (constant) electric susceptibility

Given

polarization charges, with a bulk volume density

for simplicity we consider the case in which the uniform dielectric does not contain free charges4. In this case we have

surface charges with a surface density [

where the electric field

a polarization current

The presence of surface charges implies a discontinuity of the normal component of the electric field around

Each of the candidate expressions for the momentum

where

In the next subsections I will discuss the forces associated with the Abraham or the Minkowski choice.

In Ref. [

In Equations (17) and (18)

If we apply the same strategy to determine the force acting on a dielectric, under the hypothesis that the momentum has the Minkowski form, we get, from Equation (16),

The last term in Equation (19) can be transformed as

in the absence of free charges inside the uniform insulator.

Putting together Equations (17), (19) and (20) and taking into account the discontinuity of the electric field at the dielectric surface, Equation (15), we get, for the force exerted in the Minkowski case,

Equation (21) shows that the Minkowski force is the resultant of forces which are locally orthogonal to the insulator surface.

In the presence of polarizability analytical computations are, in general, not possible. We will therefore resort to a systematic small a-expansion scheme described in [

Taking into account that

where

From Equations (23) and (16) we immediately get the corresponding expression of the force acting on the insulator under the Minkowski hypothesis, to order a

We are now ready to discuss the Snell’s law within the Minkowski scheme.

It has been shown in [

In a recent paper [

In this section I follow the strategy of Ref. [

I consider the setup, relevant for the discussion of the Snell’s law, in which an e.m. wave packet with an initial momentum

The total Minkowski momentum

where Equation (24) has been used. In Equation (25)

Equation (26) implies

where we used, according to the a-expansion,

Consistently, Equation (27) reproduces the Minkowski expression for the momentum from which we started.

Moreover Equation (26), together with Equation (28), also gives

where

The conclusion of this computation is that both the Abraham and the Minkowski forms of the momentum are compatible with the Snell’s law. Of course there is a difference, which can be experimentally detected, between the forces acting on the insulator, while crossed by the e.m. wave, in the Abraham or the Minkowski case. In particular, while in the Minkowski case the force exerted by an e.m. wave, during crossing the dielectric boundary, is orthogonal to the boundary itself, this is not true in the case of the Abraham force, Equation (23).

In this section we extend the considerations of Section 4 of [

In

The Ampère equation then becomes, in the absence of “free currents”,

which allows the introduction of the magnetic field

In the linear regime

and we have

which defines the magnetic susceptibility

At the boundary surface,

where

Equation (35) is equivalent to the discontinuity due to a surface current [

Following Ref. [

where, in order to deal with the dicontinuities, we exclude a thin region

We have

where contributions from the surfaces at infinity have been neglected, since we always consider situations in which the e.m. field differs from zero only in a finite region of space.

We have

where Equations (35) and (36) have been used.

Equation (41) shows that the total force

In situations of weak magnetization,

in which

Equation (44) shows that the Minkowski choice, Equation (38), for the momentum corresponds to a force on magnetic materials simply described as the Lorentz force acting on surface and bulk magnetization currents.

We now consider, again for the same magnetic material

again regularized by the omission of the

where we used Equation (32) and we have again neglected contributions from the surfaces at infinity.

The first term in Equation (46) only requires the knowledge of the discontinuity conditions

We have

and the expression of the force, in the Abraham case, is

Equation (50) has a simple interpretation which can be clarified considering the case of a very small

Equation (51) represents the interaction of the electric field with a “magnetic” current

In this paper I examined some questions related to the Abraham-Minkowski controversy about the momentum to be attributed to an e.m. wave packet inside a polarizable material. The analysis has been performed from the point of view of the forces experienced by the (dielectric or magnetic) material and is based on a careful treatment of the surface charges and currents and the consequent discontinuities of the electric and magnetic fields. While these results do not require any approximation, except for the idealized definition of the involved materials, we also consider the expansion in powers of the susceptibility a, which allows to show that, at least up to first order in a, the validity of the Snell’s law for the refraction of e.m. waves is not able to discriminate between the Abraham and the Minkowski proposals.

The resulting picture is that the Abraham momentum implies that the action of an e.m. field on a dielectric can be understood quite naturally as the resultant of the usual e.m. forces acting on the polarization charges and currents. In the Minkowski case the form of the force on a dielectric material is not as natural. For magnetic materials, the Abraham form of the momentum implies a picture of the force as due to the action of the magnetic field H on a distribution of magnetic charges determined by the magnetization, along the lines discussed in Ref. [

The choice between these alternatives can only come from experiments.

MassimoTesta, (2016) A Comparison between Abraham and Minkowski Momenta. Journal of Modern Physics,07,320-328. doi: 10.4236/jmp.2016.73032