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We give theoretical foundation to torque densities proposed in the past, like the one used by Beth to study experimentally the action of circularly polarized radiation on a birefringent material, or that proposed by Mansuripur to resolve a seeming paradox concerning the Lorentz force law and relativity. Our results provide new insights into electromagnetic theory, since they provide a unified and general treatment of the balance of lineal and angular momentum that permits a better assessment of some torques. Thus in this work we extend the derivations we have made of balance equations for electromagnetic linear momentum to balance equations for electromagnetic angular momentum. These balance equations are derived from the macroscopic Maxwell equations by means of vector and tensor identities and from the different ways in which these equations are written in terms of fields
* E, D, B,* and
*H,* as well as polarizations
*P,* and
*M*. Therefore these balance equations are as sound as the macroscopic Maxwell equations, with the limitations of the constitutive relations.

The classical theory of electromagnetism is a well established theory; however, there are still some conceptual problems which deserve insightful reflections, particularly in the theory of electromagnetic media.

The understanding of the interaction of electromagnetic fields and matter constitutes an open question [

A very fundamental conceptual aspect in Maxwell’s theory of electromagnetism is that the electromagnetic field is a kind of generalized mechanical system which has energy, momentum, stresses, and angular momentum. We have then balance equations for energy, momentum, angular momentum and energy-momentum- stress. The energy balance is expressed in Poynting’s theorem, derived directly as a general integral of Maxwell’s equations. The balance equation for electromagnetic momentum, however, is usually derived from the Lorentz force law [

In many works on the angular momentum of radiation it is usual to begin either with an expression for the angular momentum of radiation [

We have shown in past works [

We begin with the most usual form of writing the Maxwell equations,

∇ ⋅ D = ρ , ∇ ⋅ B = 0 , ∇ × E + ∂ t B = 0 , ∇ × H − ∂ t D = J . (1)

The corresponding momentum balance equation deduced from these equations with the aid of vector and tensor identities is

∇ ⋅ { ( D E + B H ) − 1 2 I ( D ⋅ E + B ⋅ H ) } − ∂ t ( D × B ) = ρ E + J × B + 1 2 [ ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E + ( ∇ H ) ⋅ B − ( ∇ B ) ⋅ H ] . (2)

This balance equation contains as a particular case the usual Lorentz force; we simply specialize the equation for vacuum and take the limit of a test charge.

Other form of expressing the Maxwell equations, also usual and appearing in the textbook by Panofsky and Phillips [

∇ ⋅ E = ( 1 / ε 0 ) ( ρ − ∇ ⋅ P ) , ∇ ⋅ B = 0 , ∇ × E + ∂ t B = 0 , ∇ × B − ε 0 μ 0 ∂ t E = μ 0 ( J + ∂ t P + ∇ × M ) , (3)

whose associated momentum balance equation obtained with the same method is

∇ ⋅ { ( ε 0 E E + ( 1 μ 0 ) B B ) − 1 2 I ( ε 0 E ⋅ E + ( 1 μ 0 ) B ⋅ B ) } − ε 0 ∂ t ( E × B ) = ρ E + J × B − ( ∇ ⋅ P ) E + ( ∂ t P ) × B + ( ∇ × M ) × B . (4)

Again, the Lorentz force density appears as a particular case.

A third way of expressing the Maxwell equations is

∇ ⋅ D = ρ , ∇ ⋅ B = 0 , ∇ × E + μ 0 ∂ t H = − μ 0 ∂ t M , ∇ × H − ε 0 ∂ t E = J + ∂ t P , (5)

with an associated momentum balance equation given by

∇ ⋅ { ( D E + B H ) − 1 2 I ( ε 0 E ⋅ E + μ 0 H ⋅ H ) } − ε 0 μ 0 ∂ t ( E × H ) = ρ E + μ 0 J × H + ( P ⋅ ∇ ) E + μ 0 ( ∂ t P ) × H − ε 0 μ 0 ( ∂ t M ) × E + μ 0 ( M ⋅ ∇ ) H . (6)

If fields D and B are considered as the fundamental fields, then the Maxwell equations may be expressed in the form

∇ ⋅ D = ρ , ∇ ⋅ B = 0 , ∇ × D + ε 0 ∂ t B = ∇ × P , ∇ × B − μ 0 ∂ t D = μ 0 J + μ 0 ∇ × M , (7)

and the associated momentum balance equation results

∇ ⋅ { ( 1 ε 0 D D + 1 μ 0 B B ) − 1 2 I ( 1 ε 0 D ⋅ D + 1 μ 0 B ⋅ B ) } − ∂ t ( D × B ) = 1 ε 0 ρ D + J × B + 1 ε 0 ( ∇ × P ) × D + ( ∇ × M ) × B . (8)

We have yet other form of expressing the Maxwell Equations, which is

∇ ⋅ E = ( 1 / ε 0 ) ( ρ − ∇ ⋅ P ) , ∇ ⋅ H = − ∇ ⋅ M , ∇ × E + μ 0 ∂ t H = − μ 0 ∂ t M , ∇ × H − ε 0 ∂ t E = J + ∂ t P , (9)

with the following balance equation

∇ ⋅ { ( ε 0 E E + μ 0 H H ) − 1 2 I ( ε 0 E ⋅ E + μ 0 H ⋅ H ) } − ε 0 μ 0 ∂ t ( E × H ) = ρ E + μ 0 J × H − ( ∇ ⋅ P ) E − μ 0 ( ∇ ⋅ M ) H + μ 0 ( ∂ t P ) × H + ε 0 μ 0 E × ( ∂ t M ) . (10)

There are other forms of expressing the Maxwell equations, but these five different ways of expressing these equations are sufficient to illustrate the method for deducing now balance equations of electromagnetic angular momentum, and for the analysis of the torque used by Beth [

We can observe that all the above balance equations have the structure

∇ ⋅ T ↔ − ∂ t g = f L + δ f (11)

where the tensor or dyad T ↔ has the form

T ↔ = T ↔ i − I u i (12)

(sub index i refers to the different tensors that can be constructed) while f L + δ f are force densities; f L is a force density analogous to that of Lorentz, which involves charge and current densities, and δ f is a force involving only fields [

The tensor T ↔ i is the addition of dyads of type FG, where F and G are either electric vectors (for example ED) or magnetic vectors (for example BH), while u i is an expression of type energy density, that is, the scalar product of electric vectors plus the scalar product of magnetic vectors; g is the vector product of an electric vector and a magnetic vector with units of momentum density (as the Poynting vector divided by c^{2}). With this general scheme we can analyze all the above balance equations.

Before beginning the derivation of balance equations of angular momentum it is necessary to note that in the above balance equations we have considered only isotropic media. However, if we want to analyze experiments like the one performed by Beth [^{ }

Then we have

T ↔ i = ( 1 / 2 ) ( F G + G F ) + ( 1 / 2 ) ( F G − G F ) , (13)

where the symmetrical and anti-symmetrical parts of the tensor T ↔ are defined by

T ↔ S | i = ( 1 / 2 ) ( F G + G F ) − I u i (14)

and

T ↔ A | i = ( 1 / 2 ) ( F G − G F ) , (15)

respectively. With this separation we can write the general balance Equation (11) as

∇ ⋅ ( T ↔ S | i + T ↔ A | i ) − ∂ t g = f L + δ f . (16)

In order to obtain a general balance equation of angular momentum we multiply vectorially this equation by −r on the right, getting

− ∇ ⋅ ( T ↔ S | i + T ↔ A | i ) × r − ∂ t ( r × g ) = r × ( f L + δ f ) . (17)

Now we want to rewrite the first term on the left as a divergence of a tensor which looks like ∇ ⋅ ( T ↔ × r ) , and thus it is necessary to find a relation between

∇ ⋅ ( T ↔ S | i + T ↔ A | i ) × r and ∇ ⋅ [ ( T ↔ S | i + T ↔ A | i ) × r ] . We proceed as follows.

Given the general form of the tensor T ↔ i it is necessary to analyze only a second rank tensor of type FG, and then to construct its symmetrical and anti-symmetrical parts.

Then by means of the identity

∇ ⋅ [ F η ] = ( ∇ ⋅ F ) η + ( F ⋅ ∇ ) η (18)

we obtain

( ∇ ⋅ F G ) × r = ( ∇ ⋅ F ) G × r + [ ( F ⋅ ∇ ) G ] × r . (19)

But we have that

( F ⋅ ∇ ) ( G × r ) = [ ( F ⋅ ∇ ) G ] × r + G × F (20)

since ( F ⋅ ∇ ) r = F , then

( ∇ ⋅ F G ) × r = ( ∇ ⋅ F ) G × r + ( F ⋅ ∇ ) ( G × r ) − G × F . (21)

On the other hand, by using identity (18) we get

∇ ⋅ [ F ( G × r ) ] = ( ∇ ⋅ F ) ( G × r ) + ( F ⋅ ∇ ) ( G × r ) , (22)

from which it follows that

( F ⋅ ∇ ) ( G × r ) = ∇ ⋅ [ F ( G × r ) ] − ( ∇ ⋅ F ) ( G × r ) . (23)

Substituting this result in Equation (21) we obtain

( ∇ ⋅ F G ) × r = ∇ ⋅ [ F ( G × r ) ] − G × F , (24)

which is the result we were looking for.

This equation can be expressed in terms of a dual tensor, defined as

T i j d = θ k = ( 1 / 2 ) ϵ k i j T i j . (25)

Then, if T i j = F i G j , we have

F × G = 2 T ↔ d . (26)

Using now the results expressed in Equations (24) and (26) permits to rewrite the left-hand member of Equation (17) as

[ ∇ ⋅ ( T ↔ S | i + T ↔ A | i ) ] × r = ∇ ⋅ { ( T ↔ S | i + T ↔ A | i ) × r } + 2 T ↔ d . (27)

Then Equation (17) can be written in the form

∇ ⋅ M ↔ − ∂ t l = ( τ L + δ τ ) + 2 T ↔ d , (28)

where

M ↔ = − T ↔ | i × r , l = r × g , τ L = r × f L , δ τ = r × δ f . (29)

It is important to note that the general balance equation contains a new torque of type F × G , exactly of the form considered by Beth [

We can observe that in the balance Equations (3), (5), (7), (9) and (11) only Equations (3) and (7) contain a non-symmetrical tensor (in fact the same tensor). The other equations contain different but symmetrical tensors. In a vacuum they are equal, and then in this case the balance equations of angular momentum do not contain terms of type F × G .

Now, it is possible to obtain directly the balance equations of angular momentum from Equation (29) and Equations (3), (5), (7), (8) and (11). In order to express them more compactly it is convenient to define the tensors

T ↔ 1 = ( 1 2 ) [ ( D E + B H ) + ( E D + H B ) ] + ( 1 2 ) [ ( D E + B H ) − ( E D + H B ) ] − 1 2 I ( D ⋅ E + B ⋅ H ) , (30)

T ↔ 2 = ε 0 E E + ( 1 μ 0 ) B B − 1 2 I ( ε 0 E ⋅ E + ( 1 μ 0 ) B ⋅ B ) , (31)

T ↔ 3 = ( 1 2 ) [ ( D E + B H ) + ( E D + H B ) ] + ( 1 2 ) [ ( D E + B H ) − ( E D + H B ) ] − 1 2 I ( ε 0 E ⋅ E + μ 0 H ⋅ H ) , (32)

T ↔ 4 = 1 ε 0 D D + 1 μ 0 B B − 1 2 I ( 1 ε 0 D ⋅ D + 1 μ 0 B ⋅ B ) (33)

and

T ↔ 5 = ε 0 E E + μ 0 H H − 1 2 I ( ε 0 E ⋅ E + μ 0 H ⋅ H ) . (34)

With these tensors the balance equations can be expressed in the form

− ∇ ⋅ ( T ↔ 1 × r ) − ( D × E + B × H ) − ∂ t ( r × ( D × B ) ) = r × ( ρ E + J × B + 1 2 [ ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E + ( ∇ B ) ⋅ H − ( ∇ H ) ⋅ B ] ) , (35)

− ∇ ⋅ ( T ↔ 2 × r ) − ∂ t r × ( ε 0 E × B ) = r × ( ρ E + J × B − ( ∇ ⋅ P ) E + ( ∂ t P ) × B + ( ∇ × M ) × B ) , (36)

− ∇ ⋅ ( T ↔ 3 × r ) − ( D × E + B × H ) − ∂ t ( r × ( ε 0 μ 0 E × H ) ) = r × ( ρ E + μ 0 J × H + ( P ⋅ ∇ ) E + μ 0 ( ∂ t P ) × H − ε 0 μ 0 ( ∂ t M ) × E + μ 0 ( M ⋅ ∇ ) H ) , (37)

− ∇ ⋅ ( T ↔ 4 × r ) − ∂ t ( r × ( D × B ) ) = r × ( 1 ε 0 ρ D + J × B + 1 ε 0 ( ∇ × P ) × D + ( ∇ × M ) × B ) , (38)

− ∇ ⋅ ( T ↔ 5 × r ) − ∂ t ( r × ( ε 0 μ 0 E × H ) ) = r × ( ρ E + μ 0 J × H − ( ∇ ⋅ P ) E − μ 0 ( ∇ ⋅ M ) H + μ 0 ( ∂ t P ) × H + ε 0 μ 0 E × ( ∂ t M ) ) . (39)

We have in this way the desired balance equations. We have obtained them directly from the Maxwell equations through the balance equations of electromagnetic lineal momentum, and therefore they are all as sound as the Maxwell equations. What equation is convenient to use depends on the particular physical problem we want to solve. For example, if we want to study the interaction between radiation and a non isotropic media, then we have only Equations (35) and (37) as tools for the analysis of the problem. Thus a problem will require, or will be simpler, if the convenient fields and polarizations are used.

As we mentioned in the introduction, the usual way of trying to obtain a momentum balance equation is through the density force, expressed in the Lorentz force density [

In the present case we have that the angular momentum densities associated to each of the obtained balance equations are

l 1 = r × ( D × B ) l 2 = r × ( ε 0 E × B ) l 3 = r × ( ε 0 μ 0 E × H ) l 4 = r × ( D × B ) l 5 = r × ( ε 0 μ 0 E × H ) . (40)

It is interesting that, l 1 = l 4 , consistent with Minkoski’s proposal for the definition of the electromagnetic momentum density, while l 3 = l 5 , consistent with Abraham’s definition of electromagnetic momentum density. However, l 2 is not associated with any of these two proposals. Again, we see that the Abraham-Minkowski controversy arises from considering their proposals without considering the balance equations to which they pertain, observing that there are other possible momentum densities.

We emphasize that the balance equations obtained, both of lineal and angular momentum, are as general as the Maxwell equations, from which they are derived by means of vector and tensor identities. In this way our approach permits to obtain, by taking particular or limit cases of the balance equations, different proposals of electromagnetic lineal and angular momentum densities and others that would contain P × B , as well as the force and torque densities. For example, all the balance equations of lineal momentum have in the limit of a test charge and for a vacuum the usual Lorentz force density, which is taken in the usual approach as a postulate, independent of the Maxwell equations, and as the point of departure to propose force and torque densities. We can then examine some of the proposed balance equations, or proposed lineal and angular momentum densities, or force and torque densities, as follow.

1) Jackson’s proposal of a balance equation

Since Jackson’s text [

∇ ⋅ M ↔ ′ − ∂ t ( l m e c h + l e l m ) = 0 , (41)

where

M ↔ ′ = T ↔ × r , l e l m = r × g ′ . (42)

Here T ↔ is Maxwell’s tensor,

T ↔ = D E + B H − 1 2 I ( D ⋅ E + B ⋅ H ) , (43)

and g ′ is the lineal momentum density proposed by Minkowski,

g ′ = D × B . (44)

However, Jackson considers only homogeneous and isotropic media, so that it seems that the balance equations of lineal and angular momentums are almost identical with those for vacuum, that is, they have the structure of Equation (41). If we compare Jackson’s proposal with one of the balance equations obtained with our approach, we could identify it with Equation (35) for the case in which there are not free charge and current densities, but we note that the term

δ f = 1 2 [ ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E + ( ∇ H ) ⋅ B − ( ∇ B ) ⋅ H ] (45)

is absent. This means that this balance equation is inappropriate to deal with force densities that arise in treating inhomogeneous media, since it lacks the Helmholtz force density, the radiation force, or the torque on birefringent media considered by Beth [

It important to note that the balance equations of the form of Equation (41) are valid only for vacuum, or for media of constant permittivity and permeability. The balance equations obtained with our method are more general and more complex, the extra terms being relevant for the cases mentioned above.

2) Balance equation to understand Beth’s work

In his work Beth considers a torque P × E , without indicating where it comes from. This torque is also mentioned in Bohren’s work [

Since both are equivalent, we will use Equation (35) without free charge or current densities. Then Equation (35) in this case is

− ∇ ⋅ ( T ↔ 1 × r ) − ( D × E + B × H ) − ∂ t ( r × ( D × B ) ) = r × 1 2 ( ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E + ( ∇ H ) ⋅ B − ( ∇ B ) ⋅ H ) , (46)

where the tensor T ↔ ′ 1 is defined as

T ↔ ′ 1 = ( 1 2 ) [ ( D E + B H ) + ( E D + H B ) ] + ( 1 2 ) [ D E − E D ] − 1 2 I ( D ⋅ E + B ⋅ H ) . (47)

It is enough to express it as

T ↔ ′ 1 = D E + B H − 1 2 I ( D ⋅ E + B ⋅ H ) .

In the present case we note that, with the aid of the constitutive relations, we can write

D × E + B × H = P × E + μ 0 M × H (48)

which for non magnetic media reduces to

D × E + B × H = P × E = τ Beth , (49)

where τ Beth is the torque density proposed by Beth and the tensor T ′ 1 now is

T ↔ ′ 1nonmag = D E − 1 2 I ( D ⋅ E ) . (50)

Then, for non magnetic media Equation (35) becomes

− Ñ ⋅ ( T ↔ ′ 1nonmag × r ) − ( P × E ) − ∂ t ( r × ( D × B ) ) = r × 1 2 ( ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E ) , (51)

while the integration of Equation (50), with the tensor (51), over a volume V is

∫ d 3 r ( − ∇ ⋅ { [ D E − 1 2 I ( D ⋅ E ) ] × r } ) + ∫ d 3 r ( − ( P × E ) − ∂ t ( r × ( D × B ) ) ) = ∫ d 3 r ( r × 1 2 ( ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E ) ) . (52)

Gauss’s theorem permits us to change the volume integration by a surface integration, resulting

∮ ( − d σ ⋅ { [ D E − 1 2 I ( D ⋅ E ) ] × r } ) + ∫ d 3 r ( − ( P × E ) − ∂ t ( r × ( D × B ) ) ) = ∫ d 3 r ( r × 1 2 ( ( ∇ E ) ⋅ D − ( ∇ D ) ⋅ E ) ) . . (53)

We observe that there appear integrals of type

∮ d σ ⋅ F G × r

and

∮ d σ ⋅ I u × r . (54)

In order to make the integration easier we can select the integration surface so that

d σ = ( d σ ) k ^ (55)

where k ^ is a unitary vector in the z direction. Now we use the fact that an electromagnetic wave of finite transversal section has components in the propagation direction, and then in our case a wave has a radial component as well as a component in the propagation direction. Therefore we can write

F G = ( F r r ^ + F z k ^ ) ( G r r ^ + G z k ^ ) (56)

and the integrand is of type

k ^ ⋅ [ ( F r r ^ + F z k ^ ) ( G r r ^ + G z k ^ ) ] × r = F z G z φ ^ (57)

On the other hand we have that

k ^ ⋅ ( I u ) × r = u φ ^ (58)

also in direction φ ^ . The implications of these results is that there is no contribution to the transfer of angular momentum due to the tensor T 1 , so that the balance of angular momentum becomes

− ∫ d 3 r ∂ t ( r × ( D × B ) ) = ∫ d 3 r ( P × E ) (59a)

or equivalently, for a wave incident from vacuum, to

− ∫ d 3 r ε 0 ∂ t ( r × ( E × B ) ) = ∫ d 3 r ( P × E ) . (59b)

If we interpret that the left term is the rate of change of angular momentum, then for the z component the result is

− ∫ d 3 r ε 0 ∂ t ( r × ( E × B ) ) z = ∫ d 3 r ( P × E ) z (60)

which is precisely the balance on which Beth’s work is founded.

3) Balance of angular momentum to solve Mansuripur’s paradox

Recently Mansuripur [

f | Ein-Laub = ρ E + μ 0 J × H + ( P ⋅ ∇ ) E + μ 0 ( ∂ t P ) × H − ε 0 μ 0 ( ∂ t M ) × E + μ 0 ( M ⋅ ∇ ) H . (61)

Mansuripur [

τ = r × f | Ein-Laub + P × E + μ 0 M × H . (62)

(He uses a different definition of the constitutive relation between the magnetic field and the magnetic induction. We use the standard definition, so there are some differences in μ 0 factors). We can see that these torques are proposed with plausibility arguments founded on the Lorentz force law, while we offer a general and unified approach based on balance equations derived from the Maxwell equations.

Our approach let us to obtain, among others, these force and torque densities directly from the Maxwell equations. The respective balance equations are Equation (6) and Equation (37), which gives a sounder foundation to these proposals. Then instead of taking the Lorentz force density as a postulate independent of the macroscopic Maxwell equations, we derive from these several balance equations from which several force and torque densities emerge in a natural way, among them the usual Lorentz force density in the limit of a test charge in a vacuum. Since the Maxwell equations are intrinsically relativistic, there is no conflict with relativity.

A point that must be noted is that Masuripur [

P = p δ ( r − r ′ )

or

M = m δ ( r − r ′ ) . (63)

This is a delicate point that deserves further discussion, but is not essential for the argumentation here exposed. Our point is that the Lorentz force law is not independent from the Maxwell equations, but a particular case of the balance equations deduced from these equations.

4) Open questions about these balance equations

We have shown that the structure of the Maxwell equations can be transformed into the structure of momentum and angular momentum balance equations. However, we still do not consider dispersive media, or thermodynamic effects on electromagnetism. As an example of efforts in this direction we can mention Barnett’s proposal about the flux of angular momentum [

In this work we have extended some electromagnetic linear momentum balance equations, to angular momentum balance equations, obtaining in this way different torque densities. In past works we have shown how the macroscopic Maxwell equations can be transformed into different balance equations for linear momentum. These balance equations imply different force densities that have as particular or limit cases the Lorentz force density, the Helmholtz force density, or the Einstein-Laub [

Campos-Flores, I., Jiménez-Ramírez, J.L. and Roa-Neri, J.A.E. (2017) Balance Equations of Electromagnetic Angular Momentum. Journal of Electromagnetic Analysis and Applications, 9, 203-217. https://doi.org/10.4236/jemaa.2017.912017