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although the conservation of energy and momentum of electromagnetic fields in vacuum is a very well understood problem , inside a polarizable medium it is still a matter of discussion .
a review of the literature on this subject can be found in @xcite .    in the vacuum , poynting s equation for the exchange of energy between the field and free charges is easily derived starting from lorentz s force density on charges and currents @xmath2 and from the power density on the currents @xmath3 the resulting equation @xmath4 written in terms of @xmath5 and @xmath6 is an identity valid for any solution of maxwell s equations .
it is interpreted as a local energy conservation equation by taking @xmath7 as the energy density of the electromagnetic field and poynting s vector @xmath8 as the energy flux current vector . the power exerted by the matter on the fields
is assumed to be the opposite of ( [ power - density ] ) .
electromagnetic fields also carry momentum .
the derivation of the equation for the momentum exchange requires the use of maxwell s stress tensor and a somewhat more involved calculation which do not make easy an elementary presentation of the subject . in many textbooks ( e.g @xcite , @xcite )
radiation pressure is simply introduced as a new fact , sometimes referring to it as one of maxwell s achievements . in the vacuum
the momentum density is accepted to be @xmath9 which leads to the relation @xmath10 for the energy and momentum that an electromagnetic wave releases when absorbed .
because of this in other elementary presentations ( e.g @xcite ) the density of momentum ( [ momentum - density ] ) is justified by working on the analogy with a relativistic particle , whose momentum and energy are related by @xmath11 for an ultra relativistic particle ( @xmath12 ) holds @xmath13 by assuming that this last relation is also valid for a packet of electromagnetic waves traveling in some direction , one gets ( [ deltap ] ) and therefore @xmath14 in accord with ( [ momentum - density ] ) .
this procedure has the shortcoming of having a non electromagnetic element , the relativistic particle , introduced in the discussion .
a more enlighting treatment of the subject is the one presented for example in ref.@xcite .
there , the interaction of electromagnetic wave packets coming from vacuum with electric currents in a non ideal conductor are considered .
the impulse transferred to the currents is computed and showed to be consistent with ( [ momentum - density ] ) and ( [ deltap ] ) . in this paper we show that this same idea can also be applied to the propagation of light in matter .
as we already commented , what happens with momentum inside matter is still not completely clear from the theoretical point of view . for the energy ,
poynting @xcite proposed that eq.([poynting - eq ] ) remains valid with @xmath15 and @xmath16 here , @xmath17 is the free current density , @xmath18 is the electric displacement and @xmath19 is the magnetizing field .
it may be shown that eq.([poynting - eq ] ) is consistent with maxwell s equations only for materials with linear polarizations , @xmath20 and @xmath21 .
in particular it holds for a material with permittivity @xmath22 and permeability @xmath23 , @xmath24 and @xmath25\mathbf{b}$ ] , but it also holds in the more general situation of a non - isotropic material .
poynting s density is the vacuum energy density @xmath7 plus terms that depend on the polarization @xmath26 and the magnetization @xmath27 , @xmath28 the electric term corresponds to the energy density of deformation of matter as it develops electric dipole moments .
the magnetic term is the work density made on microscopic currents by the induced electric field .
the work done on electric and magnetic moments is already included in poynting s energy density , that is the reason why in the right hand side of equation ( [ poynting - eq ] ) only the power of the free current density appears . in other words , for an electromagnetic wave in a material medium , poynting s expression corresponds to the energy density of the whole wave , including the change of energy of matter as the wave propagates in the medium .    for the momentum density of the fields in matter
the situation is less clear .
there is a dispute that has lasted more than a century @xcite on the subject .
minkowski @xcite proposed @xmath29 which for individual photons corresponds to using ( [ particle - momentum - ultra ] ) replacing @xmath30 by the speed of light in matter @xmath31 .
this was challenged by abraham @xcite who proposed instead @xmath32 which for photons corresponds to using directly ( [ particle - momentum ] ) .    to shed some light on this issue ,
in this paper we study a simple example that can be solved exactly .
it consists of a packet of plane waves propagating in a material with permeability @xmath23 and permittivity @xmath22 .
the medium being homogeneous there is no refraction and the wave travels conserving momentum and energy . inside the material there is a flat thin film of conducting material in which ohm s law holds , @xmath33 .
when the wave packet falls on the film it splits in reflected and transmitted packets , whose amplitudes are fixed by the boundary conditions ( see figure [ [ paquetes ] ] ) . the total work done on the currents in the conducting film can be calculated using ( [ power - density ] ) .
the result is consistent with poynting s energy density ( [ poynting - density ] ) .
also , the impulse on the currents of the film can be calculated using ( [ force - density ] ) .
we found that the result is consistent with momentum conservation only if the momentum carried by the waves is given by minkowski s expression ( [ minkowski ] ) .
consider a medium with electric and magnetic permeabilities @xmath22 and @xmath23 filling the space . at the plane @xmath34 there is a sheet of thickness @xmath35 made of a conductive material of resistivity @xmath36 . a packet of electromagnetic plane waves polarized in the direction @xmath37 is traveling in the @xmath38 direction .
suppose it has a spatial length @xmath39 with @xmath40 , @xmath41 a time scale , and a wave length @xmath42 such that @xmath43 and @xmath44 . in terms of a suitable function @xmath45 which vanishes outside the interval @xmath46 $ ]
the incident wave is , @xmath47 the reflected and transmitted waves are , @xmath48 and @xmath49 at @xmath34 @xmath50 is continuous , @xmath51 implying @xmath52        there is an electrical current in the sheet .
the magnetic field is discontinuous at @xmath34 .
taking a rectangular path of length @xmath53 in the @xmath54 direction and width @xmath35 in the @xmath38 direction we have , @xmath55 where @xmath56 is the plane surface with boundary @xmath57 ( see figure [ [ camino ] ] ) . the second integral vanishes because the width of the surface is negligible .
the current density is @xmath58 substituting this in the previous equation we have @xmath59 and @xmath60 introducing @xmath61 and solving the system we have , @xmath62 the energy of the wave is given by poynting s expression @xmath63 for @xmath64 we consider a cylindrical piece of the incident packet with axis parallel to @xmath38 and cross section @xmath65 .
the energy is , @xmath66 where @xmath67 for @xmath68 the energies of the reflected and transmitted waves are , @xmath69 ^ 2u_1\ , \\
u_3&=&\epsilon e_3 ^ 2 a v \bar{t}=\left[\frac{1}{1+b}\right]^2u_1\ .\end{aligned}\ ] ] the power density on the current is @xmath70 the total power transferred by the wave is , @xmath71 and the work on the current is @xmath72 ^ 2}u_1\ .\ ] ] the energy is conserved @xmath73 let us now discuss momentum conservation for this system .
we compute first the force applied on the sheet .
the force density on the current is @xmath74 in the approximation where @xmath35 is infinitesimal , the current density in the sheet , @xmath75 , is constant because the electric field is continuous .
the magnetic field then varies linearly between its values at both sides of the sheet . in computing the total force one
has to take the average on the sheet volume .
for @xmath76 we have @xmath77 where @xmath78\nonumber\\   & = & \frac{h(t)}{v}e_1\ .\end{aligned}\ ] ] substituting this in ( [ force ] ) and using ( [ currentdensity ] ) and ( [ solution ] ) we get
@xmath79 the impulse applied to the current sheet is @xmath80    the momentum carried by the wave is directed in the same direction in which the wave propagates , that is , the direction of @xmath81 .
the momentum density of the wave may be written in the form , @xmath82 with @xmath83 a constant to be determined .
for @xmath84 the momentum density is minkowski s and for @xmath85 it is abraham s .
then , the momentum carried by the incident wave is @xmath86 analogously @xmath87 for the balance equation we get , @xmath88 momentum is conserved if @xmath84 . the momentum density
( [ momden ] ) turns out to be minkowski s expression , @xmath89    the key in the previous argument is the factor @xmath90 that appears in the impulse ( [ applimp ] ) .
this can be traced back to the magnetic force in lorentz s force ( [ force - density ] ) and the relationship between the magnetic and electric amplitudes in electromagentic waves .
this relation is a consequence of maxwell s equations . for obtaining abraham s result the factor
should have been @xmath91 .
the calculation presented in this paper assumes the validity of maxwell s equations and lorentz s force density on the conduction current . using the properties of the electromagnetic wave solutions of maxwell s equations in a simple setup we showed that momentum conservation requires the use of minkowski s expression for the momentum density of the electromagnetic field in a linear medium .
although the use of the explicit solution of maxwell equations with boundary conditions puts our approach a little ahead of most introductory physics university courses it is still a very simple and elementary construction which allows the discussion of electromagnetic momentum in material media in a self - consistent way . in particular
it also may be used to treat the case when the wave propagates in the vacuum .
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we present an elementary discussion of the momentum transferred to a conducting sheet by an electromagnetic wave propagating in a polarizable medium . 
 we show that conservation of momentum is consistent with minkowski s expression for the momentum density .    
 @xmath0 + @xmath1 +