But what happens if we do account for the edge of every single atomic dipole in this way? We find that we have recovered the microscopic angular momentum density profile. Therefore, even if the angular momentum density of the beam in free space is held to be given by the macroscopic approach(Eq. 10), we find that it nevertheless interacts with matter in accordance with the microscopic angular momentum density(Eq. 11) However, to make such a distinction is superfluous as in such a case the free space angular momentum density profile may never be probed or interacted with and hence its nature is a question not of physics but ontology deso if the macroscopic angular momentum density profile is to be meaningful, we therefore find that it must cribe not only the carriage but also the transfer of momentum, and this is experimentally testable. In practice, to perform such an experiment with half-wave plates floated on a microscope slide is impractical, though it may in theory be possible to observe the transfer of angular momentum using the apparatus described above, particularly if microwaves are employed so that the angular momentum to power ratio is increase and heating problems are correspondingly reduced. Nevertheless it is interesting to discuss the theoretical onsequences of the different possible outcomes Suppose that the interaction was shown to proceed in accordance with the microscopic profile. In that case, the macroscopic integral(Eq. 8)would nevertheless continue to be a valid mathematical tool capable of calculatin the behaviour of a system provided all edge interactions are properly taken into account, as per Refs. 43, 45, 46 However, if momentum were shown to be transferred according to the macroscopic density profile(Eq. 10)then this would introduce a conflict between microscopic and macroscopic electrodynamics. No resolution to this confict is known to exist On these grounds alone it might be argued that the macroscopic expression is nothing more than a convenient analytical tool for calculating the angular momentum transfer under particular circumstances, in the same manner that when a bar magnet is modelled as a large number of circulating microscopic currents these may be reduced to a single macroscopic current for convenience. Nevertheless, the ongoing popular tendency to treat the macroscopic expression as yielding a real momentum density, and the resulting insidious errors that arise(such as the claim that a circularly polarised plane wave carries no spin, discussed above)make it desireable to demonstrate experimentally whether or not the macroscopic expression can tell us anything about the fundamental properties of an electromagnetic wave. 4. CONCLUSION In this paper we have discussed two controversies relating to the momentum of an electromagnetic wave. Now that lasers are increasingly being used for micromanipulation, it is important to resolve these issues where they may have bearing upon experimental results It has been shown that neither problem discussed here is intractable, and indeed that the Abraham-Minkowski controversy has already been resolved. Of significant note is the existence of a material counterpart to the Minkowski electromagnetic energy-momentum tensor. Awareness of the existence of this counterpart is low, and its use is vital in the analysis of experiments involving angular momentum We also believe that Khrapko's paradox should be dismissed on theoretical grounds, but on account of the large amount of popular support for the macroscopic angular momentum density profile Eq(10), an experimental discrimination is desireable If optical micromanipulation is to become a major technology in the micro- and nanotechnological revolution then it is vital that the theoretical framework underpinning its behaviour be well and widely understood. REFERENCES 1. A. Ashkin, J M. Dziedzic, J. E. Bjorkholm, and S Chu, "Observation of a single-beam gradient force optical trap for dielectric particles, " Opt. Lett. 1l, pp. 288-290, 1986 2. R. I. Khrapko, Question #79 does plane wave not carry a spin? "AmJ. Phys. 69, P. 405, 2001 3. J D. Jackson, Classical Electrodynamics, pp 605-612. John Wiley Sons, Inc, New York, 3rd ed, 1999But what happens if we do account for the edge of every single atomic dipole in this way? We find that we have recovered the microscopic angular momentum density profile. Therefore, even if the angular momentum density of the beam in free space is held to be given by the macroscopic approach (Eq. 10), we find that it nevertheless interacts with matter in accordance with the microscopic angular momentum density (Eq. 11). However, to make such a distinction is superfluous as in such a case the free space angular momentum density profile may never be probed or interacted with and hence its nature is a question not of physics but ontology. If the macroscopic angular momentum density profile is to be meaningful, we therefore find that it must describe not only the carriage but also the transfer of momentum, and this is experimentally testable. In practice, to perform such an experiment with half-wave plates floated on a microscope slide is impractical, though it may in theory be possible to observe the transfer of angular momentum using the apparatus described above, particularly if microwaves are employed so that the angular momentum to power ratio is increased and heating problems are correspondingly reduced. Nevertheless it is interesting to discuss the theoretical consequences of the different possible outcomes. 3.3. Discussion Suppose that the interaction was shown to proceed in accordance with the microscopic profile. In that case, the macroscopic integral (Eq. 8) would nevertheless continue to be a valid mathematical tool capable of calculating the behaviour of a system provided all edge interactions are properly taken into account, as per Refs. 43, 45, 46. However, if momentum were shown to be transferred according to the macroscopic density profile (Eq. 10) then this would introduce a conflict between microscopic and macroscopic electrodynamics. No resolution to this conflict is known to exist. On these grounds alone it might be argued that the macroscopic expression is nothing more than a convenient analytical tool for calculating the angular momentum transfer under particular circumstances, in the same manner that when a bar magnet is modelled as a large number of circulating microscopic currents these may be reduced to a single macroscopic current for convenience. Nevertheless, the ongoing popular tendency to treat the macroscopic expression as yielding a real momentum density, and the resulting insidious errors that arise (such as the claim that a circularly polarised plane wave carries no spin, discussed above) make it desireable to demonstrate experimentally whether or not the macroscopic expression can tell us anything about the fundamental properties of an electromagnetic wave. 4. CONCLUSION In this paper we have discussed two controversies relating to the momentum of an electromagnetic wave. Now that lasers are increasingly being used for micromanipulation, it is important to resolve these issues where they may have bearing upon experimental results. It has been shown that neither problem discussed here is intractable, and indeed that the Abraham–Minkowski controversy has already been resolved. Of significant note is the existence of a material counterpart to the Minkowski electromagnetic energy–momentum tensor. Awareness of the existence of this counterpart is low, and its use is vital in the analysis of experiments involving angular momentum. We also believe that Khrapko’s paradox should be dismissed on theoretical grounds, but on account of the large amount of popular support for the macroscopic angular momentum density profile Eq. (10), an experimental discrimination is desireable. If optical micromanipulation is to become a major technology in the micro- and nanotechnological revolution then it is vital that the theoretical framework underpinning its behaviour be well and widely understood. REFERENCES 1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, pp. 288–290, 1986. 2. R. I. Khrapko, “Question #79. does plane wave not carry a spin?,” Am. J. Phys. 69, p. 405, 2001. 3. J. D. Jackson, Classical Electrodynamics, pp. 605–612. John Wiley & Sons, Inc., New York, 3rd ed., 1999