3.9: Observations

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James R. Melcher
Massachusetts Institute of Technology via MIT OpenCourseWare

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The force densities and associated stress tensors of Table 3.10.1 are of two origins. The Kelvin force densities, the last two in the table, come from a microscopic picture of particles and dipoles subject to electric or magnetic forces which, through the agent of a kinetic equilibrium, are passed along to the ponderable continuum. The Korteweg-Helmholz force densities, all of the others in the table, are based on an energy conservation principle. The connection between micro and macro fields,needed to apply this principle is made using electrical measurements of constitutive laws to interrelate the macroscopic fields \(\overrightarrow{D}\) and \(\overrightarrow{E}\) or \(\overrightarrow{B}\) and \(\overrightarrow{H}\).

The arguments underlying each type of force density envoke certain assumptions which point to possible inadequacies. The Kelvin force densities picture the force acting on each dipole and each point charge in isolation and this force as being that transmitted to the ponderable media. This does not allow for the possibility that the micro fields of one dipole contribute to the force on a neigh-boring dipole.

This shortcoming is obviated by the energy method, which is based on a statement of energy conservation for an electromechanical subsystem. The resulting Korteweg-Helmholtz force densities\(^1\) are of course also restricted. On the one hand, they are more broadly applicable than might be concluded from the derivations. For example, the MQS continuum is viewed as "perfectly conducting," but the free current force density is certainly applicable in cases where the conductivity is finite. This is evident from its agreement with the Lorentz force density of Sec. 3.1, because the later model includes a finite mobility and hence electrical dissipation.

One way to derive a force density without ambiguity as to the validity of the result in nonconservative systems is to replace statements of energy conservation with those of power flow\(^2\). However,the principle of virtual power requires information beyond that required by the principle of virtual work used here. In addition to the constitutive laws relating the macroscopic field variables is the requirement for the power flux density, which must either be assumed or measured.

Underlying all of the discussions in this chapter has been the presumption that a clear distinction can be made between electric or magnetic force densities and those of other origins. This is tantamount to being able to isolate electromagnetic energy storage from other forms of energy storage.Piezoelectric coupling is an example where it is not fruitful to make this distinction. In that area,the stress and force density generally represent combined electric and mechanical electromechanical effects.

1. J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Co. Inc., New York, 1941, pp. 137-159.

2. P. Penfield, Jr., and H. H. Haus, Electrodynamics of Moving Media, The M.I.T. Press, Cambridge,Massachusetts, 1967, pp. 35-40.