The Magnetic and Electric Energies

as above, the contribution independent of the orientation of a and c is neglected then the electric energy density follows from analogous reasoning using equation (2.87) to give 

It is worth remarking that the forms for the electric and magnetic energies in equations (2.86), (2.98) and (2.100) may nevertheless prove useful in some instances, as mentioned in Section 2.3. 

Similarly, we may simply replace eo a in (6.49) by a/Sn to obtain the electric energy in Gaussian units (cf. (2.111)). The conversion formulae in Table D.2 may also be used if results in the literature are to be compared. 

The derivation of the equilibrium equations for SmC liquid crystals parallels that outlined in Section 2.4 for nematic and cholesteric liquid crystals, this approach being based on work by Ericksen [73, 74]. The energy density will be described in terms of the vectors a and c, and the equilibrium equations and static theory will be phrased in this formulation these vectors turn out to be particularly suitable for the mathematical description of statics and dynamics. We assume that the variation of the total energy at equihbrium satisfies a principle of virtual work for a given volume V of SmC liquid crystal of the form postulated by Leslie, Stewart and Nakagawa [173] 

The first of these conditions is due to the assumption of incompressibility and the others follow from the constraints (6.3). By carrying out calculations which parallel those required in equations (2.130) to (2.134) we arrive at the relation 

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It must be stressed that, unlike our expansion of Eq. [30], which is valid only for vanishingly small values of the order parameter, Eqs. [38] and [39] are valid for arbitrary values of the order parameter. In fact, the magnetic and electric energy density expressions of the elastic continuum theory (Eqs. [3] and [4] of Chapter 8) are special cases of Eqs. [38] and [39]. For example, consider the magnetic energy density of the low temperature, anisotropic phase of a system in which only spatial variation of ri(r) is important. S f) in Eq. [38] can then be replaced by its equilibrium value , where < > denotes thermal averaging. The term proportional to H can be neglected because of its spatial invariance, and one obtains Eq. [3] of Chapter 8 directly. 

This is the Poynting theorem S is the Poynting vector. The first two terms in Eq. (1.2.4) represent rate of change of the magnetic and electric energy densities... 

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