Field Free Energy and Torques

instead of the magnetic field, an electric field is applied at some angle to the director of a nematic liquid crystal, in analogy to (11.39), the electric polarization P is given by 

Note that, from the tensor form s = 1 + 4nx, follows = 4nXa because the unity is included in the isotropic part of tensor s. 

In each particular situation, these torques may be balanced by the elastic, surface or viscous torques. The magnetic and electric field torques may be obtained differently. Using minimisation of the free energy with respect to the director one obtains the molecular field introduced earlier, see Eq. (8.27) and then finds the torques as vector products with the director. Let us show it. The magnetic free energy density is given by 

The first term in (11.43) is independent of the director the absolute value of the second one is maximal for H II n that correspond to the minimum of magnetic free energy. Minimisation of (11.43) results in a vector of the molecular field  

For the fixed electric field applied to the sample from the electrodes, instead of Helmholtz free energy one should minimise the thermodynamic potential density  

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