A planar nematic cell of thickness is measured to have a Freedericksz threshold voltage

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Frank elasticity, defects and smectic layers

Problem

A planar nematic cell of thickness d1=10 μmd_1 = 10\,\mu\mathrm{m} is measured to have a Freedericksz threshold voltage Vc1=0.80 VV_{c1} = 0.80\,\mathrm{V}. Someone argues: "The threshold field is Ec=Vc1/d1=0.80 V/10 μm=8×104 V m−1E_c = V_{c1}/d_1 = 0.80\,\mathrm{V}/10\,\mu\mathrm{m} = 8\times10^4\,\mathrm{V\,m^{-1}}. This is the field needed to overcome the elastic torque, so a cell twice as thick, d2=20 μmd_2 = 20\,\mu\mathrm{m}, needs this same field to switch, giving a threshold voltage Vc2=Ecd2=1.60 VV_{c2} = E_c d_2 = 1.60\,\mathrm{V}, double the first." Find the error in this reasoning and give the correct threshold voltage for the 20 μm20\,\mu\mathrm{m} cell.

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