Consider a nematic liquid filling the gap between two coaxial conductive cylinders. The surfaces in contact with the liquid have been treated so that the molecules lie tangentially to it.
a) Draw the lines of the director in a plane perpendicular to the axis of the cylinders. Is it a pure bend, splay, or twist configuration?
b) I apply a voltage across the two cylinders. What is the direction of the torque per unit volume due to the electric field? What happens to the director field as I increase the voltage from zero to above the critical field for the Fredericks transition?
c) Draw the lines of the director in a plane perpendicular to the axis of the cylinders above the critical field for the Fredericks transition. Is it a pure bend, splay, or twist configuration?
d) Calculate the free energy per unit length for the configurations drawn in a) and c) assuming the radii are a and b with a < b. The free energy per unit volume is F = 0.5K * ((div(n))^2 + (curl(n))^2).
e) I want to use the coaxial cable above as a guide for laser light polarized perpendicular to the cylinder's axis. What is the optical index of the nematic liquid below and above the Fredericks transition, respectively? Use this information to design a simple electro-optical device based on the coaxial cable that can switch a laser spot between two locations.
