Consider a co$-$axial screw dislocation in a cylindrical solid of radius $100$ nm $.$ The dislocation has a burgers vector of magnitude $0\mathrm{.}2$ nm and a core radius of $0\mathrm{.}3$ nm $.$ The solid has a shear modulus of $100$ GPa. Determine the self$-$energy per unit length of the dislocation in joules.

$5\mathrm{.}81$ E$-9$

$6\mathrm{.}81$ E$-9$

$2\mathrm{.}17$ E$-9$

$1\mathrm{.}84$ E$-9$