  We prove Taylor scaling for dislocation lines characterized by line-tension
and moving by curvature under the action of an applied shear stress in a plane
containing a random array of obstacles. Specifically, we show--in the sense of
optimal scaling--that the critical applied shear stress for yielding, or
percolation-like unbounded motion of the dislocation, scales in proportion to
the square root of the obstacle density. For sufficiently small obstacle
densities, Taylor scaling dominates the linear-scaling that results from purely
energetic considerations and, therefore, characterizes the dominant
rate-limiting mechanism in that regime.
