$(13\%)$ A cylindrical hypothetical alloy $x$ was subjected to a tensile test. The original diameter and length of the cylindrical alloy rod were $20mm$ and $50mm,$ respectively. It is known that there is a $2{10}^{-3}mm$ change in diameter and the deformation is entirely elastic. The elastic modulus and Poisson's ratio for alloy $x$ are $100$GPa, and $0\mathrm{.}25,$ respectively. Please answer the following questions.

$(1)(3\%)$ Determine the lateral strain.

$(2)(5\%)$ Determine the elongation $($change in length$)$ along the tensile force direction.

$(3)(5\%)$ Determine the external force for the tensile test.