it is given that, both wires are made by same material, length of wires are same and also same force applied to the end of wire . hence, strain of both wires will be same.
Strain = stress/Y
where Y is Young’s modulus.
so, potential energy per unit volume = 1/2 × (stress)²/Y
but we know, stress is inversely proportional to square of diameter of wire.
so, potential energy per unit volume is inversely proportional to (diameter)⁴.
so, ratio of potential energy per unit volume = (d2/d1)4
Given that d1 : d2 = 1:2
so, ratio of potential energy per unit volume = 2⁴/1⁴ = 16/1