Final answer:
According to Hooke's Law and the relationship between a wire's extension under tension and its dimensions, all wires listed in the question, a, b, c, and d, will have the same extension when the same tension is applied due to the inverse square relationship with radius and direct relationship with length.
Step-by-step explanation:
The question revolves around the concept of elasticity in physics. Specifically, it involves understanding how the extension of a wire changes under tension based on its length and radius when the material of the wire and the tension applied are constant. In physics, this is described by Hooke's Law and Young's Modulus. According to Hooke's Law, the extension ΔL of a wire is directly proportional to the force F applied and the original length L, and inversely proportional to the area A of the wire's cross-section and the material's Young's Modulus E, as represented by the equation ΔL = (1/E) × (F/A) × L. The area of a wire's cross-section is πr^2, where r is the wire's radius. Thus, for a given material and tension, the extension is inversely proportional to r^2 and directly proportional to L.
By examining the provided options, we see that as the length doubles, the radius of each wire also doubles, resulting in a cross-sectional area increase by a factor of four since area A is proportional to r^2. This would suggest that the extension for all wires will actually be the same since the increase in length is offset by the increase in the cross-sectional area, maintaining the extension due to the inverse square relationship with radius and direct relationship with length.
Therefore, all wires a, b, c, and d will have the same extension, given that the tension applied to them is the same and they are made up of the same material. Thus, there is no single wire among the options given that will have the largest extension. They all will extend equally under the same tension.