Final answer:
If the volume of a wire remains constant when its length changes due to stretching, the Poisson's ratio of the material is 0.5. This indicates the material compresses transversely as much as it stretches longitudinally to maintain constant volume. The correct answer is (b) 0.5.
Step-by-step explanation:
When a wire with a constant volume is stretched, its length increases while its diameter decreases in such a way that the cross-sectional area changes to maintain the volume. The change in volume due to stretching is given by ∆V = ∆L × original area - ∆(area × original length), which should be zero if the volume is constant.
Poisson's ratio, denoted by ν (nu), is the negative ratio of transverse to longitudinal strain. In this case, the transverse strain is equal in magnitude and opposite in sign to the longitudinal strain, making their ratio -1.
If the volume doesn't change with stretching, it implies that the material is perfectly compressible in the transverse direction while it extends longitudinally, which corresponds to a Poisson's ratio of 0.5. Therefore, the answer is (b) 0.5.
To elaborate with an example, imagine stretching a cylinder of playdough. As it gets longer, it also gets narrower, but if the playdough has to maintain its volume, the amount it narrows by will be such that it compensates exactly for the increase in length, meaning the diameter decreases at a rate that keeps the volume the same. This property is quantified by the Poisson's ratio.