When a wire is stretched to double its length, we need to analyze each of the given options to determine which ones are correct. Let's break down the concepts of stress, strain, Young's modulus, and elastic potential energy to clarify the situation.
Understanding Stress and Strain
Stress is defined as the force applied per unit area, while strain is the measure of deformation representing the displacement between particles in a material body. When a wire is stretched to double its length, the strain can be calculated as:
- Strain (ε) = Change in Length / Original Length = (L - L₀) / L₀
If the wire is stretched to double its length, then:
- Original Length (L₀) = L
- New Length (L) = 2L
Thus, the strain becomes:
- ε = (2L - L) / L = L / L = 1
Evaluating the Options
Now, let's evaluate each option:
- a) Strain is unity: This is correct, as we just calculated that the strain is indeed 1 when the wire is stretched to double its length.
- b) Stress is equal to Young's modulus: This statement is not necessarily true. Young's modulus (E) is defined as the ratio of stress (σ) to strain (ε), or E = σ/ε. If strain is 1, then stress equals Young's modulus only if the stress applied is equal to the Young's modulus value. This option is misleading without additional context.
- c) Its radius is halved: This option is also misleading. When a wire is stretched, its volume remains constant (assuming incompressibility). If the length doubles, the radius does not necessarily halve; rather, it changes according to the relationship of volume. The volume before stretching (V₀ = A₀ * L₀) must equal the volume after stretching (V = A * L). Thus, if length doubles, the radius will change, but not necessarily to half.
- d) Young's modulus is equal to twice the elastic potential energy per unit volume: This statement is incorrect. The elastic potential energy per unit volume (U) for a material is given by U = (1/2) * σ * ε. Since ε = 1, U = (1/2) * σ. Young's modulus is not equal to twice this value; rather, it is a separate property defined by the ratio of stress to strain.
Final Thoughts
From this analysis, we can conclude that the only correct answer is option a, where strain is unity. Options b, c, and d do not hold true under the conditions described. It's important to carefully analyze each statement based on the definitions and relationships in material science to avoid confusion.
