The amount of elastic potential energy per unit volume (in SI unit) of a steel wire of length 100 cm to stretch it by 1 mm is (if Young's modulus of the wire = $2.0\times {10}^{11}{\mathrm{Nm}}^{-2}$) :

The amount of elastic potential energy per unit volume of a steel wire is calculated using the formula:

Elastic potential energy per unit volume (U) = (1/2) × Y × (strain)²

Here, Y is the Young's modulus of the material, and strain is defined as the ratio of elongation to the original length.

For a steel wire of length 100 cm, if it is stretched by 1 mm, we can calculate the strain as follows:

Strain = elongation / original length = (10^-3 m) / (100 × 10^-2 m) = 10^-3

Given that Young's modulus (Y) for the steel wire is 2 × 10¹¹ N/m², we can now calculate the elastic potential energy per unit volume.

Using the formula:

U = (1/2) × 2 × 10¹¹ × (10^-3)²

U = (1/2) × 2 × 10¹¹ × 10^-6

U = 10⁵ J/m³

Thus, the amount of elastic potential energy per unit volume of a steel wire of length 100 cm when stretched by 1 mm is 10⁵ J/m³. The calculated value represents the amount of elastic potential energy per unit volume that the wire possesses due to the strain caused by elongation.

In conclusion, by using Young's modulus and the strain formula, we can effectively determine the amount of elastic potential energy per unit volume of a steel wire under specified conditions. This value is crucial in understanding the behavior of materials under mechanical stress and their capacity to store elastic potential energy.