Elastic Potential Energy in a Stretched Wire:
When a wire is stretched, it stores elastic potential energy due to the deformation caused by the applied force. The amount of elastic potential energy stored per unit volume is given by the formula:

Elastic Potential Energy per unit volume (U) = (1/2) * stress * strain

Where stress is the force applied per unit area and strain is the ratio of change in length to the original length.

Comparison of Two Wires:
Let's consider two wires of the same material and length, but with diameters in the ratio 1:2. We will stretch both the wires by the same force.

Stress:
The stress applied to both wires will be the same since the force applied is the same. Stress is given by the formula:

Stress = Force / Area

The area of a wire is proportional to the square of its diameter. Therefore, if the diameters of the two wires are in the ratio 1:2, the areas will be in the ratio 1^2:2^2, which simplifies to 1:4. Hence, the stress in both wires is in the ratio 1:4.

Strain:
The strain in both wires will also be the same because they are stretched by the same force and have the same length. Strain is given by the formula:

Strain = Change in length / Original length

Since the wires have the same length, the change in length is proportional to the diameter. Therefore, if the diameters of the two wires are in the ratio 1:2, the change in length will also be in the ratio 1:2. Hence, the strain in both wires is in the ratio 1:2.

Elastic Potential Energy:
Now, let's calculate the elastic potential energy stored per unit volume for both wires.

For the first wire (with diameter ratio 1), the elastic potential energy per unit volume (U1) is given by:

U1 = (1/2) * stress * strain

For the second wire (with diameter ratio 2), the elastic potential energy per unit volume (U2) is given by:

U2 = (1/2) * stress * strain

Comparing the Ratios:
To find the ratio of U2 to U1, we divide U2 by U1:

(U2/U1) = [(1/2) * stress * strain] / [(1/2) * stress * strain]

The stress and strain cancel out, and we are left with:

(U2/U1) = 1

Therefore, the ratio of elastic potential energy per unit volume for the two wires is 1:1.

However, the question asks for the ratio of U2 to U1 in terms of their diameters. Since the diameters are in the ratio 1:2, we can square the ratio to get the ratio of their volumes.

Therefore, (U2/U1) = (2^2) : (1^2) = 4:1

Hence, the elastic potential energy stored per unit volume for the two wires is in the ratio 4:1.