To determine the extension of the composite wire made of copper and steel, we need to consider how the two materials behave under the applied load. Since they are joined end to end, they will experience the same force, but they will stretch differently due to their distinct Young's moduli. Let's break this down step by step.

Understanding Young's Modulus

Young's modulus (E) is a measure of the stiffness of a material. It is defined as the ratio of stress (force per unit area) to strain (relative change in length). The formula for Young's modulus is:

E = Stress / Strain

Where:

• Stress = Force / Area
• Strain = Change in Length / Original Length

Given Data

We have the following information:

• Length of copper wire (L_Cu) = 1 m
• Length of steel wire (L_Steel) = 0.5 m
• Extension of copper wire (ΔL_Cu) = 1 mm = 0.001 m
• Young's modulus of copper (E_Cu) = 1 x 10¹¹ N/m²
• Young's modulus of steel (E_Steel) = 2 x 10¹¹ N/m²

Calculating the Force

First, we can find the force (F) applied to the copper wire using its Young's modulus:

F = E_Cu × (ΔL_Cu / L_Cu) × A

Where A is the cross-sectional area, which will cancel out later since both wires have the same area. We can express the force as:

F = E_Cu × (0.001 / 1) × A

Thus, we can simplify this to:

F = 1 x 10¹¹ × 0.001 × A = 1 x 10⁸ × A

Finding the Extension of the Steel Wire

Next, we need to find the extension of the steel wire (ΔL_Steel) using the same force:

ΔL_Steel = (F × L_Steel) / (E_Steel × A)

Substituting the expression for F we derived earlier:

ΔL_Steel = (1 x 10⁸ × A × 0.5) / (2 x 10¹¹ × A)

Notice that A cancels out:

ΔL_Steel = (1 x 10⁸ × 0.5) / (2 x 10¹¹)

Calculating this gives:

ΔL_Steel = 0.25 x 10^-3 m = 0.00025 m = 0.25 mm

Total Extension of the Composite Wire

Now, we can find the total extension of the composite wire by adding the extensions of both wires:

Total Extension = ΔL_Cu + ΔL_Steel

Total Extension = 1 mm + 0.25 mm = 1.25 mm

Therefore, the extension of the composite wire is 1.25 mm.