A metal wire is clamped between two vertical walls. At `20^(@)C` the unstrained length of the wire is exactly equal to the separation between walls. If the temperature of the wire is decreased the graph between elastic energy density (u) and temperature (T) of the wire is

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the behavior of the metal wire when its temperature is decreased. The wire is initially unstressed at a temperature of \(20^\circ C\) and its length is equal to the separation between the two vertical walls.

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- At \(20^\circ C\), the wire is unstressed, meaning it is at its natural length, which is equal to the distance between the two walls.

2. **Effect of Temperature Decrease**:
- When the temperature of the wire decreases, the wire will contract due to thermal contraction. This means that the length of the wire will decrease.

3. **Constraints Imposed by the Walls**:
- Since the wire is clamped between the two walls, it cannot contract freely. As the temperature decreases and the wire tries to contract, it will experience tensile stress because it is held in place by the walls.

4. **Elastic Energy Density**:
- The elastic energy density \(u\) in the wire is related to the stress and strain in the wire. As the wire is forced to remain at the length of the walls while trying to contract, it will store elastic potential energy.

5. **Graph of Elastic Energy Density vs. Temperature**:
- As the temperature decreases, the elastic energy density \(u\) will initially increase because the wire is under increasing tension (stress) due to the constraints of the walls.
- The graph will show an increase in \(u\) as \(T\) decreases, indicating that the wire is storing more elastic energy as it is forced to remain at a fixed length while trying to contract.

6. **Conclusion**:
- The graph of elastic energy density \(u\) versus temperature \(T\) will slope upwards as the temperature decreases, indicating an increase in elastic energy density due to the tensile stress in the wire.