A steel wire of `2.0 mm^2` cross- section is held straight(but under no tension) by attaching it firmly to two points a distance `1.50 m` apart at `30^@ C`. If the temperature now decreases to `-10^@ C` and if the two points remain fixed, what will be the tension in the wire ? For steel, `Y = 20.0000 MPa`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given data
- Cross-sectional area of the steel wire, \( A = 2.0 \, \text{mm}^2 = 2.0 \times 10^{-6} \, \text{m}^2 \)
- Initial length of the wire, \( L = 1.50 \, \text{m} \)
- Initial temperature, \( T_1 = 30^\circ C \)
- Final temperature, \( T_2 = -10^\circ C \)
- Young's modulus for steel, \( Y = 20.0000 \, \text{MPa} = 20 \times 10^6 \, \text{Pa} \)
- Coefficient of linear expansion for steel, \( \alpha = 13 \times 10^{-6} \, \text{°C}^{-1} \)

### Step 2: Calculate the change in temperature
\[ \Delta T = T_2 - T_1 = -10^\circ C - 30^\circ C = -40^\circ C \]

### Step 3: Calculate the change in length due to thermal contraction
The change in length (\( \Delta L \)) can be calculated using the formula:
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
Substituting the values:
\[ \Delta L = 1.50 \, \text{m} \cdot 13 \times 10^{-6} \, \text{°C}^{-1} \cdot (-40^\circ C) \]
\[ \Delta L = 1.50 \cdot 13 \cdot (-40) \times 10^{-6} \, \text{m} \]
\[ \Delta L = -0.00078 \, \text{m} = -0.78 \, \text{mm} \]

### Step 4: Calculate the thermal stress in the wire
The thermal stress (\( \sigma \)) generated in the wire can be calculated using the formula:
\[ \sigma = Y \cdot \frac{\Delta L}{L} \]
Substituting the values:
\[ \sigma = 20 \times 10^6 \, \text{Pa} \cdot \frac{-0.00078}{1.50} \]
\[ \sigma = 20 \times 10^6 \cdot (-0.00052) \, \text{Pa} \]
\[ \sigma = -10400 \, \text{Pa} = -10.4 \, \text{kPa} \]

### Step 5: Calculate the tension in the wire
The tension (\( T \)) in the wire can be calculated using the formula:
\[ T = \sigma \cdot A \]
Substituting the values:
\[ T = -10400 \, \text{Pa} \cdot 2.0 \times 10^{-6} \, \text{m}^2 \]
\[ T = -0.0208 \, \text{N} \]

### Final Answer
The tension in the wire due to the decrease in temperature is approximately \( 0.0208 \, \text{N} \) (the negative sign indicates that it is a compressive force).