A steel wire of length 20 cm and uniform cross section 1 `mm^(2)` is tied rigidly at both the ends. The temperature of the wire is altered from `40^(@)` C to `20^(@)C` Coefficient of linear expansion for steel `alpha = 1.1 xx 10^(-5) l^(@)C` and Y for steel is `2.0 xx 10^(11) N//m^(2)`. The change in tension of the wire is

Options

A. `2.2xx 10^(6)` newton
B. 16 newton
C. 8 newton
D. 44 newton

Explanation

To calculate the change in tension of the steel wire when the temperature is altered from 40 °C to 20 °C, we can follow these steps:

### Step 1: Identify the given values
- Length of the wire, \( L_0 = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Cross-sectional area, \( A = 1 \, \text{mm}^2 = 1 \times 10^{-6} \, \text{m}^2 \)
- Coefficient of linear expansion for steel, \( \alpha = 1.1 \times 10^{-5} \, \text{°C}^{-1} \)
- Young's modulus for steel, \( Y = 2.0 \times 10^{11} \, \text{N/m}^2 \)
- Initial temperature, \( T_1 = 40 \, \text{°C} \)
- Final temperature, \( T_2 = 20 \, \text{°C} \)

### Step 2: Calculate the change in temperature
\[ \Delta \theta = T_2 - T_1 = 20 \, \text{°C} - 40 \, \text{°C} = -20 \, \text{°C} \]

### Step 3: Calculate the change in length due to temperature change
The change in length \( \Delta L \) due to temperature change can be calculated using the formula:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta \theta \]
Substituting the values:
\[ \Delta L = 0.2 \, \text{m} \cdot (1.1 \times 10^{-5} \, \text{°C}^{-1}) \cdot (-20 \, \text{°C}) \]
\[ \Delta L = 0.2 \cdot 1.1 \times 10^{-5} \cdot (-20) = -4.4 \times 10^{-5} \, \text{m} \]

### Step 4: Calculate the thermal stress
The thermal stress \( \sigma \) developed in the wire can be expressed as:
\[ \sigma = \frac{Y \cdot \Delta L}{L_0} \]
Substituting the values:
\[ \sigma = \frac{(2.0 \times 10^{11} \, \text{N/m}^2) \cdot (-4.4 \times 10^{-5} \, \text{m})}{0.2 \, \text{m}} \]
\[ \sigma = \frac{-8.8 \times 10^{6} \, \text{N/m}^2}{0.2} = -4.4 \times 10^{7} \, \text{N/m}^2 \]

### 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:
\[ T = (-4.4 \times 10^{7} \, \text{N/m}^2) \cdot (1 \times 10^{-6} \, \text{m}^2) \]
\[ T = -44 \, \text{N} \]
The negative sign indicates that the tension is a compressive force due to the cooling of the wire.

### Final Answer
The change in tension of the wire is \( 44 \, \text{N} \).

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