On heating the metal rod of length 2m from `0°C` to `50°C`, its length is increased by 0.1 cm. the coeffecient of linear expansion of metal rod is

Options

A. `2*10^-4 °C^-1`
B. `10^-4 °C^-1`
C. `1*10^-5 °C^-1`
D. `2*10^-5 °C^-1`

Answer

To find the coefficient of linear expansion (α) of the metal rod, we can use the formula:

\[ \alpha = \frac{\Delta L}{L_0 \Delta T} \]

Where:
- \( \Delta L \) = change in length
- \( L_0 \) = initial length
- \( \Delta T \) = change in temperature

### Step-by-Step Solution:

1. **Identify the values given in the problem:**
- Initial length \( L_0 = 2 \, \text{m} \)
- Change in length \( \Delta L = 0.1 \, \text{cm} \)
- Change in temperature \( \Delta T = 50°C - 0°C = 50°C \)

2. **Convert the change in length from centimeters to meters:**
\[ \Delta L = 0.1 \, \text{cm} = 0.1 \times 10^{-2} \, \text{m} = 0.001 \, \text{m} \]

3. **Substitute the values into the formula:**
\[ \alpha = \frac{0.001 \, \text{m}}{2 \, \text{m} \times 50 \, \text{°C}} \]

4. **Calculate the denominator:**
\[ 2 \, \text{m} \times 50 \, \text{°C} = 100 \, \text{m°C} \]

5. **Now substitute this back into the equation for α:**
\[ \alpha = \frac{0.001 \, \text{m}}{100 \, \text{m°C}} = \frac{0.001}{100} \, \text{°C}^{-1} \]

6. **Calculate α:**
\[ \alpha = 0.00001 \, \text{°C}^{-1} = 1 \times 10^{-5} \, \text{°C}^{-1} \]

### Final Answer:
The coefficient of linear expansion of the metal rod is:
\[ \alpha = 1 \times 10^{-5} \, \text{°C}^{-1} \]