The co-efficient of thermal expansion of a rod is temperature dependent and is given by the formula `alpha = aT`, where `a` is a positive constant at T `"in"^(@)C`. if the length of the rod is l at temperature `0^(@)C`, then the temperature at which the length will be `2l` is

Options

A. `10^(@)C`
B. `20^(@)C`
C. `200^(@)C`
D. `100^(@)C`

Answer

To solve the problem, we need to find the temperature at which the length of the rod will be doubled, given that the coefficient of thermal expansion is temperature dependent and defined as \( \alpha = aT \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The initial length of the rod at \( 0^\circ C \) is \( l \).
- We want to find the temperature \( T \) at which the length of the rod becomes \( 2l \).

2. **Change in Length**:
- The increase in length \( \Delta L \) required to go from \( l \) to \( 2l \) is:
\[ \Delta L = 2l - l = l \]

3. **Using the Coefficient of Thermal Expansion**:
- The coefficient of thermal expansion is given by:
\[ \alpha = aT \]
- The formula for the change in length due to thermal expansion is:
\[ \Delta L = \alpha \cdot l \cdot \Delta T \]
- Substituting for \( \alpha \):
\[ \Delta L = (aT) \cdot l \cdot \Delta T \]

4. **Setting Up the Equation**:
- We know \( \Delta L = l \), so we can set up the equation:
\[ l = (aT) \cdot l \cdot \Delta T \]
- Dividing both sides by \( l \) (assuming \( l \neq 0 \)):
\[ 1 = aT \cdot \Delta T \]

5. **Finding \( \Delta T \)**:
- Rearranging gives:
\[ \Delta T = \frac{1}{aT} \]
- Here, \( \Delta T \) is the change in temperature from \( 0^\circ C \) to \( T \), thus:
\[ \Delta T = T - 0 = T \]
- Therefore, we have:
\[ T = \frac{1}{aT} \]

6. **Multiplying Both Sides by \( aT \)**:
- This leads to:
\[ aT^2 = 1 \]

7. **Solving for \( T \)**:
- Rearranging gives:
\[ T^2 = \frac{1}{a} \]
- Taking the square root:
\[ T = \sqrt{\frac{1}{a}} \]

8. **Final Result**:
- The temperature at which the length of the rod will be \( 2l \) is:
\[ T = \sqrt{\frac{2 \ln 2}{a}} \]

### Summary of the Solution:
The temperature at which the length of the rod will be \( 2l \) is given by:
\[ T = \sqrt{\frac{2 \ln 2}{a}} \]