There are two rods of length `l_1` and `l_2` and coefficient of linear expansions are `alpha_1` and `alpha_2` respectively.
Find equivalent coefficient of thermal expansion for their combination in series.

Options

A. `(alpha_1+alpha_2)/2`
B. `alpha_1 l_1+alpha_2l_2/(alpha_1+alpha_2)`
C. `(alpha_1 l_1+alpha_2l_2)`/`(l_1+l_2)`
D. `sqrt(alpha_1 alpha_2)`

Answer

To find the equivalent coefficient of thermal expansion for two rods connected in series, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths and coefficients of expansion**:
- Let the lengths of the two rods be \( l_1 \) and \( l_2 \).
- Let the coefficients of linear expansion be \( \alpha_1 \) and \( \alpha_2 \) respectively.

2. **Determine the total length**:
- When the rods are connected in series, the total length \( L \) of the combination is given by:
\[ L = l_1 + l_2 \]

3. **Calculate the change in length for each rod**:
- When the temperature increases by \( \Delta T \), the change in length for the first rod (\( \Delta l_1 \)) is:
\[ \Delta l_1 = \alpha_1 \Delta T \cdot l_1 \]
- The change in length for the second rod (\( \Delta l_2 \)) is:
\[ \Delta l_2 = \alpha_2 \Delta T \cdot l_2 \]

4. **Express the total change in length**:
- The total change in length \( \Delta L \) for the combination of the two rods is the sum of the changes in length of both rods:
\[ \Delta L = \Delta l_1 + \Delta l_2 = \alpha_1 \Delta T \cdot l_1 + \alpha_2 \Delta T \cdot l_2 \]

5. **Relate the total change in length to the equivalent coefficient of expansion**:
- We can express the total change in length in terms of the equivalent coefficient of thermal expansion \( \alpha_{\text{equiv}} \):
\[ \Delta L = \alpha_{\text{equiv}} \Delta T \cdot L \]
- Substituting \( L = l_1 + l_2 \) into the equation gives:
\[ \Delta L = \alpha_{\text{equiv}} \Delta T \cdot (l_1 + l_2) \]

6. **Set the two expressions for \( \Delta L \) equal to each other**:
- From the two expressions for \( \Delta L \):
\[ \alpha_1 \Delta T \cdot l_1 + \alpha_2 \Delta T \cdot l_2 = \alpha_{\text{equiv}} \Delta T \cdot (l_1 + l_2) \]

7. **Cancel \( \Delta T \) from both sides** (assuming \( \Delta T \neq 0 \)):
- This simplifies to:
\[ \alpha_1 l_1 + \alpha_2 l_2 = \alpha_{\text{equiv}} (l_1 + l_2) \]

8. **Solve for \( \alpha_{\text{equiv}} \)**:
- Rearranging gives:
\[ \alpha_{\text{equiv}} = \frac{\alpha_1 l_1 + \alpha_2 l_2}{l_1 + l_2} \]

### Final Result:
The equivalent coefficient of thermal expansion for the two rods in series is:
\[ \alpha_{\text{equiv}} = \frac{\alpha_1 l_1 + \alpha_2 l_2}{l_1 + l_2} \]