Two metallic rods of length l and 3l have coefficient of linear expansion `alpha` and `3alpha` respectively. The coefficient of linear expansion ofr their series combinations, is

Options

A. `alpha`
B. `2.5alpha`
C. `4alpha`
D. `2alpha`

Answer

To find the coefficient of linear expansion for the series combination of two metallic rods with different lengths and coefficients of linear expansion, we can follow these steps:

### Step-by-step Solution:

1. **Identify the Given Values**:
- Length of the first rod, \( L_1 = l \)
- Coefficient of linear expansion of the first rod, \( \alpha_1 = \alpha \)
- Length of the second rod, \( L_2 = 3l \)
- Coefficient of linear expansion of the second rod, \( \alpha_2 = 3\alpha \)

2. **Calculate the Change in Length for Each Rod**:
- The change in length for the first rod when the temperature changes by \( \Delta T \) is given by:
\[ \Delta L_1 = L_1 \cdot \alpha_1 \cdot \Delta T = l \cdot \alpha \cdot \Delta T \]
- The change in length for the second rod is:
\[ \Delta L_2 = L_2 \cdot \alpha_2 \cdot \Delta T = 3l \cdot (3\alpha) \cdot \Delta T = 9l\alpha \cdot \Delta T \]

3. **Total Change in Length**:
- When the rods are connected in series, the total change in length, \( \Delta L \), is the sum of the changes in length of both rods:
\[ \Delta L = \Delta L_1 + \Delta L_2 = (l \cdot \alpha \cdot \Delta T) + (9l\alpha \cdot \Delta T) = 10l\alpha \cdot \Delta T \]

4. **Total Length of the Combined Rods**:
- The total length of the two rods combined is:
\[ L_{\text{total}} = L_1 + L_2 = l + 3l = 4l \]

5. **Finding the Equivalent Coefficient of Linear Expansion**:
- The equivalent coefficient of linear expansion \( \alpha_{\text{eq}} \) for the combined length can be expressed as:
\[ \Delta L = L_{\text{total}} \cdot \alpha_{\text{eq}} \cdot \Delta T \]
- Substituting the values we have:
\[ 10l\alpha \cdot \Delta T = 4l \cdot \alpha_{\text{eq}} \cdot \Delta T \]
- Dividing both sides by \( l \cdot \Delta T \) (assuming \( l \) and \( \Delta T \) are not zero):
\[ 10\alpha = 4\alpha_{\text{eq}} \]
- Solving for \( \alpha_{\text{eq}} \):
\[ \alpha_{\text{eq}} = \frac{10\alpha}{4} = \frac{5\alpha}{2} = 2.5\alpha \]

### Final Answer:
The coefficient of linear expansion for their series combination is \( \frac{5\alpha}{2} \) or \( 2.5\alpha \).