There are two rods of length `l_1` `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(aplha_1 alpha_2)`

Answer

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

### Step 1: Understand the Problem
We have two rods with lengths \( l_1 \) and \( l_2 \), and their coefficients of linear expansion are \( \alpha_1 \) and \( \alpha_2 \) respectively. We need to find the equivalent coefficient of thermal expansion \( \alpha \) when these rods are combined in series.

### Step 2: Length Change Due to Temperature Increase
When the temperature increases by \( \Delta T \), the change in length for each rod can be expressed using the formula for linear expansion:
- For rod 1:
\[ L_1' = L_1 (1 + \alpha_1 \Delta T) \]
- For rod 2:
\[ L_2' = L_2 (1 + \alpha_2 \Delta T) \]

### Step 3: Total Length After Temperature Increase
The total length of the two rods after the temperature increase will be:
\[ L' = L_1' + L_2' = L_1 (1 + \alpha_1 \Delta T) + L_2 (1 + \alpha_2 \Delta T) \]

### Step 4: Simplifying the Total Length Expression
Substituting the expressions for \( L_1' \) and \( L_2' \):
\[ L' = L_1 + L_2 + L_1 \alpha_1 \Delta T + L_2 \alpha_2 \Delta T \]
This can be rewritten as:
\[ L' = (L_1 + L_2) + \Delta T (L_1 \alpha_1 + L_2 \alpha_2) \]

### Step 5: Relate the Total Length to the Equivalent Coefficient
We can express the total length after the temperature increase in terms of the original length \( L = L_1 + L_2 \) and the equivalent coefficient \( \alpha \):
\[ L' = L (1 + \alpha \Delta T) \]

### Step 6: Equating the Two Expressions
Now, we equate the two expressions for \( L' \):
\[ L (1 + \alpha \Delta T) = (L_1 + L_2) + \Delta T (L_1 \alpha_1 + L_2 \alpha_2) \]

### Step 7: Isolate the Equivalent Coefficient
Dividing both sides by \( L \):
\[ 1 + \alpha \Delta T = 1 + \frac{\Delta T (L_1 \alpha_1 + L_2 \alpha_2)}{L} \]
From this, we can isolate \( \alpha \):
\[ \alpha = \frac{L_1 \alpha_1 + L_2 \alpha_2}{L_1 + L_2} \]

### Final Result
Thus, the equivalent coefficient of thermal expansion \( \alpha \) for the two rods in series is given by:
\[ \alpha = \frac{L_1 \alpha_1 + L_2 \alpha_2}{L_1 + L_2} \]

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