Three rods of same dimension are joined in series having thermal conductivity `k_1` , `k_2` and `k_3`. The equivalent thermal conductivity is

Options

A. `k_1``k_2``k_3`/`k_2``k_3`+`k_1``k_2`+`k_1``k_3`
B. 3(`k_1``k_2``k_3`)/`k_2``k_3`+`k_1``k_2`+`k_1``k_3`
C. `k_1`+`k_2`+`k_3`
D. (`k_1`+`k_2`+`k_3`)/ `3`

Answer

To find the equivalent thermal conductivity of three rods joined in series with thermal conductivities \( k_1 \), \( k_2 \), and \( k_3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have three rods of the same dimensions (length \( L \) and cross-sectional area \( A \)) connected in series. The thermal conductivities of the rods are \( k_1 \), \( k_2 \), and \( k_3 \).

2. **Thermal Resistance Calculation**:
The thermal resistance \( R \) for each rod can be calculated using the formula:
\[ R = \frac{L}{kA} \]
Therefore, the thermal resistances for the three rods will be:
- For rod 1: \( R_1 = \frac{L}{k_1 A} \)
- For rod 2: \( R_2 = \frac{L}{k_2 A} \)
- For rod 3: \( R_3 = \frac{L}{k_3 A} \)

3. **Total Thermal Resistance in Series**:
Since the rods are in series, the total thermal resistance \( R_{\text{eq}} \) is the sum of the individual resistances:
\[ R_{\text{eq}} = R_1 + R_2 + R_3 = \frac{L}{k_1 A} + \frac{L}{k_2 A} + \frac{L}{k_3 A} \]

4. **Factor Out Common Terms**:
We can factor out \( \frac{L}{A} \) from the equation:
\[ R_{\text{eq}} = \frac{L}{A} \left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right) \]

5. **Finding Equivalent Conductivity**:
The equivalent thermal conductivity \( k_{\text{eq}} \) can be defined in terms of the equivalent resistance:
\[ R_{\text{eq}} = \frac{L}{k_{\text{eq}} A} \]
Setting the two expressions for \( R_{\text{eq}} \) equal gives:
\[ \frac{L}{k_{\text{eq}} A} = \frac{L}{A} \left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right) \]

6. **Solving for \( k_{\text{eq}} \)**:
Canceling \( \frac{L}{A} \) from both sides, we have:
\[ \frac{1}{k_{\text{eq}}} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \]
Taking the reciprocal gives:
\[ k_{\text{eq}} = \frac{1}{\left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right)} \]

7. **Finding a Common Denominator**:
To express \( k_{\text{eq}} \) in a more usable form, we can find a common denominator:
\[ k_{\text{eq}} = \frac{k_1 k_2 k_3}{k_2 k_3 + k_1 k_3 + k_1 k_2} \]

### Final Answer:
Thus, the equivalent thermal conductivity \( k_{\text{eq}} \) for the three rods in series is:
\[ k_{\text{eq}} = \frac{3 k_1 k_2 k_3}{k_1 k_2 + k_2 k_3 + k_3 k_1} \]