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 1: Understand the Concept of Thermal Resistance
Thermal resistance \( R \) for a rod can be defined as:
\[ R = \frac{L}{kA} \]
where:
- \( L \) is the length of the rod,
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area.

### Step 2: Calculate Individual Resistances
For each rod, the thermal resistance can be expressed as:
- For rod 1: \( R_1 = \frac{L}{k_1 A} \)
- For rod 2: \( R_2 = \frac{L}{k_2 A} \)
- For rod 3: \( R_3 = \frac{L}{k_3 A} \)

### Step 3: Find Total Resistance in Series
When these rods are connected in series, the total thermal resistance \( R_{\text{net}} \) is the sum of the individual resistances:
\[ R_{\text{net}} = R_1 + R_2 + R_3 = \frac{L}{k_1 A} + \frac{L}{k_2 A} + \frac{L}{k_3 A} \]

### Step 4: Simplify the Total Resistance
Factoring out \( \frac{L}{A} \):
\[ R_{\text{net}} = \frac{L}{A} \left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right) \]

### Step 5: Relate Total Resistance to Equivalent Conductivity
The equivalent thermal resistance can also be expressed in terms of the equivalent thermal conductivity \( k' \):
\[ R_{\text{net}} = \frac{L}{k' A} \]

### Step 6: Set the Two Expressions for Resistance Equal
Setting the two expressions for \( R_{\text{net}} \) equal gives:
\[ \frac{L}{k' A} = \frac{L}{A} \left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right) \]

### Step 7: Cancel Common Terms
Cancel \( \frac{L}{A} \) from both sides:
\[ \frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \]

### Step 8: Solve for Equivalent Conductivity
Taking the reciprocal gives us the equivalent thermal conductivity:
\[ k' = \frac{1}{\left( \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \right)} \]

### Step 9: Find a Common Denominator
To express this in a more usable form, we can find a common denominator:
\[ k' = \frac{k_1 k_2 k_3}{k_2 k_3 + k_3 k_1 + k_1 k_2} \]

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