Two rods of equal length and diameter have thermal conductivity 3 and 4 units respectively. If they are joined in series, the thermal conductivity of the combination in the given units would be

Options

A. `3.43` units
B. `4.43` units
C. `5.43` units
D. `2.43` units

Explanation

To solve the problem of finding the equivalent thermal conductivity of two rods joined in series, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Thermal conductivity of rod 1, \( k_1 = 3 \) units
- Thermal conductivity of rod 2, \( k_2 = 4 \) units
- Length of both rods, \( l_1 = l_2 = l \) (equal length)
- Diameter of both rods is equal, which implies that their cross-sectional areas are also equal, \( A_1 = A_2 = A \).

2. **Calculate the Thermal Resistance of Each Rod:**
- The thermal resistance \( R \) for each rod can be calculated using the formula:
\[ R = \frac{l}{kA} \]
- For rod 1:
\[ R_1 = \frac{l}{k_1 A} = \frac{l}{3A} \]
- For rod 2:
\[ R_2 = \frac{l}{k_2 A} = \frac{l}{4A} \]

3. **Calculate the Total Thermal Resistance in Series:**
- When the rods are connected in series, the total resistance \( R \) is the sum of the individual resistances:
\[ R = R_1 + R_2 = \frac{l}{3A} + \frac{l}{4A} \]
- To combine these fractions, find a common denominator (which is 12A):
\[ R = \frac{4l}{12A} + \frac{3l}{12A} = \frac{(4 + 3)l}{12A} = \frac{7l}{12A} \]

4. **Relate the Total Resistance to the Equivalent Thermal Conductivity:**
- The equivalent thermal resistance can also be expressed as:
\[ R = \frac{l_{total}}{k_{eq} A} \]
- Since the total length when connected in series is \( l_{total} = 2l \):
\[ R = \frac{2l}{k_{eq} A} \]

5. **Set the Two Expressions for Resistance Equal:**
- Equating the two expressions for resistance:
\[ \frac{7l}{12A} = \frac{2l}{k_{eq} A} \]
- Cancel \( l \) and \( A \) from both sides:
\[ \frac{7}{12} = \frac{2}{k_{eq}} \]

6. **Solve for the Equivalent Thermal Conductivity \( k_{eq} \):**
- Rearranging gives:
\[ k_{eq} = \frac{2 \times 12}{7} = \frac{24}{7} \approx 3.43 \text{ units} \]

### Final Answer:
The equivalent thermal conductivity of the combination of the two rods in series is approximately \( 3.43 \) units.

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