Two rods of equal length and diameter but of thermal conductivities 2 and 3 unites respectively are joined in parallel. The thermal conductivity of the combination is:

Options

A. 1
B. 1.5
C. 2.5
D. 5

Explanation

To find the thermal conductivity of two rods joined in parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data**:
- Thermal conductivity of rod 1, \( K_1 = 2 \) units
- Thermal conductivity of rod 2, \( K_2 = 3 \) units
- Length of both rods, \( L \) (same for both)
- Diameter of both rods is the same, hence their cross-sectional area \( A \) is the same.

2. **Understanding Parallel Connection**:
- When two rods are connected in parallel, the total area available for heat transfer is the sum of the areas of both rods. Since the diameter is the same, the area will be \( 2A \) for both rods combined.

3. **Heat Resistance Calculation**:
- The heat resistance \( R \) for a rod can be expressed as:
\[ R = \frac{L}{KA} \]
- For rod 1:
\[ R_1 = \frac{L}{K_1 A} = \frac{L}{2A} \]
- For rod 2:
\[ R_2 = \frac{L}{K_2 A} = \frac{L}{3A} \]

4. **Equivalent Resistance in Parallel**:
- The equivalent resistance \( R_{eq} \) for resistances in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
- Substituting the values:
\[ \frac{1}{R_{eq}} = \frac{1}{\frac{L}{2A}} + \frac{1}{\frac{L}{3A}} = \frac{2A}{L} + \frac{3A}{L} \]
- Combining the fractions:
\[ \frac{1}{R_{eq}} = \frac{(2 + 3)A}{L} = \frac{5A}{L} \]
- Therefore, the equivalent resistance is:
\[ R_{eq} = \frac{L}{5A} \]

5. **Finding Equivalent Thermal Conductivity**:
- The equivalent thermal conductivity \( K_{eq} \) can be calculated using the equivalent resistance:
\[ R_{eq} = \frac{L}{K_{eq} \cdot 2A} \]
- Setting the two expressions for \( R_{eq} \) equal:
\[ \frac{L}{5A} = \frac{L}{K_{eq} \cdot 2A} \]
- Canceling \( L \) and \( A \) from both sides:
\[ \frac{1}{5} = \frac{1}{K_{eq} \cdot 2} \]
- Rearranging gives:
\[ K_{eq} \cdot 2 = 5 \quad \Rightarrow \quad K_{eq} = \frac{5}{2} = 2.5 \]

### Final Answer:
The thermal conductivity of the combination is \( K_{eq} = 2.5 \) units.

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