Two rods of the same length and diameter, having thermal conductivities `K_(1)` and `K_(2)`, are joined in parallel. The equivalent thermal conductivity to the combinationk is

Options

A. `(K_(1) K_(2))/(K_(1) + K_(2))`
B. `K_(1) K_(2)`
C. `(K_(1) + K_(2))/(2)`
D. `sqrt(K_(1) K_(2))`

Explanation

To find the equivalent thermal conductivity \( k \) of two rods joined in parallel with thermal conductivities \( K_1 \) and \( K_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have two rods of the same length \( L \) and diameter \( d \) which are joined in parallel. The thermal conductivities of the rods are \( K_1 \) and \( K_2 \).

2. **Heat Resistance Calculation**:
The heat resistance \( R \) for a rod can be expressed as:
\[ R = \frac{L}{K \cdot A} \]
where \( A \) is the cross-sectional area of the rod. Since both rods have the same diameter, their cross-sectional areas are equal.

3. **Resistance of Each Rod**:
For the first rod:
\[ R_1 = \frac{L}{K_1 \cdot A} \]
For the second rod:
\[ R_2 = \frac{L}{K_2 \cdot A} \]

4. **Equivalent Resistance in Parallel**:
When resistances are in parallel, the equivalent resistance \( R_{eq} \) can be calculated using:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Substituting the expressions for \( R_1 \) and \( R_2 \):
\[ \frac{1}{R_{eq}} = \frac{K_1 \cdot A}{L} + \frac{K_2 \cdot A}{L} \]

5. **Simplifying the Equation**:
Factoring out \( \frac{A}{L} \):
\[ \frac{1}{R_{eq}} = \frac{A}{L} \left( K_1 + K_2 \right) \]
Thus:
\[ R_{eq} = \frac{L}{A \left( K_1 + K_2 \right)} \]

6. **Finding Equivalent Thermal Conductivity**:
The equivalent thermal conductivity \( K_{eq} \) can be defined as:
\[ R_{eq} = \frac{L}{K_{eq} \cdot A} \]
Setting the two expressions for \( R_{eq} \) equal to each other:
\[ \frac{L}{K_{eq} \cdot A} = \frac{L}{A \left( K_1 + K_2 \right)} \]

7. **Solving for \( K_{eq} \)**:
Canceling \( L \) and \( A \) from both sides:
\[ K_{eq} = K_1 + K_2 \]

### Final Result:
The equivalent thermal conductivity \( K_{eq} \) for the two rods joined in parallel is:
\[ K_{eq} = K_1 + K_2 \]