The wax melts upto 2 cm and 6 cm lengths for two identical rods of different metals in Ingen Hausz's experiment. The ratio of their thermal conductivities is

Options

A. `1:9`
B. `1:4`
C. `1:3`
D. `1:2`

Explanation

To solve the problem, we need to determine the ratio of the thermal conductivities of two different metals based on the lengths of wax melted in Ingen Hausz's experiment.

### Step-by-Step Solution:

1. **Understanding the Experiment**:
- We have two identical rods made of different metals.
- The wax melts to different lengths: 2 cm for the first rod and 6 cm for the second rod.

2. **Identifying Variables**:
- Let the thermal conductivity of the first metal be \( K_1 \) and the second metal be \( K_2 \).
- The lengths of wax melted are \( L_1 = 2 \, \text{cm} \) and \( L_2 = 6 \, \text{cm} \).

3. **Using the Relationship**:
- According to the principles of heat conduction, the thermal conductivity \( K \) is directly proportional to the square of the length of wax melted. This can be expressed as:
\[ K \propto L^2 \]
- Therefore, we can write the relationship for the two metals as:
\[ \frac{K_1}{K_2} = \frac{L_1^2}{L_2^2} \]

4. **Substituting the Values**:
- Substitute the values of \( L_1 \) and \( L_2 \) into the equation:
\[ \frac{K_1}{K_2} = \frac{(2 \, \text{cm})^2}{(6 \, \text{cm})^2} \]
- This simplifies to:
\[ \frac{K_1}{K_2} = \frac{4}{36} \]

5. **Simplifying the Ratio**:
- Simplifying \( \frac{4}{36} \) gives:
\[ \frac{K_1}{K_2} = \frac{1}{9} \]

6. **Conclusion**:
- The ratio of the thermal conductivities \( K_1 : K_2 \) is \( 1 : 9 \).

### Final Answer:
The ratio of their thermal conductivities is \( \frac{K_1}{K_2} = \frac{1}{9} \).