Two identical rods of copper and iron are coated with wax uniformly. When one end of each is kept at temperature of boiling water, the length upto which wax melts are 8.4 cm and 4.2 cm respectively. If thermal conductivity of copper is 0.92, then thermal conductivity of iron is

Options

A. 0.23
B. 0.46
C. 0.115
D. 0.69

Answer

To solve the problem, we will use the relationship between thermal conductivity and the length of the melted wax on the rods. The key idea is that the thermal conductivity (K) is inversely proportional to the square of the length of the wax melted (L) for the two rods.

### Step-by-Step Solution:

1. **Identify the given values:**
- Length of wax melted on copper rod (L1) = 8.4 cm
- Length of wax melted on iron rod (L2) = 4.2 cm
- Thermal conductivity of copper (K1) = 0.92 W/m·K
- Thermal conductivity of iron (K2) = ?

2. **Understand the relationship:**
The relationship between thermal conductivity and the length of melted wax can be expressed as:
\[ \frac{K_1}{K_2} = \frac{L_2^2}{L_1^2} \]

3. **Substituting the known values:**
Substitute L1 and L2 into the equation:
\[ \frac{0.92}{K_2} = \frac{(4.2)^2}{(8.4)^2} \]

4. **Calculate the squares of lengths:**
\[ (4.2)^2 = 17.64 \quad \text{and} \quad (8.4)^2 = 70.56 \]
Thus, the equation becomes:
\[ \frac{0.92}{K_2} = \frac{17.64}{70.56} \]

5. **Simplifying the fraction:**
Calculate the fraction:
\[ \frac{17.64}{70.56} = \frac{1}{4} \quad \text{(since } 17.64 \text{ is half of } 35.28 \text{ and } 70.56 \text{ is double of } 35.28\text{)} \]

6. **Setting up the equation:**
Now we have:
\[ \frac{0.92}{K_2} = \frac{1}{4} \]

7. **Cross-multiplying to find K2:**
Cross-multiply to solve for K2:
\[ 0.92 \cdot 4 = K_2 \]
\[ K_2 = 3.68 \]

### Final Answer:
The thermal conductivity of iron (K2) is **3.68 W/m·K**.