The thermal conductivity of two materials are in the ratio 1 : 2. What will be the ratio of thermal resistances of rods of these materials having length in the ratio 1 :2 and area of cross-section in the ratio 1:2 :

Options

A. `2 : 1`
B. `1 : 4`
C. `1 : 8`
D. `1 : 16`

Explanation

To solve the problem, we need to find the ratio of thermal resistances of two rods made from different materials, given their thermal conductivities, lengths, and areas of cross-section.

### Step-by-Step Solution:

1. **Understanding Thermal Resistance**:
The thermal resistance \( R \) of a material is given by the formula:
\[ R = \frac{L}{kA} \]
where:
- \( L \) is the length of the rod,
- \( k \) is the thermal conductivity of the material,
- \( A \) is the area of cross-section.

2. **Assigning Values**:
Let's denote the thermal conductivity of the first material as \( k \) and the second material as \( 2k \) (since the ratio of thermal conductivities is 1:2).
- For the first rod:
- Length \( L_1 = L \)
- Area \( A_1 = A \)
- Thermal conductivity \( k_1 = k \)
- For the second rod:
- Length \( L_2 = 2L \)
- Area \( A_2 = 2A \)
- Thermal conductivity \( k_2 = 2k \)

3. **Calculating Thermal Resistances**:
- For the first rod (Material 1):
\[ R_1 = \frac{L_1}{k_1 A_1} = \frac{L}{k \cdot A} \]
- For the second rod (Material 2):
\[ R_2 = \frac{L_2}{k_2 A_2} = \frac{2L}{2k \cdot 2A} = \frac{2L}{4kA} = \frac{L}{2kA} \]

4. **Finding the Ratio of Thermal Resistances**:
Now, we can find the ratio \( R_1 : R_2 \):
\[ \frac{R_1}{R_2} = \frac{\frac{L}{kA}}{\frac{L}{2kA}} = \frac{L}{kA} \cdot \frac{2kA}{L} = 2 \]
Thus, the ratio of thermal resistances \( R_1 : R_2 = 2 : 1 \).

### Final Answer:
The ratio of thermal resistances of the two rods is \( 2 : 1 \).

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