A wall has two layers A and B, each made of different material. Both the layers have the same thickness. The thermal conductivity of the material of A is twice that of B . Under thermal equilibrium, the temperature difference across the wall is `36^@C.` The temperature difference across the layer A is

Options

A. `6^(@)`C
B. `12^(@)`C
C. `24^(@)`C
D. `18^(@)`C

Explanation

To solve the problem, we will use the concept of thermal resistance in series and the relationship between heat transfer, temperature difference, and thermal conductivity.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Let the thermal conductivity of layer B be \( K \).
- Therefore, the thermal conductivity of layer A is \( 2K \) (since it is twice that of B).
- Let the thickness of each layer be \( d \).
- The total temperature difference across the wall is \( \Delta T = 36^\circ C \).

2. **Calculate the Thermal Resistance of Each Layer:**
- The thermal resistance \( R \) for a layer can be calculated using the formula:
\[ R = \frac{d}{kA} \]
- For layer A (with thermal conductivity \( 2K \)):
\[ R_A = \frac{d}{2KA} \]
- For layer B (with thermal conductivity \( K \)):
\[ R_B = \frac{d}{KA} \]

3. **Express the Total Thermal Resistance:**
- Since the layers are in series, the total thermal resistance \( R_{total} \) is:
\[ R_{total} = R_A + R_B = \frac{d}{2KA} + \frac{d}{KA} = \frac{d}{2KA} + \frac{2d}{2KA} = \frac{3d}{2KA} \]

4. **Use the Heat Transfer Equation:**
- The heat transfer \( Q \) through the wall can be expressed as:
\[ Q = \frac{\Delta T}{R_{total}} = \frac{36}{\frac{3d}{2KA}} = \frac{36 \cdot 2KA}{3d} = \frac{72KA}{3d} = \frac{24KA}{d} \]

5. **Determine the Temperature Difference Across Layer A:**
- The temperature difference across layer A (\( \Delta T_A \)) can be calculated using its thermal resistance:
\[ \Delta T_A = Q \cdot R_A = Q \cdot \frac{d}{2KA} \]
- Substituting for \( Q \):
\[ \Delta T_A = \left(\frac{24KA}{d}\right) \cdot \frac{d}{2KA} = \frac{24KA}{d} \cdot \frac{d}{2KA} = \frac{24}{2} = 12^\circ C \]

### Final Answer:
The temperature difference across layer A is \( 12^\circ C \).

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