A wall has two layers A and B, each made of different materials. 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 differeence across the wall is `36^(@)C`. The temperature difference across the layer A is

Options

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

Explanation

To solve the problem, we need to find the temperature difference across layer A of the wall, given that the thermal conductivity of layer A is twice that of layer B and the total temperature difference across the wall is 36°C.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the thermal conductivity of layer B be \( K \).
- Then, the thermal conductivity of layer A will be \( 2K \).
- Let the thickness of each layer be \( L \).
- Let the temperature difference across layer A be \( \Delta T_A \).
- Let the temperature difference across layer B be \( \Delta T_B \).

2. **Use the Concept of Thermal Resistance**:
- The thermal resistance \( R \) for conduction through a layer can be expressed as:
\[ R = \frac{L}{K \cdot A} \]
- For layer A:
\[ R_A = \frac{L}{2K \cdot A} \]
- For layer B:
\[ R_B = \frac{L}{K \cdot A} \]

3. **Relate Temperature Differences and Thermal Resistances**:
- According to the principle of thermal equilibrium, the heat current \( I \) through both layers is the same:
\[ I = \frac{\Delta T_A}{R_A} = \frac{\Delta T_B}{R_B} \]
- This leads to the equations:
\[ I = \frac{\Delta T_A \cdot 2K \cdot A}{L} \]
\[ I = \frac{\Delta T_B \cdot K \cdot A}{L} \]

4. **Set Up the Equation for Total Temperature Difference**:
- The total temperature difference across the wall is given as:
\[ \Delta T_A + \Delta T_B = 36^\circ C \]

5. **Express \( \Delta T_B \) in Terms of \( \Delta T_A \)**:
- From the equations for \( I \):
\[ \Delta T_B = \frac{I \cdot L}{K \cdot A} \]
- Substituting \( I \) from layer A:
\[ \Delta T_B = \frac{2 \Delta T_A \cdot K \cdot A}{L} \cdot \frac{L}{K \cdot A} = 2 \Delta T_A \]

6. **Substitute into Total Temperature Difference**:
- Now substitute \( \Delta T_B \) into the total temperature difference equation:
\[ \Delta T_A + 2 \Delta T_A = 36^\circ C \]
\[ 3 \Delta T_A = 36^\circ C \]
\[ \Delta T_A = \frac{36^\circ C}{3} = 12^\circ C \]

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

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