A wall has two layers A and B, each made of different materials. Both layers are of same thickness. But, the thermal conductivity of material A is twice that of B. If, in the steady state, the temperature difference across the wall is `24^(@)C`, then the temperature difference across the layer B is :

Options

A. `8^(@)C`
B. `12^(@)C`
C. `16^(@)C`
D. `20^(@)C`

Explanation

To solve the problem, we need to determine the temperature difference across layer B of a wall that consists of two layers A and B, where the thermal conductivity of layer A is twice that of layer B.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Let the thermal conductivity of layer B be \( k \).
- Then, the thermal conductivity of layer A is \( 2k \).
- Both layers have the same thickness \( D \).
- The total temperature difference across the wall is \( 24^\circ C \).

2. **Define the Resistance of Each Layer**:
- The thermal resistance \( R \) of a layer can be calculated using the formula:
\[ R = \frac{L}{kA} \]
where \( L \) is the thickness, \( k \) is the thermal conductivity, and \( A \) is the area.
- For layer A:
\[ R_A = \frac{D}{2kA} \]
- For layer B:
\[ R_B = \frac{D}{kA} \]

3. **Calculate the Total Resistance**:
- The total resistance \( R_{total} \) of the two layers in series is:
\[ R_{total} = R_A + R_B = \frac{D}{2kA} + \frac{D}{kA} = \frac{D}{2kA} + \frac{2D}{2kA} = \frac{3D}{2kA} \]

4. **Calculate the Thermal Current**:
- The thermal current \( I \) through the wall can be expressed as:
\[ I = \frac{\Delta T}{R_{total}} = \frac{24^\circ C}{\frac{3D}{2kA}} = \frac{48kA}{3D} = \frac{16kA}{D} \]

5. **Determine the Temperature Drop Across Each Layer**:
- The temperature drop across layer A (\( \Delta T_A \)) can be calculated as:
\[ \Delta T_A = I \cdot R_A = I \cdot \frac{D}{2kA} \]
- Substituting \( I \):
\[ \Delta T_A = \left(\frac{16kA}{D}\right) \cdot \frac{D}{2kA} = \frac{16}{2} = 8^\circ C \]

6. **Calculate the Temperature Drop Across Layer B**:
- The temperature drop across layer B (\( \Delta T_B \)) is:
\[ \Delta T_B = \Delta T - \Delta T_A = 24^\circ C - 8^\circ C = 16^\circ C \]

### Final Answer:
The temperature difference across layer B is \( 16^\circ C \).

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