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 meterial 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. `18^@C`
D. `24^@C`

Explanation

To solve the problem, we need to find the temperature difference across layer A of the wall consisting of two layers A and B with different thermal conductivities. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have two layers A and B, each with the same thickness (L). The thermal conductivity of layer A (K_A) is twice that of layer B (K_B). The total temperature difference across the wall is given as 36°C.

### Step 2: Define Variables
Let:
- K_A = 2K_B (thermal conductivity of layer A)
- K_B = K_B (thermal conductivity of layer B)
- ΔT_total = θ_A - θ_B = 36°C (total temperature difference across the wall)
- θ_C = temperature at the junction between layers A and B.

### Step 3: Use the Heat Conduction Equation
According to the principle of thermal equilibrium, the heat conducted through both layers must be equal. The heat flow (Q) through a material is given by:
\[ Q = \frac{K \cdot A \cdot \Delta T}{L} \]
Since both layers have the same area (A) and thickness (L), we can ignore these factors for our calculations.

For layer A:
\[ Q_A = K_A \cdot \Delta T_A = K_A \cdot (θ_A - θ_C) \]

For layer B:
\[ Q_B = K_B \cdot \Delta T_B = K_B \cdot (θ_C - θ_B) \]

### Step 4: Set the Heat Flow Equal
Since \( Q_A = Q_B \):
\[ K_A \cdot (θ_A - θ_C) = K_B \cdot (θ_C - θ_B) \]

Substituting \( K_A = 2K_B \):
\[ 2K_B \cdot (θ_A - θ_C) = K_B \cdot (θ_C - θ_B) \]

### Step 5: Simplify the Equation
Dividing through by \( K_B \) (assuming \( K_B \neq 0 \)):
\[ 2(θ_A - θ_C) = θ_C - θ_B \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 2θ_A - 2θ_C = θ_C - θ_B \]
\[ 2θ_A + θ_B = 3θ_C \]

### Step 7: Express θ_B in Terms of θ_A
From the total temperature difference:
\[ θ_B = θ_A - 36 \]

### Step 8: Substitute θ_B into the Equation
Substituting \( θ_B \) into the equation:
\[ 2θ_A + (θ_A - 36) = 3θ_C \]
\[ 3θ_A - 36 = 3θ_C \]

### Step 9: Solve for θ_C
Dividing through by 3:
\[ θ_C = θ_A - 12 \]

### Step 10: Find the Temperature Difference Across Layer A
The temperature difference across layer A is:
\[ ΔT_A = θ_A - θ_C \]
Substituting for \( θ_C \):
\[ ΔT_A = θ_A - (θ_A - 12) = 12°C \]

### Final Answer
The temperature difference across layer A is **12°C**.