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

Explanation

To solve the problem step by step, we will analyze the thermal properties of the two layers A and B of the wall.

### Step 1: Define the Variables
Let:
- \( k \) = thermal conductivity of material B
- \( 2k \) = thermal conductivity of material A (since it is twice that of B)
- \( T_A \) = temperature at the left side of layer A
- \( T_B \) = temperature at the right side of layer B
- \( T \) = temperature at the junction between layers A and B
- The thickness of both layers = \( t \)
- The total temperature difference across the wall = \( T_B - T_A = 36^\circ C \)

### Step 2: Write the Heat Flow Equations
According to Fourier's law of heat conduction, the heat flow \( Q \) through each layer can be expressed as:
\[ Q = \frac{k \cdot A \cdot (T_A - T)}{t} \quad \text{(for layer A)} \]
\[ Q = \frac{2k \cdot A \cdot (T - T_B)}{t} \quad \text{(for layer B)} \]

Since the heat flow is constant in steady state, we can set the two equations equal to each other:
\[ \frac{2k \cdot A \cdot (T_A - T)}{t} = \frac{k \cdot A \cdot (T - T_B)}{t} \]

### Step 3: Simplify the Equation
We can cancel out \( k \), \( A \), and \( t \) from both sides:
\[ 2(T_A - T) = (T - T_B) \]
This simplifies to:
\[ 2T_A - 2T = T - T_B \]
Rearranging gives:
\[ 3T = T_B + 2T_A \]

### Step 4: Use the Total Temperature Difference
We know from the problem statement that:
\[ T_B - T_A = 36^\circ C \]
We can express \( T_B \) in terms of \( T_A \):
\[ T_B = T_A + 36 \]

### Step 5: Substitute and Solve for \( T_A \)
Substituting \( T_B \) into the equation \( 3T = T_B + 2T_A \):
\[ 3T = (T_A + 36) + 2T_A \]
This simplifies to:
\[ 3T = 3T_A + 36 \]
Thus, we can express \( T \) in terms of \( T_A \):
\[ T = T_A + 12 \]

### Step 6: Find the Temperature Difference Across Layer A
Now, we can find the temperature difference across layer A:
\[ \Delta T_A = T_A - T = T_A - (T_A + 12) = -12^\circ C \]
This means the temperature difference across layer A is:
\[ \Delta T_A = 12^\circ C \]

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