A wall has two layers A and B each made of different materials. The layer A is 10cm thick and B is 20 cm thick. The thermal conductivity of A is thrice that of B. Under thermal equilibrium temperature difference across the wall is `35^@C`. The difference of temperature across the layer A is

Options

A. `20^(@)C`
B. `10^(@)C`
C. `15^(@)C`
D. `5^(@)C`

Explanation

To solve the problem, we need to find the temperature difference across layer A of a wall that consists of two layers A and B, where the thermal conductivity of layer A is three times that of layer B. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Identify Given Data:**
- Thickness of layer A, \( L_A = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Thickness of layer B, \( L_B = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Thermal conductivity of layer B, \( K_B = K \)
- Thermal conductivity of layer A, \( K_A = 3K \)
- Total temperature difference across the wall, \( \Delta T = T_1 - T_2 = 35^\circ C \)

2. **Set Up Heat Transfer Equations:**
- The rate of heat transfer through layer A (\( H_A \)) can be expressed as:
\[ H_A = \frac{K_A \cdot A \cdot (T_1 - T)}{L_A} = \frac{3K \cdot A \cdot (T_1 - T)}{0.1} \]
- The rate of heat transfer through layer B (\( H_B \)) can be expressed as:
\[ H_B = \frac{K_B \cdot A \cdot (T - T_2)}{L_B} = \frac{K \cdot A \cdot (T - T_2)}{0.2} \]

3. **Apply Thermal Equilibrium Condition:**
- Under thermal equilibrium, the heat transfer rates through both layers are equal:
\[ H_A = H_B \]
- Substituting the expressions for \( H_A \) and \( H_B \):
\[ \frac{3K \cdot A \cdot (T_1 - T)}{0.1} = \frac{K \cdot A \cdot (T - T_2)}{0.2} \]

4. **Simplify the Equation:**
- Cancel out \( K \) and \( A \) (assuming they are non-zero):
\[ \frac{3(T_1 - T)}{0.1} = \frac{(T - T_2)}{0.2} \]
- Cross-multiplying gives:
\[ 6(T_1 - T) = T - T_2 \]

5. **Rearranging the Equation:**
- Rearranging the equation, we have:
\[ 6T_1 - 6T = T - T_2 \]
\[ 6T_1 - T + T_2 = 6T \]
- This can be rewritten as:
\[ 7T = 6T_1 + T_2 \]
- Therefore:
\[ T = \frac{6T_1 + T_2}{7} \]

6. **Calculate Temperature Difference Across Layer A:**
- The temperature difference across layer A is \( T_1 - T \):
\[ T_1 - T = T_1 - \frac{6T_1 + T_2}{7} \]
- Simplifying this gives:
\[ T_1 - T = \frac{7T_1 - 6T_1 - T_2}{7} = \frac{T_1 - T_2}{7} \]

7. **Substituting the Total Temperature Difference:**
- We know \( T_1 - T_2 = 35^\circ C \):
\[ T_1 - T = \frac{35}{7} = 5^\circ C \]

### Final Answer:
The temperature difference across layer A is \( 5^\circ C \).