A composite slab consists of two parts of equal thickness. The thermal conductivity of one is twice that of the other. What will be the ratio of temperature difference across the two layers in the state of equilibrium ?

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the ratio of the temperature differences across two layers of a composite slab in thermal equilibrium. Let's denote the two layers as Layer 1 and Layer 2.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Both layers have equal thickness \( L \).
- Let the thermal conductivity of Layer 1 be \( K_1 \).
- The thermal conductivity of Layer 2 is \( K_2 = 2K_1 \) (as given, one is twice the other).

2. **Understand the Heat Transfer in Steady State:**
- In thermal equilibrium, the rate of heat transfer through both layers must be equal. This can be expressed as:
\[ \frac{dQ}{dt} = \frac{K_1 \cdot A \cdot \Delta T_1}{L} = \frac{K_2 \cdot A \cdot \Delta T_2}{L} \]
- Here, \( \Delta T_1 \) is the temperature difference across Layer 1, and \( \Delta T_2 \) is the temperature difference across Layer 2.

3. **Set Up the Equation:**
- Since \( K_2 = 2K_1 \), we can substitute \( K_2 \) into the equation:
\[ \frac{K_1 \cdot A \cdot \Delta T_1}{L} = \frac{2K_1 \cdot A \cdot \Delta T_2}{L} \]

4. **Cancel Common Terms:**
- The area \( A \) and thickness \( L \) are common on both sides and can be canceled out:
\[ K_1 \cdot \Delta T_1 = 2K_1 \cdot \Delta T_2 \]
- Dividing both sides by \( K_1 \) (assuming \( K_1 \neq 0 \)):
\[ \Delta T_1 = 2 \Delta T_2 \]

5. **Find the Ratio of Temperature Differences:**
- Rearranging gives:
\[ \frac{\Delta T_1}{\Delta T_2} = 2 \]

### Final Answer:
The ratio of the temperature difference across the two layers in the state of equilibrium is:
\[ \frac{\Delta T_1}{\Delta T_2} = 2:1 \]