Two cylinders P and Q have the same length and diameter and are made of different materials having thermal conductivities in the ratio `2 : 3`. These two cyinders ar combined to make a cylinder. One end of P is kept at `100^(@)C` and another end of Q at `0^(@)C`. THe temperature at the interface of P and Q is

Options

A. `30^(@)C `
B. `40^(@)C `
C. `50^(@)C `
D. `60^(@)C `

Explanation

To solve the problem, we need to find the temperature at the interface of two cylinders P and Q, which are made of different materials with thermal conductivities in the ratio of 2:3.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Length of both cylinders (P and Q): L
- Diameter of both cylinders: same (not needed for calculations as it cancels out)
- Thermal conductivity of cylinder P: \( k_P = 2k \)
- Thermal conductivity of cylinder Q: \( k_Q = 3k \)
- Temperature at one end of P: \( T_P = 100^\circ C \)
- Temperature at one end of Q: \( T_Q = 0^\circ C \)

2. **Set Up the Heat Transfer Equations:**
Since both cylinders are in series, the heat transfer rate (Q/T) through both cylinders must be equal:
\[ \frac{Q}{T} = \frac{k_P \cdot A \cdot (T_P - T_{\text{interface}})}{L} = \frac{k_Q \cdot A \cdot (T_{\text{interface}} - T_Q)}{L} \]

3. **Substituting the Known Values:**
Substitute \( k_P \) and \( k_Q \) into the equation:
\[ \frac{2k \cdot A \cdot (100 - T_{\text{interface}})}{L} = \frac{3k \cdot A \cdot (T_{\text{interface}} - 0)}{L} \]

4. **Cancel Common Terms:**
Since \( k \), \( A \), and \( L \) are common on both sides, we can cancel them out:
\[ 2(100 - T_{\text{interface}}) = 3T_{\text{interface}} \]

5. **Expand and Rearrange the Equation:**
Expanding the left side:
\[ 200 - 2T_{\text{interface}} = 3T_{\text{interface}} \]
Rearranging gives:
\[ 200 = 3T_{\text{interface}} + 2T_{\text{interface}} \implies 200 = 5T_{\text{interface}} \]

6. **Solve for the Temperature at the Interface:**
Divide both sides by 5:
\[ T_{\text{interface}} = \frac{200}{5} = 40^\circ C \]

### Final Answer:
The temperature at the interface of cylinders P and Q is \( 40^\circ C \).