In a steady state of thermal conduction, temperature of the ends A and B of a 20 cm long rod are `100^(@)C` and `0^(@)C` respectively. What will be the temperature of the rod at a point at a distance of 9 cm from the end A of the rod

Options

A. `45^(@) C`
B. `55^(@) C`
C. `5^(@) C`
D. `65^(@) C`

Explanation

To solve the problem, we need to find the temperature at a point located 9 cm from end A of a 20 cm long rod, where the temperatures at the ends A and B are 100°C and 0°C, respectively.

### Step-by-Step Solution:

1. **Identify the Lengths and Temperatures**:
- Length of the rod (L) = 20 cm
- Temperature at end A (T_A) = 100°C
- Temperature at end B (T_B) = 0°C
- Distance from end A to point C (d) = 9 cm

2. **Understand the Concept of Steady State**:
- In a steady state of thermal conduction, the heat current (rate of heat transfer) is constant throughout the rod.
- The temperature gradient is linear in a uniform rod, meaning the temperature changes uniformly along the length of the rod.

3. **Set Up the Temperature Gradient**:
- The temperature difference between the two ends of the rod is:
\[ \Delta T = T_A - T_B = 100°C - 0°C = 100°C \]
- The total length of the rod is 20 cm, so the temperature gradient (change in temperature per unit length) is:
\[ \text{Temperature Gradient} = \frac{\Delta T}{L} = \frac{100°C}{20 \text{ cm}} = 5°C/\text{cm} \]

4. **Calculate the Temperature at Point C**:
- The distance from end A to point C is 9 cm. Therefore, the temperature drop from end A to point C is:
\[ \text{Temperature Drop} = \text{Distance} \times \text{Temperature Gradient} = 9 \text{ cm} \times 5°C/\text{cm} = 45°C \]
- Now, subtract this temperature drop from the temperature at end A to find the temperature at point C:
\[ T_C = T_A - \text{Temperature Drop} = 100°C - 45°C = 55°C \]

5. **Conclusion**:
- The temperature at the point located 9 cm from end A is **55°C**.