Temperature of hot end and cold end of a rod, which is in steady state are `100^(@)C` and `40^(@)C` respectively. The area of cross section of rod and its thermal conductivity are uniform. The temperature of rod at its mid point is (Assume no heat loss thorugh lateral surface)

Options

A. `70^(@)C`
B. `60^(@)C`
C. `80^(@)C`
D. None of these

Answer

To find the temperature at the midpoint of the rod, we can use the concept of steady-state heat conduction. Here’s the step-by-step solution:

### Step 1: Understand the Setup
We have a rod with two ends at different temperatures:
- Hot end (Point P): \( T_1 = 100^\circ C \)
- Cold end (Point Q): \( T_2 = 40^\circ C \)

The rod is in steady state, meaning the temperature distribution along the rod does not change with time.

### Step 2: Identify the Midpoint
Let’s denote the midpoint of the rod as Point C. We need to find the temperature at this point, \( T_C \).

### Step 3: Apply the Concept of Steady-State Heat Conduction
In steady state, the rate of heat transfer through the rod is constant. The heat transfer can be expressed using Fourier's law of heat conduction:
\[ Q = \frac{k \cdot A \cdot (T_1 - T_2)}{L} \]
Where:
- \( Q \) is the rate of heat transfer,
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( L \) is the length of the rod.

### Step 4: Set Up the Equation for the Midpoint
Since the rod is uniform, we can assume that the temperature gradient is linear between the two ends. Therefore, the temperature at the midpoint can be calculated using the formula:
\[ T_C = \frac{T_1 + T_2}{2} \]

### Step 5: Substitute the Values
Substituting the given temperatures:
\[ T_C = \frac{100^\circ C + 40^\circ C}{2} = \frac{140^\circ C}{2} = 70^\circ C \]

### Conclusion
The temperature at the midpoint of the rod is:
\[ \boxed{70^\circ C} \]