A conductor of length L is placed along the x - axis , with one of its ends at x = 0 and the other at x = L. If the rate of flow of heat energy through the conduct is constant and its thermal resistance per unit length is also constant , then Which of the following graphs is/are correct ?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the situation of heat transfer through a conductor and how it relates to temperature distribution along the length of the conductor.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a conductor of length \( L \) placed along the x-axis, with one end at \( x = 0 \) and the other at \( x = L \).
- The rate of heat flow (H) through the conductor is constant, and the thermal resistance per unit length is also constant.

2. **Heat Transfer Equation**:
- The rate of heat transfer \( H \) can be expressed using Fourier's law of heat conduction:
\[ H = K A \frac{dT}{dx} \]
where:
- \( K \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \frac{dT}{dx} \) is the temperature gradient.

3. **Constant Thermal Resistance**:
- The thermal resistance \( R \) of the conductor can be defined as:
\[ R = \frac{L}{K A} \]
- Given that the thermal resistance per unit length is constant, it implies that \( K A \) is also constant.

4. **Finding the Temperature Gradient**:
- Since \( H \) is constant and \( K A \) is constant, it follows that \( \frac{dT}{dx} \) must also be constant.
- Therefore, we can express this as:
\[ \frac{dT}{dx} = -\text{constant} \]
- The negative sign indicates that as \( x \) increases, the temperature \( T \) decreases.

5. **Graphical Representation**:
- The relationship \( \frac{dT}{dx} = -\text{constant} \) implies that the temperature \( T \) decreases linearly with \( x \).
- This results in a straight line graph with a negative slope.

6. **Conclusion**:
- The correct graph representing the temperature distribution along the conductor will show a straight line with a negative slope, indicating a linear decrease in temperature from \( T(0) \) at \( x = 0 \) to \( T(L) \) at \( x = L \).