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 heat flow through a conductor and how temperature varies along its length.

### Step-by-Step Solution:

1. **Understanding the Problem**:
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 through the conductor is constant, and the thermal resistance per unit length is also constant.

2. **Using the Heat Flow Equation**:
The rate of heat flow \( \frac{dq}{dt} \) through the conductor can be expressed using Fourier's law of heat conduction:
\[ \frac{dq}{dt} = -kA \frac{(T - T_0)}{L} \]
where:
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( T \) is the temperature at position \( x \),
- \( T_0 \) is the temperature at the other end of the conductor.

3. **Rearranging the Equation**:
We can rearrange the equation to express temperature \( T \) as a function of position \( x \):
\[ T(x) = -\frac{dq}{dt} \frac{x}{kA} + T_0 \]
This equation shows that \( T(x) \) is a linear function of \( x \).

4. **Identifying the Graph Characteristics**:
- The slope of the temperature vs. position graph is negative, indicating that as we move along the conductor from \( x = 0 \) to \( x = L \), the temperature decreases.
- The y-intercept of the graph is \( T_0 \), which is positive.

5. **Analyzing the Given Graph Options**:
- **Option 1**: Slope is positive (incorrect).
- **Option 2**: No intercept (incorrect).
- **Option 3**: Negative slope with a positive intercept (correct).
- **Option 4**: Not a straight line (incorrect).

6. **Conclusion**:
The correct graph that represents the relationship between temperature and position along the conductor is **Option 3**.