The ends of a long bar are maintained at different temperatures and there is no loss of heat from the sides of the bar due to conduction or radiation. The graph of temperature against distance of the bar when it has attained steady state is shown here. The graph shows (i) the temperature gradient is not uniform.
(ii) the bar has uniform cross-sectional area.
(iii) the cross-sectional area of the bar increases as the distance from the hot end increases.
(iv) the cross-sectional area of the bar becreases as the distance from the hot end increases

Options

A. i & ii
B. ii & iii
C. i & iv
D. ii & iv

Answer

To solve the problem, we need to analyze the given information about the long bar that is maintained at different temperatures at its ends and understand the implications of the temperature gradient and the cross-sectional area.

### Step-by-Step Solution:

1. **Understanding the Steady State Condition**:
- In steady state, the temperature distribution along the bar does not change with time. The heat transfer through the bar reaches a constant value.
- Since there is no heat loss from the sides, the heat current (rate of heat transfer) remains constant throughout the bar.

2. **Analyzing the Temperature Gradient**:
- The graph shows temperature (T) plotted against distance (x) from the hot end of the bar.
- If the graph is not a straight line (for example, it is parabolic), it indicates that the temperature gradient (dT/dx) is not uniform. This means that the rate of temperature change with respect to distance varies along the length of the bar.

3. **Using Fourier's Law of Heat Conduction**:
- According to Fourier's law, the heat current (dq/dt) can be expressed as:
\[ \frac{dq}{dt} = kA \frac{dT}{dx} \]
- Here, \(k\) is the thermal conductivity, \(A\) is the cross-sectional area, and \(dT/dx\) is the temperature gradient.
- Since \(dq/dt\) is constant, if \(dT/dx\) changes, then \(A\) must also change to keep the product \(A \cdot \frac{dT}{dx}\) constant.

4. **Determining the Cross-Sectional Area**:
- If the temperature gradient (dT/dx) increases as we move from the hot end to the cold end, then the cross-sectional area \(A\) must decrease to maintain a constant heat current.
- This leads us to conclude that the cross-sectional area of the bar decreases as the distance from the hot end increases.

5. **Evaluating the Statements**:
- (i) The temperature gradient is not uniform: **True** (as indicated by the graph).
- (ii) The bar has a uniform cross-sectional area: **False** (as we concluded that the area decreases).
- (iii) The cross-sectional area of the bar increases as the distance from the hot end increases: **False**.
- (iv) The cross-sectional area of the bar decreases as the distance from the hot end increases: **True**.

6. **Final Conclusion**:
- The true statements are (i) and (iv). Therefore, the correct answer is that statements 1 and 4 are true.