The two ends of a uniform rod of thermal conductivity k are maintained at different but constant temperatures. The temperature gradientat any point on the rod is `(d theta)/(dl)`(equal to the difference in temperature per unit length). The heat flow per unit time per unit cross-section of the rod is `l` then which of the following statements is/are correct:

Options

A. `(d theta)/(dl)` is the same for all points on the rod.
B. l will decrease as we move from higher to lower temperature.
C. `l=k. (d theta)/(dl)`
D. All the above options are incorrect.

Answer

To solve the problem, we need to analyze the heat flow through a uniform rod with constant thermal conductivity \( k \) and a temperature gradient defined by \( \frac{dT}{dL} \). Let's break down the steps:

### Step 1: Understand the Heat Flow Equation
The heat flow \( Q \) through a rod can be described by Fourier's law of heat conduction, which states:

\[ Q = k \cdot A \cdot \frac{dT}{dL} \]

where:
- \( Q \) is the heat flow per unit time,
- \( k \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area of the rod,
- \( \frac{dT}{dL} \) is the temperature gradient.

### Step 2: Define the Cross-Sectional Area
In this problem, it is mentioned that the heat flow is per unit time per unit cross-section. This implies that we can consider the cross-sectional area \( A \) to be 1 (unit area). Therefore, the equation simplifies to:

\[ Q = k \cdot \frac{dT}{dL} \]

### Step 3: Relate Temperature Gradient to Heat Flow
From the simplified equation, we can express the temperature gradient as:

\[ \frac{dT}{dL} = \frac{Q}{k} \]

### Step 4: Analyze the Given Statements
Now, we need to evaluate the statements provided in the question. The problem states that the heat flow per unit time per unit cross-section of the rod is \( L \). Therefore, we can write:

\[ L = k \cdot \frac{dT}{dL} \]

This implies that:

\[ \frac{dT}{dL} = \frac{L}{k} \]

### Step 5: Check the Validity of Statements
1. **Statement A**: If it states that \( L \) is proportional to \( k \) and \( \frac{dT}{dL} \), then it is correct.
2. **Statement B**: If it states that \( \frac{dT}{dL} \) is constant, then it is also correct, as the rod is uniform and the temperature gradient remains the same throughout its length.
3. **Statement C**: If it states that \( L = k \cdot \frac{dT}{dL} \), then it is correct based on our earlier derivation.
4. **Statement D**: If it states that \( \frac{dT}{dL} \) is the same for all points on the rod, it is also correct due to the uniformity of the rod.

### Conclusion
Based on the analysis, the correct statements are those that align with the derived equations and the properties of the uniform rod.

### Final Answer
The correct statements are A, C, and D.