The ends of a metal rod are kept at temperatures `theta_1` and `theta_2` with `theta_2gttheta_1`. The rate of flow of heat along the rod is directly proportional to

Options

A. the length of the rod
B. the diameter of the rod
C. the cross sectional area of the rod
D. the temperature difference `(theta_2-theta_1)` between the ends of the rod

Explanation

To solve the problem, we need to analyze the relationship between the rate of heat flow along a metal rod and various factors such as temperature difference, length of the rod, and cross-sectional area.

### Step-by-Step Solution:

1. **Understanding Heat Flow**:
Heat flows from a region of higher temperature to a region of lower temperature. In this case, the ends of the rod are at temperatures \( \theta_2 \) and \( \theta_1 \) with \( \theta_2 > \theta_1 \).

2. **Using Fourier's Law of Heat Conduction**:
According to Fourier's law, the rate of heat flow \( \frac{Q}{T} \) (where \( Q \) is the heat transferred and \( T \) is the time) is given by the formula:
\[ \frac{Q}{T} = k \cdot A \cdot \frac{\theta_2 - \theta_1}{L} \]
where:
- \( k \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area of the rod,
- \( \theta_2 - \theta_1 \) is the temperature difference,
- \( L \) is the length of the rod.

3. **Identifying Proportional Relationships**:
- From the equation, we can see that the rate of heat flow \( \frac{Q}{T} \) is **directly proportional** to:
- The temperature difference \( \theta_2 - \theta_1 \) (Option 4).
- The cross-sectional area \( A \) of the rod (Option 3).
- The rate of heat flow is **inversely proportional** to the length \( L \) of the rod (Option 1), meaning that as the length increases, the rate of heat flow decreases.

4. **Analyzing the Options**:
- **Option 1**: Length of the rod - **Incorrect** (inversely proportional).
- **Option 2**: Diameter of the rod - **Incorrect** (directly proportional to the area, not diameter).
- **Option 3**: Cross-sectional area of the rod - **Correct** (directly proportional).
- **Option 4**: Temperature difference \( \theta_2 - \theta_1 \) - **Correct** (directly proportional).

5. **Conclusion**:
The rate of flow of heat along the rod is directly proportional to the cross-sectional area of the rod and the temperature difference between the ends of the rod. Therefore, the correct answers are **Option 3** and **Option 4**.