The ends of a metal rod are kept at temperature `theta_(1) and theta_(2)` with `theta_(2) gt theta_(1)`. At steady state 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

Answer

To solve the problem, we need to analyze the heat transfer through a metal rod that has its ends maintained at different temperatures. The question asks what the rate of heat flow is directly proportional to at steady state.

### Step-by-Step Solution:

1. **Understand the Concept of Heat Transfer**:
The heat transfer through a solid rod can be described by Fourier's law of heat conduction. According to this law, the rate of heat transfer (dQ/dt) through a material is proportional to the temperature difference across the material.

2. **Identify the Variables**:
- Let \( \theta_1 \) be the temperature at one end of the rod.
- Let \( \theta_2 \) be the temperature at the other end of the rod, where \( \theta_2 > \theta_1 \).
- The temperature difference \( \Delta T \) can be expressed as \( \Delta T = \theta_2 - \theta_1 \).
- Let \( A \) be the cross-sectional area of the rod.
- Let \( L \) be the length of the rod.
- Let \( k \) be the thermal conductivity of the material.

3. **Apply Fourier's Law**:
According to Fourier's law, the rate of heat transfer \( \frac{dQ}{dt} \) is given by:
\[ \frac{dQ}{dt} = k \cdot A \cdot \frac{\Delta T}{L} \]
Here, \( \frac{dQ}{dt} \) is the rate of heat flow, \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, \( \Delta T \) is the temperature difference, and \( L \) is the length of the rod.

4. **Determine Proportional Relationships**:
From the equation, we can see that:
- The rate of heat flow \( \frac{dQ}{dt} \) is directly proportional to the cross-sectional area \( A \).
- The rate of heat flow \( \frac{dQ}{dt} \) is also directly proportional to the temperature difference \( \Delta T \).
- The rate of heat flow \( \frac{dQ}{dt} \) is inversely proportional to the length \( L \) of the rod.

5. **Conclusion**:
Therefore, at steady state, the rate of flow of heat along the rod is directly proportional to:
- The cross-sectional area of the rod \( A \)
- The temperature difference \( \Delta T = \theta_2 - \theta_1 \)

### Final Answer:
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.