The ends of a rod of uniform thermal conductivity are maintained at different (constant) temperatures. Afer the steady state is achieved:

Options

A. heat flows in the rod from high temperature to low temperature even if the rod has the non-unform cross sectional area
B. temperature gradient along length is same even if the rod has the non-unform cross sectional area
C. heat current is same even if the rod hasnon-unform cross sectional area
D. If the rod has uniform cross-sectional area the temperature is same all the point at rod.

Answer

To solve the problem regarding the heat conduction in a rod of uniform thermal conductivity maintained at different constant temperatures, we can break it down into the following steps:

### Step 1: Understand the Setup
We have a rod with two ends maintained at different constant temperatures, \( T_H \) (hot end) and \( T_C \) (cold end). The rod has uniform thermal conductivity \( k \).

**Hint:** Identify the temperatures at both ends of the rod and recognize that steady state means a constant temperature distribution along the rod.

### Step 2: Define Steady State
In steady state, the temperature at each point along the rod does not change over time. This means that while heat is flowing from the hot end to the cold end, the temperature at the hot end remains \( T_H \) and the temperature at the cold end remains \( T_C \).

**Hint:** Remember that steady state implies a constant temperature distribution, not thermal equilibrium.

### Step 3: Apply Fourier's Law of Heat Conduction
According to Fourier's law, the heat transfer \( Q \) through the rod can be expressed as:
\[ Q = k \cdot A \cdot \frac{T_H - T_C}{L} \]
where \( A \) is the cross-sectional area of the rod, and \( L \) is the length of the rod.

**Hint:** Recognize that heat flows from the higher temperature to the lower temperature.

### Step 4: Analyze Heat Flow
Heat will always flow from the hot end to the cold end, regardless of whether the rod has a uniform or non-uniform cross-sectional area. Therefore, the first statement that heat flows from high temperature to low temperature is correct.

**Hint:** Focus on the direction of heat flow and the implications of temperature differences.

### Step 5: Temperature Gradient
The temperature gradient \( \frac{dT}{dx} \) is defined as:
\[ \frac{dT}{dx} = \frac{T_H - T_C}{L} \]
If the rod has a non-uniform cross-sectional area, the temperature gradient will not remain constant because it is inversely proportional to the cross-sectional area. Thus, the second statement is incorrect.

**Hint:** Consider how changes in cross-sectional area affect the temperature gradient.

### Step 6: Heat Current Consistency
The heat current (rate of heat transfer) remains constant throughout the rod in steady state, even if the cross-sectional area varies. This is due to the conservation of energy, which states that the amount of heat entering one end of the rod must equal the amount of heat leaving the other end.

**Hint:** Think about the principle of conservation of energy in the context of heat flow.

### Step 7: Temperature Uniformity
At steady state, the temperature will not be the same at all points along the rod if there is heat flow. Therefore, the fourth statement is incorrect.

**Hint:** Remember that heat flow implies a temperature gradient, which means temperatures will vary along the length of the rod.

### Conclusion
Based on the analysis:
- The correct options are 1 (heat flows from high to low temperature) and 3 (heat current is the same throughout the rod).
- The incorrect options are 2 (temperature gradient remains the same) and 4 (temperature is the same at all points).

**Final Answer:** The correct options are 1 and 3.

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