On heating one end of a rod the temperature of the whole rod will be uniform when .

Options

A. K = 1
B. K = 0
C. K = 100
D. `K = oo`

Answer

To solve the problem of when the temperature of a rod will become uniform upon heating one end, we can break down the explanation into clear steps:

### Step-by-Step Solution:

1. **Understanding Heat Transfer**:
When one end of a rod is heated, heat begins to transfer from the hot end to the cooler end. The rate of heat transfer depends on the thermal conductivity of the material.

2. **Thermal Conductivity (K)**:
The thermal conductivity (K) of a material is a measure of its ability to conduct heat. Higher values of K indicate that the material can transfer heat more efficiently.

3. **Temperature Gradient**:
The temperature gradient (dθ/dx) is the change in temperature (dθ) per unit length (dx) along the rod. If the temperature gradient is steep, it means there is a significant difference in temperature over a short distance, leading to a rapid heat transfer.

4. **Condition for Uniform Temperature**:
For the entire rod to reach a uniform temperature, the heat must be conducted through the rod without any significant temperature difference remaining. This means that the temperature gradient must approach zero (dθ/dx = 0).

5. **Relation Between Heat Transfer and Thermal Conductivity**:
The rate of heat transfer (Q) can be expressed as:
\[ Q = K \cdot A \cdot \frac{dθ}{dx} \]
where A is the cross-sectional area. For the temperature to become uniform, we need to minimize the temperature gradient, which implies that the thermal conductivity (K) must be very high.

6. **Conclusion**:
If the thermal conductivity (K) approaches infinity, the heat transfer will occur instantaneously, leading to a uniform temperature throughout the rod. Thus, the condition for the temperature of the whole rod to be uniform is that the thermal conductivity must be infinite.

### Final Answer:
The temperature of the whole rod will be uniform when the thermal conductivity (K) is infinite.

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