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

Explanation

To determine when the temperature of the whole rod will be uniform upon heating one end, we can follow these steps:

### Step 1: Understand the concept of thermal conductivity
Thermal conductivity (K) is a measure of a material's ability to conduct heat. For a rod to have a uniform temperature throughout, it must conduct heat perfectly.

**Hint:** Recall that thermal conductivity is a property of materials that indicates how well they can transfer heat.

### Step 2: Define the conditions for uniform temperature
For the temperature to be uniform across the rod, the heat supplied must be distributed evenly throughout the rod. This means that the temperature difference (dT) between any two points in the rod must be zero.

**Hint:** Think about what it means for temperature to be uniform. If dT = 0, then every point in the rod is at the same temperature.

### Step 3: Apply the heat conduction equation
The heat conduction equation is given by:
\[ Q = K \cdot A \cdot \frac{dT}{dx} \]
where:
- \( Q \) is the heat transferred,
- \( K \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \frac{dT}{dx} \) is the temperature gradient.

For uniform temperature, \( dT = 0 \), which leads to:
\[ Q = K \cdot A \cdot 0 \]
This means that no heat is lost or gained across the rod.

**Hint:** Consider what happens to the equation when dT approaches zero.

### Step 4: Analyze the implications of dT = 0
If \( dT = 0 \), then the equation implies that the thermal conductivity \( K \) must be infinite for the heat transfer to occur without any temperature difference. This is a theoretical scenario, as no real material can achieve infinite thermal conductivity.

**Hint:** Think about the physical meaning of infinite thermal conductivity. What would that imply about the material?

### Step 5: Conclusion
Thus, for the temperature of the whole rod to be uniform when one end is heated, the thermal conductivity of the rod must be infinite. In practical terms, this means that the rod would need to be an ideal conductor of heat.

**Hint:** Reflect on the options given in the question and select the one that corresponds to this conclusion.

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