One end of a rod of length `20cm` is inserted in a furnace at `800K` The sides of the rod are covered with an insulating material and the other end emits radiation like a black body. The temperature of this end is `750K` in the steady state The temperature of the surrounding air is `300K` Assuming radiation to be the only important mode of energy transfer between the surrounding and the open end of the rod, find the thermal conductivity of the rod Stefan's constant `sigma = 6.0 xx 10^(-8) W m^(-2) K^(-4)` .

Explanation

To find the thermal conductivity of the rod, we can follow these steps:

### Step 1: Understand the Problem
We have a rod of length \( L = 20 \, \text{cm} = 0.2 \, \text{m} \) with one end in a furnace at \( T = 800 \, \text{K} \) and the other end at \( T_1 = 750 \, \text{K} \). The surrounding air temperature is \( T_0 = 300 \, \text{K} \). The rod is insulated on the sides, and the end emits radiation like a black body.

### Step 2: Write the Heat Conduction Equation
The heat conducted through the rod can be expressed as:
\[
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