A rod of length `L` and uniform cross-sectional area has varying thermal conductivity which changes linearly from `2K` at end `A` to `K` at the other end `B`. The ends `A` and `B` of the rod are maintained at constant temperture `100^(@)C` and `0^(@)C`, respectively. At steady state, the graph of temperture : `T=T(x)` where `x=` distance from end `A` will be

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the heat transfer through a rod with varying thermal conductivity. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a rod of length `L` with thermal conductivity that varies linearly from `2K` at one end (A) to `K` at the other end (B). The temperatures at the ends are maintained at `100°C` (at A) and `0°C` (at B). We need to find the temperature distribution `T(x)` along the length of the rod.

### Step 2: Define the Thermal Conductivity as a Function of Position
Since the thermal conductivity changes linearly, we can express it as:
\[ K(x) = K_A + \left( \frac{K_B - K_A}{L} \right) x \]
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