A uniform steel rod of length 50 cm is insulated on its sides. There is a layer of water of thick ness 0.2 mm at each end of the rod. If the ends of the rod are exposed to ice at `0^(@)C`, and steam at `100^(@)C,` calculate the temperature gradient in the rod. Thermal conductivities of steel and water are `0.11calcm^(-1)s^(-1)``""^(@)C^(-1) and 1.5xx10^(-3)calcm^(-1)s^(-1)``""^(@)C^(-1)`

Explanation

To solve the problem of finding the temperature gradient in a uniform steel rod with water layers at both ends, we can follow these steps:

### Step 1: Understand the Setup
We have a steel rod of length \( L = 50 \, \text{cm} = 0.5 \, \text{m} \) insulated on its sides. There are water layers of thickness \( x = 0.2 \, \text{mm} = 0.0002 \, \text{m} \) at each end. One end of the rod is at \( 0^\circ C \) (ice) and the other end is at \( 100^\circ C \) (steam).

### Step 2: Identify Thermal Conductivities
The thermal conductivities are given as:
- Steel: \( k_s = 0.11 \, \text{cal} \, \text{cm}^{-1} \, \text{s}^{-1} \, ^\circ C^{-1} = 0.11 \times 10^4 \, \text{cal} \, \text{m}^{-1} \, \text{s}^{-1} \, ^\circ C^{-1} \) (since \( 1 \, \text{cm} = 0.01 \, \text{m} \))
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