Calculate the difference in temperature between two sides of an iron plate 20 m m thick, when heat is conducted at the rate of `6xx10^(5)` cal/min/`m^(2)`. K for metal is 0.2 cal`s^(-1)cm^(-1).^(@)C^(-1)`

Answer

To solve the problem, we will use the formula for heat conduction, which is given by Fourier's law:

\[ Q = \frac{K \cdot A \cdot (T_1 - T_2)}{X} \]

Where:
- \( Q \) = rate of heat transfer (cal/min)
- \( K \) = thermal conductivity (cal/s·cm·°C)
- \( A \) = area (cm²)
- \( T_1 - T_2 \) = temperature difference (°C)
- \( X \) = thickness of the material (cm)

### Step 1: Convert the given values to the appropriate units
- Thickness \( X = 20 \, \text{mm} = 2 \, \text{cm} \)
- Rate of heat transfer \( Q = 6 \times 10^5 \, \text{cal/min} \)
- Area \( A = 1 \, \text{m}^2 = 10^4 \, \text{cm}^2 \)
- Thermal conductivity \( K = 0.2 \, \text{cal/s·cm·°C} \)
- Convert \( Q \) to cal/s:
\[ Q = \frac{6 \times 10^5 \, \text{cal/min}}{60 \, \text{s/min}} = 1 \times 10^4 \, \text{cal/s} \]

### Step 2: Rearrange the formula to find the temperature difference
We need to find \( T_1 - T_2 \):
\[ T_1 - T_2 = \frac{Q \cdot X}{K \cdot A} \]

### Step 3: Substitute the values into the equation
Substituting the known values:
\[ T_1 - T_2 = \frac{(1 \times 10^4 \, \text{cal/s}) \cdot (2 \, \text{cm})}{(0.2 \, \text{cal/s·cm·°C}) \cdot (10^4 \, \text{cm}^2)} \]

### Step 4: Calculate the temperature difference
Calculating the numerator:
\[ 1 \times 10^4 \cdot 2 = 2 \times 10^4 \]
Calculating the denominator:
\[ 0.2 \cdot 10^4 = 2 \times 10^3 \]
Now substituting back:
\[ T_1 - T_2 = \frac{2 \times 10^4}{2 \times 10^3} = 10 \, \text{°C} \]

### Final Answer
The difference in temperature between the two sides of the iron plate is \( 10 \, \text{°C} \).

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