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)`

Options

A. `20^(@)C`
B. `10^(@)C`
C. `30^(@)C`
D. `12^(@)C`

Answer

To solve the problem of calculating the temperature difference between two sides of an iron plate, we will use the formula for heat conduction. Let's break down the steps:

### Step 1: Understand the Given Data
- Thickness of the iron plate (x) = 20 mm = 2 cm (conversion from mm to cm)
- Rate of heat conduction (q) = \(6 \times 10^5\) cal/min/m²
- Conductivity of iron (k) = 0.2 cal/s·cm·°C
- Area (A) = 1 m² = \(10^4\) cm² (conversion from m² to cm²)
- Time (t) = 1 min = 60 s (conversion from minutes to seconds)

### Step 2: Write the Formula for Heat Conduction
The formula for heat conduction is given by:
\[ q = k \cdot A \cdot \frac{(T_1 - T_2)}{x \cdot t} \]
Where:
- \(T_1 - T_2\) is the temperature difference we want to find.

### Step 3: Rearrange the Formula to Find \(T_1 - T_2\)
Rearranging the formula to isolate \(T_1 - T_2\):
\[ T_1 - T_2 = \frac{q \cdot x \cdot t}{k \cdot A} \]

### Step 4: Substitute the Known Values
Now, substitute the known values into the rearranged formula:
- \(q = 6 \times 10^5\) cal/min/m² = \(6 \times 10^5\) cal/min = \(6 \times 10^5 \div 60\) cal/s = \(1 \times 10^4\) cal/s (conversion from min to s)
- \(x = 2\) cm
- \(t = 60\) s
- \(k = 0.2\) cal/s·cm·°C
- \(A = 10^4\) cm²

Substituting these values:
\[ T_1 - T_2 = \frac{(1 \times 10^4) \cdot (2) \cdot (60)}{0.2 \cdot (10^4)} \]

### Step 5: Simplify the Expression
Calculating the numerator:
\[ 1 \times 10^4 \cdot 2 \cdot 60 = 1.2 \times 10^6 \]
Now, calculate the denominator:
\[ 0.2 \cdot 10^4 = 2 \times 10^3 \]

Now substituting back:
\[ T_1 - T_2 = \frac{1.2 \times 10^6}{2 \times 10^3} = \frac{1.2 \times 10^6}{2 \times 10^3} = 600 \]

### Step 6: Final Calculation
Thus, the temperature difference is:
\[ T_1 - T_2 = 600 \text{ °C} \]

### Conclusion
The difference in temperature between the two sides of the iron plate is **600 °C**.

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