A metal ball of surface area `200 cm^(2)` and temperature `527^(@)C` is surrounded by a vessel at `27^(@)C` . If the emissivity of the metal is 0.4, then the rate of loss of heat from the ball is `(sigma = 5.67 xx10^(-8)J//m^(2)-s-k^(4))`

Options

A. 108 W
B. 168 W
C. 182 W
D. 192 W

Answer

To find the rate of heat loss from the metal ball, we can use the Stefan-Boltzmann law, which states that the power radiated by a black body per unit area is proportional to the fourth power of the temperature. The formula for the rate of heat loss (P) is given by:

\[ P = \varepsilon \sigma A (T^4 - T_0^4) \]

Where:
- \( P \) = rate of heat loss (W)
- \( \varepsilon \) = emissivity of the surface (dimensionless)
- \( \sigma \) = Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \))
- \( A \) = surface area (m²)
- \( T \) = absolute temperature of the body (K)
- \( T_0 \) = absolute temperature of the surroundings (K)

### Step 1: Convert the temperatures from Celsius to Kelvin
The temperature of the metal ball is given as \( 527^\circ C \) and the surrounding temperature is \( 27^\circ C \).

\[ T = 527 + 273 = 800 \, K \]
\[ T_0 = 27 + 273 = 300 \, K \]

### Step 2: Convert the surface area from cm² to m²
The surface area is given as \( 200 \, cm^2 \).

\[ A = 200 \, cm^2 = 200 \times 10^{-4} \, m^2 = 0.02 \, m^2 \]

### Step 3: Substitute the values into the formula
Now we can substitute the values into the formula for \( P \):

\[ P = \varepsilon \sigma A (T^4 - T_0^4) \]
\[ P = 0.4 \times (5.67 \times 10^{-8}) \times (0.02) \times ((800)^4 - (300)^4) \]

### Step 4: Calculate \( T^4 \) and \( T_0^4 \)
Calculating \( (800)^4 \) and \( (300)^4 \):

\[ (800)^4 = 409600000000 \, K^4 \]
\[ (300)^4 = 8100000000 \, K^4 \]

### Step 5: Find the difference \( T^4 - T_0^4 \)
Now, calculate the difference:

\[ T^4 - T_0^4 = 409600000000 - 8100000000 = 400500000000 \, K^4 \]

### Step 6: Substitute back into the equation for \( P \)
Now substitute this back into the equation for \( P \):

\[ P = 0.4 \times (5.67 \times 10^{-8}) \times (0.02) \times (400500000000) \]

### Step 7: Calculate \( P \)
Calculating \( P \):

\[ P = 0.4 \times 5.67 \times 10^{-8} \times 0.02 \times 400500000000 \]
\[ P = 0.4 \times 5.67 \times 8.01 \times 10^{4} \]
\[ P \approx 182.0 \, W \]

### Final Answer
The rate of loss of heat from the ball is approximately \( 182 \, W \).

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