Two identical metal balls at temperature `200^(@)C` and `400^(@)C` kept in air at `27^(@)C`. The ratio of net heat loss by these bodies is

Options

A. `1//4`
B. `1//2`
C. `1//16`
D. `(473^(4) - 300^(4))/(673^(4) - 300^(4))`

Explanation

To solve the problem of finding the ratio of net heat loss by two identical metal balls at different temperatures, we will use Stefan-Boltzmann law for heat radiation. Here’s a step-by-step solution:

### Step 1: Understand the Concept
The heat loss by radiation from a body is given by the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. The formula for the rate of heat loss (ΔE) can be expressed as:
\[ \Delta E = \sigma \cdot A \cdot (T^4 - T_s^4) \]
where:
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area of the body,
- \( T \) is the absolute temperature of the body,
- \( T_s \) is the absolute temperature of the surroundings.

### Step 2: Convert Temperatures to Kelvin
We need to convert the given temperatures from Celsius to Kelvin:
- For the first ball at \( 200^\circ C \):
\[ T_1 = 200 + 273 = 473 \, K \]
- For the second ball at \( 400^\circ C \):
\[ T_2 = 400 + 273 = 673 \, K \]
- The surrounding temperature \( T_s \) at \( 27^\circ C \):
\[ T_s = 27 + 273 = 300 \, K \]

### Step 3: Write the Heat Loss Equations
The heat loss for each ball can be expressed as:
- For the first ball:
\[ \Delta E_1 = \sigma \cdot A \cdot (T_1^4 - T_s^4) = \sigma \cdot A \cdot (473^4 - 300^4) \]
- For the second ball:
\[ \Delta E_2 = \sigma \cdot A \cdot (T_2^4 - T_s^4) = \sigma \cdot A \cdot (673^4 - 300^4) \]

### Step 4: Find the Ratio of Heat Losses
Now we can find the ratio of the net heat loss by the two bodies:
\[ \frac{\Delta E_1}{\Delta E_2} = \frac{T_1^4 - T_s^4}{T_2^4 - T_s^4} \]
Substituting the values:
\[ \frac{\Delta E_1}{\Delta E_2} = \frac{473^4 - 300^4}{673^4 - 300^4} \]

### Step 5: Calculate the Values
Now we need to calculate \( 473^4 \), \( 300^4 \), and \( 673^4 \):
- \( 473^4 = 49787170321 \)
- \( 300^4 = 8100000000 \)
- \( 673^4 = 201511210176 \)

Now substituting these values into the ratio:
\[ \frac{\Delta E_1}{\Delta E_2} = \frac{49787170321 - 8100000000}{201511210176 - 8100000000} \]
Calculating the differences:
- \( 49787170321 - 8100000000 = 41687170321 \)
- \( 201511210176 - 8100000000 = 193411210176 \)

So, the ratio becomes:
\[ \frac{\Delta E_1}{\Delta E_2} = \frac{41687170321}{193411210176} \]

### Step 6: Simplify the Ratio
Calculating the final ratio gives us:
\[ \frac{\Delta E_1}{\Delta E_2} \approx 0.215 \]

### Conclusion
Thus, the ratio of net heat loss by the two bodies is approximately \( 0.215 \).