The original temperature of a black body is . `727^(@) C` . The temperature at which this black body must be raised so as to double the total radiant energy, is

Options

A. 971 K
B. 1190 K
C. 2001 K
D. 1458 K

Answer

To solve the problem of determining the temperature at which a black body must be raised to double its total radiant energy, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know the original temperature of the black body is \(727^\circ C\). We need to find the new temperature \(T\) at which the total radiant energy is doubled.

2. **Convert Celsius to Kelvin**:
The temperature in Kelvin is given by:
\[ T(K) = T(°C) + 273 \]
So, for \(727^\circ C\):
\[ T_1 = 727 + 273 = 1000 \, K \]

3. **Use Stefan-Boltzmann Law**:
According to Stefan-Boltzmann law, the energy emitted per unit area per unit time \(E\) is proportional to the fourth power of the absolute temperature \(T\):
\[ E \propto T^4 \]
If \(E_1\) is the energy at \(T_1\), then:
\[ E_1 = \sigma A T_1^4 \]
where \(\sigma\) is the Stefan-Boltzmann constant and \(A\) is the area.

4. **Set Up the Equation for Doubled Energy**:
We want to find the temperature \(T\) such that the energy \(E_2\) is double that of \(E_1\):
\[ E_2 = 2E_1 = \sigma A T^4 \]

5. **Relate the Energies**:
Setting the equations for \(E_1\) and \(E_2\) gives:
\[ 2E_1 = \sigma A T^4 \implies 2(\sigma A T_1^4) = \sigma A T^4 \]
Dividing both sides by \(\sigma A\):
\[ 2T_1^4 = T^4 \]

6. **Substitute \(T_1\)**:
Substitute \(T_1 = 1000 \, K\):
\[ 2(1000^4) = T^4 \]

7. **Solve for \(T\)**:
Taking the fourth root of both sides:
\[ T = (2 \cdot 1000^4)^{1/4} = 1000 \cdot (2^{1/4}) \]
Calculating \(2^{1/4}\):
\[ 2^{1/4} \approx 1.189207 \]
Therefore:
\[ T \approx 1000 \cdot 1.189207 \approx 1189.207 \, K \]

8. **Final Answer**:
Rounding gives us:
\[ T \approx 1190 \, K \]

### Conclusion:
The temperature at which the black body must be raised to double the total radiant energy is approximately **1190 K**.