The amount of heat energy radiated by a metal at temperature T is E. When the temperature is increased to 3T, energy radiated is

Options

A. `81 E`
B. `9 E`
C. `3 E`
D. `27 E`

Answer

To solve the problem, we will use Stefan's Law, which states that the energy radiated by a black body is directly proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Understanding Stefan's Law**:
According to Stefan's Law, the energy radiated (E) by a body is given by the formula:
\[ E \propto T^4 \]
where \(T\) is the absolute temperature of the body.

2. **Initial Condition**:
At temperature \(T\), the energy radiated is given as \(E\). We can express this as:
\[ E = kT^4 \]
where \(k\) is a constant of proportionality.

3. **New Condition**:
Now, we need to find the energy radiated when the temperature is increased to \(3T\). Using Stefan's Law again, we can express the new energy radiated \(E'\) as:
\[ E' = k(3T)^4 \]

4. **Calculating \(E'\)**:
Expanding the expression for \(E'\):
\[ E' = k(3^4)(T^4) = k(81)(T^4) = 81kT^4 \]

5. **Relating \(E'\) to \(E\)**:
Since we know that \(E = kT^4\), we can substitute this into our expression for \(E'\):
\[ E' = 81E \]

6. **Final Result**:
Therefore, the energy radiated when the temperature is increased to \(3T\) is:
\[ E' = 81E \]

### Conclusion:
The amount of heat energy radiated by the metal when the temperature is increased to \(3T\) is \(81E\).

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