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 of radiation, which states that the amount of heat energy radiated by a black body is directly proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Understand the relationship from Stefan's Law**:
According to Stefan's Law, the energy radiated \( E \) by a body is given by:
\[ E = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the absolute temperature.

2. **Initial Condition**:
At the initial temperature \( T \), the energy radiated is given as \( E \). Therefore, we can express it as:
\[ E = K T^4 \]
where \( K = \sigma A \) is a constant.

3. **New Condition**:
When the temperature is increased to \( 3T \), we need to find the new energy radiated, which we will denote as \( E' \). Using the same formula, we can express \( E' \) as:
\[ E' = K (3T)^4 \]

4. **Calculate \( (3T)^4 \)**:
\[ (3T)^4 = 3^4 T^4 = 81 T^4 \]

5. **Substituting back into the equation for \( E' \)**:
\[ E' = K \cdot 81 T^4 \]

6. **Relate \( E' \) to \( E \)**:
Since we know \( E = K T^4 \), we can substitute this into our equation for \( E' \):
\[ E' = 81 E \]

### Final Answer:
The energy radiated when the temperature is increased to \( 3T \) is:
\[ E' = 81E \]