A ball of surface temperature T is in thermal equilibrium with its environment. Which of the curve gives the radiating power E radiated by the sphere as a function of time 't' ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the relationship between the radiating power \( E \) of a sphere and its surface temperature \( T \) in thermal equilibrium with its environment.

### Step-by-Step Solution:

1. **Understanding Thermal Equilibrium**:
- When an object is in thermal equilibrium with its environment, it means that the temperature of the object remains constant over time. In this case, the ball has a surface temperature \( T \).

2. **Applying Stefan-Boltzmann Law**:
- The power radiated by a black body (or a sphere in this case) is given by the Stefan-Boltzmann law:
\[ E = \sigma A T^4 \]
where:
- \( E \) is the radiating power,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area of the sphere,
- \( T \) is the absolute temperature of the sphere.

3. **Identifying the Relationship**:
- Since the ball is in thermal equilibrium, the temperature \( T \) does not change over time. Therefore, the radiating power \( E \) remains constant as well.

4. **Graph Analysis**:
- We need to determine how \( E \) behaves as a function of time \( t \). Since \( E \) is constant (it does not change with time), the graph of \( E \) versus \( t \) will be a horizontal line.

5. **Selecting the Correct Curve**:
- Among the given options (curves), the only curve that represents a constant value of \( E \) over time \( t \) is a horizontal line. This corresponds to option A.

### Conclusion:
The correct answer is that the radiating power \( E \) remains constant over time \( t \), and thus the curve that represents this relationship is a horizontal line. Therefore, the correct option is **A**.