A planet is at an average distance d from the sun and its average surface temeperature is T. Assume that the planet receives energy only from the sun and loses energy only through radiation from the surface. Neglect atmospheric effects. If `Tpropd^(-n)`, the value of n is

Options

A. 2
B. 1
C. `(1)/(2)`
D. `(1)/(4)`

Explanation

To solve the problem, we need to analyze the energy balance between the energy received by the planet from the Sun and the energy radiated by the planet itself.

### Step-by-Step Solution:

1. **Understanding Energy Reception**:
The power radiated by the Sun can be denoted as \( P \). The planet receives this energy over a certain area. The area through which the energy from the Sun reaches the planet is given by the formula for the surface area of a sphere, which is \( 4\pi d^2 \), where \( d \) is the distance from the Sun to the planet.

The rate of energy received by the planet can be expressed as:
\[ \text{Rate of energy received} = \frac{P}{4\pi d^2} \]

2. **Understanding Energy Radiation**:
The planet radiates energy according to the Stefan-Boltzmann law, which states that the power radiated per unit area of a black body is proportional to the fourth power of its temperature \( T \). The total power radiated by the planet can be expressed as:
\[ \text{Rate of energy radiated} = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant and \( A \) is the surface area of the planet, which can be approximated as \( 4\pi r^2 \) (where \( r \) is the radius of the planet).

Therefore, the total power radiated by the planet is:
\[ \text{Rate of energy radiated} = \sigma (4\pi r^2) T^4 \]

3. **Setting Up the Thermal Equilibrium**:
At thermal equilibrium, the rate of energy received by the planet equals the rate of energy radiated by the planet:
\[ \frac{P}{4\pi d^2} = \sigma (4\pi r^2) T^4 \]

4. **Simplifying the Equation**:
We can cancel \( 4\pi \) from both sides:
\[ \frac{P}{d^2} = 4\sigma r^2 T^4 \]

5. **Rearranging for Temperature**:
Rearranging the equation to express \( T^4 \):
\[ T^4 = \frac{P}{4\sigma r^2 d^2} \]

6. **Identifying Proportional Relationships**:
Since \( P \), \( \sigma \), and \( r \) are constants, we can express \( T^4 \) in terms of \( d^2 \):
\[ T^4 \propto \frac{1}{d^2} \]

Taking the fourth root gives:
\[ T \propto d^{-1/2} \]

7. **Conclusion**:
From the relationship \( T \propto d^{-n} \), we can identify that \( n = \frac{1}{2} \).

### Final Answer:
The value of \( n \) is \( \frac{1}{2} \).