If the radius of a star is `R` and it acts as a black body, what would b the temperature of the star, in which the rate of energy production is `Q`?

Options

A. a. `(4pi R^(2)Q//sigma)^(1//4)`
B. b. `(Q//4piR^(2)sigma)^(1//4)`
C. c. `Q//4piR^(2)sigma`
D. d. `(Q//4piR^(2)sigma)^(-1//2)`

Explanation

To find the temperature of a star that acts as a black body with a given radius \( R \) and a rate of energy production \( Q \), we can use Stefan-Boltzmann's law. Here’s a step-by-step solution:

### Step 1: Understand the relationship given by Stefan-Boltzmann's Law
Stefan-Boltzmann's law states that the total energy radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature. Mathematically, it is expressed as:
\[ E = \sigma T^4 \]
where:
- \( E \) is the energy emitted per unit area,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( T \) is the absolute temperature of the black body.

### Step 2: Relate the rate of energy production to the total energy emitted
The total rate of energy production \( Q \) of the star can be expressed as:
\[ Q = E \cdot A \]
where \( A \) is the surface area of the star.

### Step 3: Calculate the surface area of the star
The surface area \( A \) of a sphere (the shape of the star) is given by:
\[ A = 4\pi R^2 \]

### Step 4: Substitute the surface area into the energy equation
Substituting the expression for \( A \) into the equation for \( Q \):
\[ Q = E \cdot 4\pi R^2 \]

### Step 5: Substitute \( E \) from Stefan-Boltzmann's law
From Stefan-Boltzmann's law, we can substitute \( E \):
\[ Q = \sigma T^4 \cdot 4\pi R^2 \]

### Step 6: Solve for temperature \( T \)
Rearranging the equation to solve for \( T^4 \):
\[ T^4 = \frac{Q}{4\pi R^2 \sigma} \]
Now, taking the fourth root to find \( T \):
\[ T = \left(\frac{Q}{4\pi R^2 \sigma}\right)^{1/4} \]

### Final Answer
Thus, the temperature \( T \) of the star is given by:
\[ T = \left(\frac{Q}{4\pi R^2 \sigma}\right)^{1/4} \]