A spherical black body of radius n radiates power p and its rate of cooling is R. then.

Options

A. `p prop n`
B. `p prop n^(2)`
C. `R prop n^(2)`
D. `R prop (1)/(n)`

Answer

To solve the problem, we need to analyze the relationships involving the power radiated by a spherical black body and its rate of cooling.

### Step-by-Step Solution:

1. **Understanding the Power Radiated**:
The power radiated by a black body is given by the Stefan-Boltzmann Law:
\[ P = \sigma A (T^4 - T_0^4) \]
where \( P \) is the power, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area of the body, \( T \) is the temperature of the body, and \( T_0 \) is the temperature of the surroundings.

2. **Calculating the Surface Area**:
The surface area \( A \) of a sphere with radius \( n \) is:
\[ A = 4\pi n^2 \]

3. **Substituting the Area into the Power Equation**:
Substituting the expression for \( A \) into the power equation gives:
\[ P = \sigma (4\pi n^2) (T^4 - T_0^4) \]
This can be simplified to:
\[ P = 4\pi \sigma n^2 (T^4 - T_0^4) \]

4. **Finding the Relationship of Power with Radius**:
If we consider \( T \) and \( T_0 \) as constants, we can conclude that:
\[ P \propto n^2 \]
This means that the power radiated is proportional to the square of the radius of the sphere.

5. **Understanding the Rate of Cooling**:
The rate of cooling \( R \) can be expressed as:
\[ R = \frac{dT}{dt} = \frac{P}{m c} \]
where \( m \) is the mass of the body and \( c \) is the specific heat capacity.

6. **Calculating the Mass**:
The mass \( m \) of the spherical body can be expressed as:
\[ m = \rho V = \rho \left(\frac{4}{3} \pi n^3\right) \]
where \( \rho \) is the density of the material.

7. **Substituting Mass into the Rate of Cooling**:
Substituting \( m \) into the rate of cooling equation gives:
\[ R = \frac{P}{\rho \left(\frac{4}{3} \pi n^3\right) c} \]

8. **Substituting Power into the Rate of Cooling**:
We can substitute the expression for \( P \) into the rate of cooling:
\[ R = \frac{4\pi \sigma n^2 (T^4 - T_0^4)}{\rho \left(\frac{4}{3} \pi n^3\right) c} \]
Simplifying this expression leads to:
\[ R = \frac{3\sigma (T^4 - T_0^4)}{\rho c n} \]

9. **Finding the Relationship of Rate of Cooling with Radius**:
From the above equation, we can see that:
\[ R \propto \frac{1}{n} \]
This indicates that the rate of cooling is inversely proportional to the radius of the sphere.

### Conclusion:
- The power radiated \( P \) is proportional to \( n^2 \).
- The rate of cooling \( R \) is inversely proportional to \( n \).