A spherical solid blakc body of radius 'r' radiates power 'H' and its rate of cooling is 'C'. If density is constant then which of the following is/are true.

Options

A. `H prop r` and `c prop r^(2)`
B. `H prop r^(2)` and `c prop (1)/(r )`
C. `H prop r` and `c prop (1)/(r^(2))`
D. `H prop r^(2)` and `c prop r^(2)`

Answer

To solve the problem, we need to analyze the relationships between the given parameters: the radius of the spherical black body (r), the power it radiates (H), and its rate of cooling (C).

### Step-by-Step Solution:

1. **Understanding Power Radiation**:
The power radiated by a black body is given by the Stefan-Boltzmann Law:
\[ H = \sigma \cdot e \cdot A \cdot T^4 \]
where:
- \( H \) is the power radiated,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( e \) is the emissivity (for a black body, \( e = 1 \)),
- \( A \) is the surface area of the sphere,
- \( T \) is the absolute temperature of the body.

2. **Calculating Surface Area**:
The surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
Substituting this into the power equation gives:
\[ H = \sigma \cdot 1 \cdot (4\pi r^2) \cdot T^4 = 4\pi \sigma r^2 T^4 \]
This shows that \( H \) is proportional to \( r^2 \).

3. **Understanding Rate of Cooling**:
The rate of cooling \( C \) can be defined as:
\[ C = \frac{dT}{dt} \]
The power radiated can also be expressed in terms of the mass \( m \) and specific heat \( s \):
\[ H = \frac{dq}{dt} = m \cdot s \cdot \frac{dT}{dt} \]
where \( dq \) is the heat lost.

4. **Relating Mass and Volume**:
The mass \( m \) of the sphere can be expressed as:
\[ m = \rho \cdot V = \rho \cdot \left(\frac{4}{3}\pi r^3\right) \]
Substituting this into the power equation gives:
\[ H = \rho \cdot \left(\frac{4}{3}\pi r^3\right) \cdot s \cdot \frac{dT}{dt} \]

5. **Setting the Equations Equal**:
Equating the two expressions for power \( H \):
\[ 4\pi \sigma r^2 T^4 = \rho \cdot \left(\frac{4}{3}\pi r^3\right) \cdot s \cdot \frac{dT}{dt} \]
Simplifying this equation leads to:
\[ C \propto \frac{H}{\rho \cdot \frac{4}{3}\pi r^3 \cdot s} \]

6. **Finding the Proportionality**:
From the above equation, we can see that the rate of cooling \( C \) is inversely proportional to the radius \( r \):
\[ C \propto \frac{1}{r} \]

### Conclusion:
From the analysis, we conclude that:
- The power \( H \) radiated by the spherical black body is proportional to \( r^2 \).
- The rate of cooling \( C \) is inversely proportional to \( r \).