Two spheres of the same material have radii 1m and 4m and temperatures 4000K and 2000K respectively. The ratio of the energy radiated per second by the first sphere to that by the second is

Options

A. 4 : 1
B. 1 : 1
C. 0.042361111111111
D. 0.044444444444444

Explanation

To solve the problem, we will use the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature and its surface area.

### Step-by-Step Solution:

1. **Identify the given parameters:**
- Radius of the first sphere, \( r_1 = 1 \, \text{m} \)
- Radius of the second sphere, \( r_2 = 4 \, \text{m} \)
- Temperature of the first sphere, \( T_1 = 4000 \, \text{K} \)
- Temperature of the second sphere, \( T_2 = 2000 \, \text{K} \)

2. **Recall the formula for power radiated:**
The power radiated by a sphere is given by:
\[ P = E \sigma A T^4 \]
where \( A \) is the surface area of the sphere, \( E \) is the emissivity (which is the same for both spheres since they are of the same material), and \( \sigma \) is the Stefan-Boltzmann constant.

3. **Calculate the surface area of each sphere:**
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Therefore, for the first sphere:
\[ A_1 = 4 \pi (r_1^2) = 4 \pi (1^2) = 4 \pi \]
For the second sphere:
\[ A_2 = 4 \pi (r_2^2) = 4 \pi (4^2) = 4 \pi (16) = 64 \pi \]

4. **Write the expressions for power radiated by each sphere:**
For the first sphere:
\[ P_1 = E \sigma (4 \pi) (4000^4) \]
For the second sphere:
\[ P_2 = E \sigma (64 \pi) (2000^4) \]

5. **Set up the ratio of the powers:**
The ratio of the power radiated by the first sphere to that by the second sphere is:
\[ \frac{P_1}{P_2} = \frac{E \sigma (4 \pi) (4000^4)}{E \sigma (64 \pi) (2000^4)} \]
The \( E \), \( \sigma \), and \( \pi \) terms cancel out:
\[ \frac{P_1}{P_2} = \frac{4 (4000^4)}{64 (2000^4)} = \frac{4}{64} \cdot \frac{4000^4}{2000^4} \]

6. **Simplify the ratio:**
\[ \frac{P_1}{P_2} = \frac{1}{16} \cdot \left( \frac{4000}{2000} \right)^4 = \frac{1}{16} \cdot (2)^4 = \frac{1}{16} \cdot 16 = 1 \]

7. **Conclusion:**
The ratio of the energy radiated per second by the first sphere to that by the second sphere is:
\[ \frac{P_1}{P_2} = 1 \]

### Final Answer:
The ratio of the energy radiated per second by the first sphere to that by the second is \( 1:1 \).