The sphere of radii 8 cm and 2 cm are cooling. Their temperatures are `127^(@)C` and `527^(@)C` respectively . Find the ratio of energy radiated by them in the same time

Options

A. 0.06
B. 0.5
C. 1
D. 2

Answer

To find the ratio of energy radiated by two spheres with different radii and temperatures, we can apply Stefan-Boltzmann's 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 data:**
- Radius of sphere 1, \( r_1 = 8 \, \text{cm} \)
- Radius of sphere 2, \( r_2 = 2 \, \text{cm} \)
- Temperature of sphere 1, \( T_1 = 127^\circ C = 127 + 273 = 400 \, K \)
- Temperature of sphere 2, \( T_2 = 527^\circ C = 527 + 273 = 800 \, K \)

2. **Calculate the surface area of both spheres:**
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
- For sphere 1:
\[ A_1 = 4\pi (r_1^2) = 4\pi (8^2) = 4\pi (64) = 256\pi \, \text{cm}^2 \]
- For sphere 2:
\[ A_2 = 4\pi (r_2^2) = 4\pi (2^2) = 4\pi (4) = 16\pi \, \text{cm}^2 \]

3. **Apply Stefan-Boltzmann's law:**
The energy radiated per unit time (power) is given by:
\[ P = \epsilon A \sigma T^4 \]
where \( \epsilon \) is the emissivity (assumed to be the same for both spheres), and \( \sigma \) is the Stefan-Boltzmann constant.

Since we are interested in the ratio of energy radiated by both spheres, we can ignore \( \epsilon \) and \( \sigma \) as they are constants for both spheres.

4. **Write the power equations for both spheres:**
- For sphere 1:
\[ P_1 = A_1 T_1^4 = (256\pi)(400^4) \]
- For sphere 2:
\[ P_2 = A_2 T_2^4 = (16\pi)(800^4) \]

5. **Find the ratio of the powers:**
\[ \frac{P_1}{P_2} = \frac{256\pi (400^4)}{16\pi (800^4)} \]
The \( \pi \) cancels out:
\[ \frac{P_1}{P_2} = \frac{256 (400^4)}{16 (800^4)} = \frac{256}{16} \cdot \frac{(400^4)}{(800^4)} = 16 \cdot \left(\frac{400}{800}\right)^4 \]
\[ = 16 \cdot \left(\frac{1}{2}\right)^4 = 16 \cdot \frac{1}{16} = 1 \]

6. **Conclusion:**
The ratio of energy radiated by the two spheres in the same time is:
\[ \frac{Q_1}{Q_2} = 1 \]

### Final Answer:
The ratio of energy radiated by the two spheres is \( 1:1 \).