Two identical solid spheres have the same temperature.One of the spheres is cut into two identical pieces.These two hemispheres are then separated. The intact sphere radiates an energy Q during a given time interval. During the same interval, the two hemispheres radiate a total energy Q'. What is the ratio `Q' // Q `?

Options

A. `2.0`
B. `4.0`
C. `0.50`
D. `1.5`

Answer

To solve the problem, we need to analyze the energy radiated by the intact sphere and the two hemispheres using Stefan-Boltzmann's Law of Radiation.

### Step 1: Understand the energy radiated by the intact sphere
According to Stefan-Boltzmann's Law, the energy radiated by a black body is given by the formula:
\[ Q = \sigma A T^4 \]
where:
- \( Q \) is the energy radiated,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T \) is the absolute temperature.

For the intact sphere, the surface area \( A \) is:
\[ A = 4\pi r^2 \]
Thus, the energy radiated \( Q \) by the intact sphere is:
\[ Q = \sigma (4\pi r^2) T^4 \]

### Step 2: Calculate the energy radiated by the two hemispheres
When the sphere is cut into two hemispheres, we need to calculate the total surface area of the two hemispheres. Each hemisphere has a curved surface area and a flat circular base.

The curved surface area of one hemisphere is:
\[ A_{curved} = 2\pi r^2 \]
The flat circular area is:
\[ A_{flat} = \pi r^2 \]
Thus, the total surface area \( A' \) of one hemisphere is:
\[ A' = A_{curved} + A_{flat} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \]

Since there are two hemispheres, the total surface area for both is:
\[ A_{total} = 2 \times (3\pi r^2) = 6\pi r^2 \]

Using Stefan-Boltzmann's Law, the total energy radiated \( Q' \) by the two hemispheres is:
\[ Q' = \sigma (6\pi r^2) T^4 \]

### Step 3: Find the ratio \( \frac{Q'}{Q} \)
Now we can find the ratio of the energy radiated by the hemispheres to that radiated by the intact sphere:
\[ \frac{Q'}{Q} = \frac{\sigma (6\pi r^2) T^4}{\sigma (4\pi r^2) T^4} \]
The \( \sigma \), \( \pi \), and \( r^2 \) terms cancel out:
\[ \frac{Q'}{Q} = \frac{6}{4} = \frac{3}{2} = 1.5 \]

### Conclusion
The ratio \( \frac{Q'}{Q} \) is:
\[ \frac{Q'}{Q} = 1.5 \]