Two spheres made of same material have radii in the ratio 1: 2 Both are at same temperature. Ratio of heat radiation energy emitted per second by them is

Options

A. `1 : 2`
B. `1 : 8`
C. `1 : 4`
D. `1 : 16`

Answer

To solve the problem of finding the ratio of heat radiation energy emitted per second by two spheres made of the same material and at the same temperature, we can follow these steps:

### Step 1: Understand the formula for heat radiation
The heat radiation emitted per second (u) by a body is given by the formula:
\[ u = e \cdot \sigma \cdot A \cdot T^4 \]
where:
- \( e \) is the emissivity (constant for the same material),
- \( \sigma \) is the Stefan-Boltzmann constant (constant for all bodies),
- \( A \) is the surface area of the sphere,
- \( T \) is the absolute temperature of the body.

### Step 2: Determine the surface area of the spheres
The surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 3: Write the surface areas for both spheres
Let the radius of the first sphere be \( r_1 \) and the radius of the second sphere be \( r_2 \). Given that the ratio of their radii is \( r_1 : r_2 = 1 : 2 \), we can express this as:
\[ r_2 = 2r_1 \]

Now, we can find the surface areas:
- For the first sphere:
\[ A_1 = 4\pi r_1^2 \]
- For the second sphere:
\[ A_2 = 4\pi r_2^2 = 4\pi (2r_1)^2 = 4\pi \cdot 4r_1^2 = 16\pi r_1^2 \]

### Step 4: Calculate the ratio of the surface areas
Now, we can find the ratio of the surface areas \( A_1 \) and \( A_2 \):
\[ \frac{A_1}{A_2} = \frac{4\pi r_1^2}{16\pi r_1^2} = \frac{4}{16} = \frac{1}{4} \]

### Step 5: Relate the heat radiation emitted by both spheres
Since both spheres are made of the same material and are at the same temperature, the heat radiation emitted per second by each sphere can be expressed as:
\[ u_1 \propto A_1 \]
\[ u_2 \propto A_2 \]

Thus, the ratio of heat radiation emitted by the two spheres is:
\[ \frac{u_1}{u_2} = \frac{A_1}{A_2} = \frac{1}{4} \]

### Conclusion
The ratio of heat radiation energy emitted per second by the two spheres is:
\[ \frac{u_1}{u_2} = \frac{1}{4} \]