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. `1:1`
B. `16:1`
C. `4:1`
D. `1:9`

Explanation

To find the ratio of the energy radiated per second by the first sphere to that by the second, we can use the Stefan-Boltzmann law, which states that the power (energy radiated per second) emitted by a black body is proportional to the fourth power of its absolute temperature and its surface area.

The formula for the power radiated by a sphere is given by:

\[ P = \epsilon \sigma A T^4 \]

Where:
- \( P \) is the power radiated,
- \( \epsilon \) is the emissivity (which is the same for both spheres since they are made of the same material),
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area of the sphere,
- \( T \) is the absolute temperature of the sphere.

### Step 1: 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 (radius \( r_1 = 1 \, m \)):
\[ A_1 = 4 \pi (1)^2 = 4 \pi \, m^2 \]

For Sphere 2 (radius \( r_2 = 4 \, m \)):
\[ A_2 = 4 \pi (4)^2 = 4 \pi (16) = 64 \pi \, m^2 \]

### Step 2: Write the expressions for power radiated by both spheres.

For Sphere 1:
\[ P_1 = \epsilon \sigma A_1 T_1^4 = \epsilon \sigma (4 \pi) (4000)^4 \]

For Sphere 2:
\[ P_2 = \epsilon \sigma A_2 T_2^4 = \epsilon \sigma (64 \pi) (2000)^4 \]

### Step 3: Calculate the ratio of the power radiated by the two spheres.

The ratio of the energy radiated per second by the first sphere to that by the second is:

\[ \frac{P_1}{P_2} = \frac{\epsilon \sigma (4 \pi) (4000)^4}{\epsilon \sigma (64 \pi) (2000)^4} \]

### Step 4: Simplify the ratio.

The \( \epsilon \), \( \sigma \), and \( \pi \) terms cancel out:

\[ \frac{P_1}{P_2} = \frac{4 (4000)^4}{64 (2000)^4} \]

Now, simplify the fraction:

\[ \frac{P_1}{P_2} = \frac{4}{64} \cdot \frac{(4000)^4}{(2000)^4} = \frac{1}{16} \cdot \left(\frac{4000}{2000}\right)^4 \]

Since \( \frac{4000}{2000} = 2 \):

\[ \frac{P_1}{P_2} = \frac{1}{16} \cdot (2)^4 = \frac{1}{16} \cdot 16 = 1 \]

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

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