Two spheres of same material have radius 1 m and 4 m and temperature 4000 K and 2000 K respectively. The energy radiated per second by the first sphere is

Options

A. greater than that by the second
B. less than that by the second
C. equal in both cases
D. the information is incomplete to draw any conclusion

Explanation

To solve the problem of finding the energy radiated per second by the first sphere, we can follow these steps:

### Step 1: Understand the formula for energy radiated
The energy radiated per second by a body is given by the Stefan-Boltzmann law:
\[ U = e \sigma A T^4 \]
where:
- \( U \) is the energy radiated per second,
- \( e \) is the emissivity (which is constant for the same material),
- \( \sigma \) is the Stefan-Boltzmann constant (also constant for the same material),
- \( A \) is the surface area,
- \( T \) is the absolute temperature in Kelvin.

### Step 2: Calculate the surface area of each sphere
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
For the first sphere with radius \( r_1 = 1 \, \text{m} \):
\[ A_1 = 4 \pi (1)^2 = 4 \pi \, \text{m}^2 \]
For the second sphere with radius \( r_2 = 4 \, \text{m} \):
\[ A_2 = 4 \pi (4)^2 = 4 \pi (16) = 64 \pi \, \text{m}^2 \]

### Step 3: Write the expressions for energy radiated by both spheres
For the first sphere at temperature \( T_1 = 4000 \, \text{K} \):
\[ U_1 = e \sigma A_1 T_1^4 = e \sigma (4 \pi) (4000)^4 \]
For the second sphere at temperature \( T_2 = 2000 \, \text{K} \):
\[ U_2 = e \sigma A_2 T_2^4 = e \sigma (64 \pi) (2000)^4 \]

### Step 4: Simplify the expressions
Since \( e \) and \( \sigma \) are constants, we can denote \( K = e \sigma \):
\[ U_1 = K (4 \pi) (4000)^4 \]
\[ U_2 = K (64 \pi) (2000)^4 \]

### Step 5: Calculate \( (4000)^4 \) and \( (2000)^4 \)
Calculating \( (4000)^4 \):
\[ (4000)^4 = 256 \times 10^{12} \]
Calculating \( (2000)^4 \):
\[ (2000)^4 = 16 \times 10^{12} \]

### Step 6: Substitute back into the expressions
Now substituting these values back:
\[ U_1 = K (4 \pi) (256 \times 10^{12}) = K (1024 \pi \times 10^{12}) \]
\[ U_2 = K (64 \pi) (16 \times 10^{12}) = K (1024 \pi \times 10^{12}) \]

### Step 7: Compare \( U_1 \) and \( U_2 \)
From the calculations:
\[ U_1 = U_2 \]
Thus, the energy radiated per second by the first sphere is equal to that of the second sphere.

### Final Answer
The energy radiated per second by the first sphere is equal to that of the second sphere.

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