Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in same atmosphere. The temperature of the smaller body is 800 K and temperature of bigger body is 400 K. If the energy radiated from the smaller body is E, the energy radiated from the bigger body is (assume, effect of the surrounding to be negligible)

The energy radiated by a body is given by Stefan-Boltzmann law, which states:

$$P = \sigma \cdot A \cdot e \cdot T^4$$

where:

• $P$ is the power radiated,
• $\sigma$ is Stefan-Boltzmann constant,
• $A$ is the surface area of the body,
• $e$ is the emissivity of the material (same for both bodies, as they are made of the same material),
• $T$ is the absolute temperature of the body.

Step 1: Energy radiated from the smaller sphere ($P_s$):

For the smaller sphere:

• Radius, $r_s = 0.2 \, \text{m}$,
• Temperature, $T_s = 800 \, \text{K}$,
• Surface area, $A_s=4\pi r_s^2=4\pi (0.2{)}^2$

The power radiated is:

$$P_s = \sigma \cdot (4 \pi r_s^2) \cdot e \cdot T_s^4 = \sigma \cdot 4 \pi \cdot (0.2)^2 \cdot e \cdot (800)^4$$

Step 2: Energy radiated from the bigger sphere ($P_b$):

For the larger sphere:

• Radius, $r_b = 0.8 \, \text{m}$,
• Temperature, $T_b = 400 \, \text{K}$,
• Surface area, $A_b=4\pi r_b^2=4\pi (0.8{)}^2$.

The power radiated is:

$$P_b = \sigma \cdot (4 \pi r_b^2) \cdot e \cdot T_b^4 = \sigma \cdot 4 \pi \cdot (0.8)^2 \cdot e \cdot (400)^4$$

Step 3: Ratio of energy radiated ($P_b / P_s$):

Take the ratio of the powers radiated by the bigger and smaller spheres:

$$\frac{P_b}{P_s} = \frac{\sigma \cdot 4 \pi \cdot (0.8)^2 \cdot e \cdot (400)^4}{\sigma \cdot 4 \pi \cdot (0.2)^2 \cdot e \cdot (800)^4}$$

Simplify:

$\frac{P_b}{P_s}=\frac{(0.8{)}^2}{(0.2{)}^2}\cdot \frac{(400{)}^4}{(800{)}^4}=1$

So:

The energy radiated from the bigger body is equal to the energy radiated from the smaller body:

$\boxed{P_b=P_s=E}$