**30.** Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in the same atmosphere. The temperature of the smaller body is 800 K and the temperature of the bigger body is 400 K. If the energy radiated from the smaller body is \( E \), the energy radiated from the bigger body is (assume the effect of the surrounding to be negligible): - (1) 256 \( E \) - (2) \( E \) - (3) 64 \( E \) - (4) 16 \( E \)

Key concept – Stefan–Boltzmann law

For a perfect emitter (black body), the power radiated is

where

• $\sigma$ is the Stefan–Boltzmann constant,
• $A$ is the surface area,
• $T$ is the absolute temperature in kelvin.

Because both spheres are of the same material and the surroundings are neglected, the emissivity factor is the same for both, so it cancels out in any ratio.

How a student should think step-by-step

1. Recognise the formula → Radiation power $P$ depends on surface area and $T^4$.
2. Write the surface area for a sphere → $A = 4\pi r^2$.
3. Set up a ratio of powers to avoid writing nasty constants.
   $\frac{P_2}{P_1} = \frac{\sigma (4\pi r_2^2) T_2^4}{\sigma (4\pi r_1^2) T_1^4} = \frac{r_2^2}{r_1^2}\;\frac{T_2^4}{T_1^4}$
4. Substitute the numbers ($r_1 = 0.2\,\text{m},\; T_1 = 800\,\text{K};\; r_2 = 0.8\,\text{m},\; T_2 = 400\,\text{K}$) and simplify.
5. Interpret the result → Compare with the given options.

Imagine two hot iron balls

• One ball is tiny (radius of 0.2 m) but very hot at 800 K.
• The other is big (radius of 0.8 m) but only warm at 400 K.

Both balls glow and throw out light and heat just like an electric bulb does.
How much heat they throw out each second depends on how large their glowing skin is and how hot that skin is.

Think of it like:

1. Skin size → the larger the skin, the more spots to shine from.
2. Temperature → the hotter it is, the brighter each spot shines, and this brightness grows very, very fast (temperature to the power four!).

Even though the big ball has more skin, it is cooler. The question is: does the big cool ball shine more, less, or the same as the small hot ball?

Step-by-step calculation

Surface area of a sphere:

Power radiated by sphere:

Ratio of powers

Let the smaller sphere be (1) and the bigger sphere be (2).

Substitute values:

Compute each factor separately:

Multiply:

Therefore

Hence, the energy radiated from the bigger body is $E$.

Correct option: (2)