Stefan-Boltzmann law — calculate power radiated by a black body at 600K

Question

A spherical black body of radius 10 cm is maintained at a temperature of 600 K. Calculate the total power radiated by the body. Stefan-Boltzmann constant $\sigma = 5.67 \times 10^{-8}$ W m $^{-2}$ K $^{-4}$ .

(JEE Main 2023, similar pattern)

Solution — Step by Step

For a perfect black body (emissivity $\varepsilon = 1$ ), the total power radiated is:

P = \sigma A T^4

where $A$ is the surface area and $T$ is the absolute temperature in Kelvin.

For a sphere of radius $r = 0.10$ m:

A = 4\pi r^2 = 4\pi (0.10)^2 = 0.04\pi \text{ m}^2 \approx 0.1257 \text{ m}^2

T^4 = (600)^4 = 600^2 \times 600^2 = 3.6 \times 10^5 \times 3.6 \times 10^5 = 1.296 \times 10^{11} \text{ K}^4

P = 5.67 \times 10^{-8} \times 0.1257 \times 1.296 \times 10^{11}

P = 5.67 \times 0.1257 \times 1.296 \times 10^{3}

P = 0.9236 \times 10^{3}

\boxed{P \approx 924 \text{ W}}

Why This Works

Every object with temperature above absolute zero emits thermal radiation. The Stefan-Boltzmann law tells us that the radiated power scales as $T^4$ — a remarkably steep dependence. Doubling the temperature increases the radiation by a factor of 16. This is why extremely hot objects (like stars) radiate enormously more power than warm objects.

A “black body” is a perfect emitter and absorber. Real objects emit less, captured by the emissivity factor $\varepsilon$ (0 to 1): $P = \varepsilon \sigma A T^4$ .

Alternative Method

If the surroundings are at temperature $T_0$ (not negligible), the net power radiated is:

P_{net} = \sigma A (T^4 - T_0^4)

In our problem, the question asks for total power radiated (not net), so we use just $\sigma A T^4$ .

In JEE numericals, computing $T^4$ can be tedious. A quick approach: $(600)^4 = (6)^4 \times 10^8 = 1296 \times 10^8$ . Breaking the base into a small number times a power of 10 saves time and reduces arithmetic errors.

Common Mistake

Students sometimes use the temperature in Celsius instead of Kelvin. The Stefan-Boltzmann law requires absolute temperature. Using 600°C instead of 600 K would mean the actual temperature is 873 K, and $T^4$ would be off by a factor of $(873/600)^4 \approx 4.5$ . Always check your units — temperature must be in Kelvin for radiation laws.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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