Q.A blackened metal sphere of surface area $0.1\ \text{m}^2$ is maintained at a steady temperature of $727\,^\circ\text{C}$ ($1000\ \text{K}$) inside an enclosure whose walls are at $27\,^\circ\text{C}$ ($300\ \text{K}$). Taking the Stefan-Boltzmann constant as $\sigma = 5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$, find

From a Hot Plate to a Law of Nature

Imagine you hold your hand near a hot iron. You feel warmth — that's radiant energy leaving the metal and reaching your skin. Now imagine the iron is twice as hot. Does it feel twice as warm? No — it feels much more intense. That jump is not linear; it's dramatic. The Stefan–Boltzmann law captures exactly how dramatic.

Every object above absolute zero emits electromagnetic radiation. The hotter it is, the more energy it radiates per second from each square metre of its surface. For a perfect emitter — called a black body — the relationship is astonishingly simple.

Here $E$ is the energy radiated per unit area per unit time (power per area, in $\text{W/m}^2$), $T$ is the absolute temperature in kelvin, and $\sigma$ is the Stefan–Boltzmann constant:

The key point: $T$ is raised to the fourth power. Double the temperature, and the radiated power increases by a factor of $2^4 = 16$. That is why a red-hot object (say 1000 K) radiates far more intensely than a warm object at 300 K — about $(1000/300)^4 \approx 123$ times more.

Why the Fourth Power?

The $T^4$ dependence is not arbitrary. It emerges from the deeper Planck radiation law, which describes how energy is distributed across different wavelengths. When you integrate Planck's law over all wavelengths, the total power comes out proportional to $T^4$. In other words, the fourth power is a consequence of the shape of the black-body spectrum — not a guess.

A Quick Check with Numbers

Take a black body at room temperature, 300 K. Its radiated power per square metre is:

That is substantial — about five 100-watt bulbs per square metre. But we don't feel it because our surroundings radiate back at us at nearly the same rate. The net energy exchange depends on the temperature difference.

Now heat the same surface to 600 K. The power becomes:

That is 16 times larger — exactly $2^4$.

Why It Matters

The Stefan–Boltzmann law is the backbone of thermal radiation physics. It explains: …