For most practical temperature, we may take it that all...

For most practical temperature, we may take it that all bodies radiate heat in proportion to the fourth power of temperature $T$. If a body (coated black) at $400 \mathrm{~K}$ surrounded by atmosphere at $300 \mathrm{~K}$ has cooling rate $r_{o}$, the same body at $900 \mathrm{~K}$, surrounded by the same atmosphere, will have cooling rate closer to (a) $16 r / 3$ (b) $81 r_{o} / 16$ (c) $16 r$ (d) $4 r_{\text {o }}$

For most practical temperature, we may take it that all bodies radiate heat in proportion to the fourth power of temperature $T$. If a body (coated black) at $400 \mathrm{~K}$ surrounded by atmosphere at $300 \mathrm{~K}$ has cooling rate $r_{o}$, the same body at $900 \mathrm{~K}$, surrounded by the same atmosphere, will have cooling rate closer to
(a) $16 r / 3$
(b) $81 r_{o} / 16$
(c) $16 r$
(d) $4 r_{\text {o }}$

Key Concepts

Role of Ambient Temperature

When evaluating radiative cooling, the temperature of the surrounding environment cannot be neglected. In many cases, the net cooling rate is calculated by subtracting the fourth power of the ambient temperature from that of the body. This correction is particularly important at lower temperatures or when the object’s temperature is not far above ambient, as it can substantially reduce the effective cooling rate.

Temperature Scaling

Because radiative power scales as the fourth power of the temperature, even moderate increases in temperature can lead to very large increases in radiated energy. Understanding this strong temperature dependence is crucial when comparing cooling rates under different conditions, as the relative change in the T4 term can dramatically affect the net emission.

Net Radiative Heat Transfer

In many practical situations, the cooling or heating of an object by radiation is determined by the net power, which is the difference between the energy emitted by the object and the energy absorbed from its surroundings. This net effect is often computed as the difference of the fourth powers of the temperatures, reflecting both the object’s emission and the incoming radiation from its environment.

Stefan–Boltzmann Law

This law states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its absolute temperature. It is a cornerstone in thermal physics and provides the basis for understanding radiative heat transfer between bodies at different temperatures.

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