Stefan–Boltzmann law: For an ideal black body, the total emissive power per unit area is directly proportional to which function of absolute temperature?

Introduction / Context:
Thermal radiation from surfaces is central to high-temperature heat transfer. The Stefan–Boltzmann law provides the fundamental relation for the total emissive power of a black body.

Given Data / Assumptions:

• Ideal black body surface (emissivity = 1).
• Absolute temperature T in Kelvin.
• Stefan–Boltzmann constant sigma is a universal constant.

Concept / Approach:
The law states: E_b = sigma * T^4 where E_b is the emissive power (W/m^2). The fourth-power dependence means small increases in temperature cause large increases in radiative heat emission.

Step-by-Step Solution:
Recognize black body assumption → emissivity = 1.Apply Stefan–Boltzmann equation: E_b proportional to T^4.Conclude the correct functional dependence is the fourth power.

Verification / Alternative check:
Compare with gray body: E = epsilon * sigma * T^4; the exponent remains 4; epsilon only scales magnitude.

Why Other Options Are Wrong:

• T, T^2, T^3, log(T): none capture the steep growth rate observed and proven in radiation physics; only T^4 matches experiments and theory.

Common Pitfalls:
Confusing the Stefan–Boltzmann law with Wien’s displacement law (which relates peak wavelength and temperature).

Final Answer:
T^4 (fourth power of absolute temperature)