Understanding Black Body Energy Radiation

When we talk about heat transfer, radiation is one important mode. All objects above absolute zero temperature emit thermal radiation. A black body is an idealized object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It is also a perfect emitter of thermal radiation.

The question asks about the rate at which energy is radiated by a black body based on its absolute temperature. This specific relationship is described by a fundamental law in physics.

Laws Related to Heat Transfer and Thermodynamics

Let's look at the options provided:

• Grashof law: This law is primarily used in natural convection heat transfer. It relates to buoyancy-driven flow.
• Prandtl law: Prandtl number is a dimensionless number used in the study of fluid mechanics and heat transfer. It relates momentum diffusivity to thermal diffusivity. While important in convection, it doesn't directly give the rate of radiation from a black body.
• Nusselt law: Nusselt number is a dimensionless number used in the study of convective heat transfer. It represents the ratio of convective to conductive heat transfer across a boundary. Like Prandtl law, it's related to convection, not black body radiation rate.
• Stefan-Boltzmann law: This law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body's absolute temperature (\(T\)).

The Stefan-Boltzmann Law Explained

The Stefan-Boltzmann law is precisely what governs the rate of energy radiation from an ideal black body. The formula for the energy radiated per unit time per unit area (often called emissive power, \(E\)) of a black body is given by:

\(E = \sigma T^4\)

Where:

• \(E\) is the total energy radiated per unit time per unit surface area (\(W/m^2\)).
• \(\sigma\) is the Stefan-Boltzmann constant, approximately \(5.67 \times 10^{-8} W/(m^2 \cdot K^4)\).
• \(T\) is the absolute temperature of the black body in Kelvin (\(K\)).

For a real object (not a perfect black body), the rate of radiation is also dependent on its emissivity (\(\epsilon\)), which is a value between 0 and 1. The rate of energy radiated (\(\dot{Q}\)) by a real body with surface area \(A\) is given by:

\(\dot{Q} = \epsilon \sigma A T^4\)

For a perfect black body, \(\epsilon = 1\), so the formula simplifies to \(\dot{Q} = \sigma A T^4\), or the emissive power \(E = \sigma T^4\).

Therefore, the rate at which energy is radiated by a black body at an absolute temperature is directly given by the Stefan-Boltzmann law.

Comparing the Laws

Based on this comparison, only the Stefan-Boltzmann law directly describes the rate of energy radiated by a black body based on its absolute temperature.