Thermal Radiation

Thermal radiation

Thermal radiation is the transfer of heat energy through electromagnetic waves, without requiring any material medium. This means heat can be transferred even through vacuum. Thermal radiation consists of a spectrum of electromagnetic waves, predominantly in the infrared and visible regions. All objects emit and absorb thermal radiation continuously, depending on their temperature.

Key Concept: A body at a higher temperature than its surroundings emits more radiation than it absorbs, leading to a net loss of heat and a drop in temperature. Conversely, a cooler body absorbs more radiation than it emits, causing its temperature to rise.

The black body

A perfectly black body is an idealized object that absorbs all incident electromagnetic radiation, regardless of wavelength or angle. It reflects no radiation. Since a perfect black body absorbs 100% of radiation, it is also a perfect emitter of thermal radiation.

Examples of nearly perfect black bodies:

• The Sun (approximate black body)
• A surface coated with lamp black (absorbs about 96-98% of incident radiation)

How to realize a black body

A simple experimental black body can be made by punching a small hole in a closed empty metal container. Although the surface may be reflective, the hole appears black because incoming radiation bounces inside, being partially absorbed at each reflection, and thus almost no radiation escapes.

Black body radiation (BBR)

Black body radiation refers to the thermal radiation emitted by a black body at a certain absolute temperature $T$. This radiation spans all wavelengths and its intensity distribution depends solely on temperature.

Intensity of radiation

Intensity, $I$: Rate of radiant energy transfer per unit area, measured in watts per square meter $\mathrm{W/m^2}$.

$I = \frac{\text{Power}}{\text{Area}}$

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Laws of black body radiation

a. Wien's Displacement Law

The wavelength $\lambda_{\max}$ at which the radiation intensity is maximum is inversely proportional to the absolute temperature $T$ of the black body:

$\lambda_{\max} T = b$

where $b = 2.9 \times 10^{-3} \text{ m·K}$

This explains why objects glow different colors at different temperatures. As $T$ increases, $\lambda_{\max}$ shifts towards shorter wavelengths (from infrared to visible light and beyond).

b. Stefan-Boltzmann Law

The total power radiated per unit surface area of a black body per unit time, $E$, is proportional to the fourth power of its absolute temperature:

$E = \sigma T^4$

where $\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$ (Stefan-Boltzmann constant)

This law quantifies the total energy output from a black body given its temperature.

Emissivity $\epsilon$

Real surfaces do not emit as much radiation as a perfect black body. Emissivity $\epsilon$ is a dimensionless number (0 to 1) that quantifies this efficiency:

$P = \epsilon \sigma A T^4$

where

• $P$ = power radiated,
• $A$ = surface area,
• $\epsilon$ = emissivity of the surface.

Problem example: Radiation from human body

Given:

• Surface area $A = 1.25\, m^2$
• Temperature $T = 30^\circ C = 303\,K$
• Stefan constant $\sigma = 5.67 \times 10^{-8}\, W/m^2 K^4$

Find the total power radiated:

$P = \epsilon \sigma A T^4$

Assuming $\epsilon = 1$ for simplicity,

$P = 1 \times 5.67 \times 10^{-8} \times 1.25 \times (303)^4 \approx \text{Calculate value}$

Provost's theory of heat exchange

When a body is placed in an environment at temperature $T_0$, it radiates energy at a rate proportional to its surface area $A$ and $T^4$, and absorbs energy proportional to $A$ and $T_0^4$. Net radiated energy per second:

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

If $T > T_0$, the body loses heat; if $T < T_0$, it gains heat.

Energy distribution in black body spectrum

The spectral radiance of a black body (energy per unit wavelength) is described by Planck's Law:

$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$

Where:

• $h$ = Planck's constant ($6.626 \times 10^{-34} \,J\cdot s$)
• $c$ = speed of light ($3 \times 10^8 \, m/s$)
• $k_B$ = Boltzmann constant ($1.381 \times 10^{-23} \, J/K$)

This formula precisely predicts the intensity distribution across wavelengths for a black body at temperature $T$, improving upon Wien's and Stefan's laws.

Kirchhoff's law of radiation

The ratio of emissive power to absorptive power at a given wavelength is the same for all bodies at the same temperature and equals the emissive power of a perfect black body at that temperature:

$\frac{e_\lambda}{a_\lambda} = \text{constant} = E_{\lambda, \text{black body}}$

Good absorbers are good emitters at the same wavelength.

Solar luminosity and solar constant

Solar luminosity $L_s$: total energy emitted by the sun per second.

Solar constant $C$: energy per unit area received from the sun at Earth's distance:

$C = \frac{L_s}{4 \pi r^2}$

where $r$ is the Earth-Sun distance.

Using Stefan-Boltzmann law for the sun's surface:

$L_s = 4 \pi R_s^2 \sigma T_s^4$

where $R_s$ is the radius of the sun and $T_s$ its surface temperature.

Albedo

Albedo: the fraction of solar energy reflected by a planet back into space.

Important