Class 12 > Unit # 25: Quantum Physics > Laws of Black Body Radiations

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Black Body Radiations, Laws, and Planck's Hypothesis

Video Lecture: Black Body Radiations & Planck's Hypothesis

Watch the complete video lecture explaining black body radiation, its laws, and Planck's quantum theory:

Black Body Radiation

An ideal body that absorbs the entire radiation incident upon it, regardless of frequency, is called a "Black Body". The advantage of introducing the concept of a black body is that we can disregard the precise nature of whatever is radiating in the discussion of thermal radiation because all black bodies behave identically.

A black body can be approximated by a hollow object with a very fine hole leading to its interior. Any radiation striking the hole enters the cavity where it is trapped by repeated reflections back and forth until it is completely absorbed. The cavity walls are constantly emitting and absorbing radiation. Just as a black body is nearly a perfect absorber, it is also the most effective emitter of radiation when heated. Radiations emitted from the cavity are known as "Black Body Radiations".

Fig: Cavity black body and intensity vs wavelength distribution curves at different temperatures.

It is experimentally observed that a black body radiates more when hot than when it is cold. Furthermore, the spectrum of a hot black body has its peak at a higher frequency (shorter wavelength) than the peak in the spectrum of a cooler one.

Laws of Black Body Radiation

I. Wien's Law (Wien's Displacement Law)

According to Wien's law, “The wavelength for which maximum radiation is emitted is inversely proportional to the absolute temperature of the black body.”
Alternatively: “The product of the wavelength for maximum radiation emitted and the absolute temperature of the black body remains constant.”

Mathematically, if $\lambda_m$ is the maximum wavelength of emitted radiation, then:

$\lambda_m T = \text{constant}$ or $\lambda_m \propto \frac{1}{T}$

Note: Wien's law holds well at short wavelengths (high frequencies), but it fails at long wavelengths (low frequencies, infrared region), where it underestimates radiation intensity.

II. Stefan's Law (Stefan-Boltzmann Law)

According to Stefan's law, “The total amount of energy radiated per second per unit area of a black body is directly proportional to the fourth power of its absolute temperature.”

Mathematically, if $E$ is the energy radiated per second per unit area and $T$ is the temperature:

$E \propto T^4$
$E = \sigma T^4$

Where $\sigma$ is known as the Stefan-Boltzmann constant.

III. Rayleigh-Jeans Law

Rayleigh and Jeans considered that radiation inside a cavity at absolute temperature $T$, whose walls are perfect reflectors, consists of a series of standing electromagnetic waves, and the density of standing waves in the cavity is independent of the shape of the cavity. The higher the frequency, the shorter the wavelength and the greater the number of possible standing waves.

According to the Rayleigh-Jeans law, “The energy radiated with a particular wavelength is inversely proportional to the fourth power of the wavelength.”

$E_\lambda \propto \frac{1}{\lambda^4}$

At short wavelengths (high frequencies, ultraviolet region), the formula predicts infinite intensity, which is physically impossible. This major discrepancy is known as the Ultraviolet Catastrophe.

Planck's Hypothesis (Planck's Law)

To resolve the Ultraviolet Catastrophe, Max Planck proposed a revolutionary hypothesis:

“Energy is emitted or absorbed in discrete amounts or in the form of bundles of energy, each bundle being called a quantum (plural: quanta).”

The energy of each quantum is directly proportional to the frequency of the radiation:

$E \propto f$
$E = hf$

Where $h$ is a constant known as Planck's constant, and its value is $6.63 \times 10^{-34}\text{ J}\cdot\text{s}$.

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