LEVELJEE Main

Animated Solution for Physics - Properties of Solids and Liquids: The earth radiates in the infra-red region of the spectrum. The spectrum is correctly given by

Select Answer:

Visualized Solution

$T_{Earth} \approx 300 K$

Earth acts as a blackbody radiator.
Average temperature $T \approx 300 K$ .

$λ_{m} T = b$

Wien's Displacement Law gives the peak wavelength.
For $T = 300 K$ , $λ_{m}$ is in the infrared region.

$E_{λ} = \frac{8 π h c}{λ ^{5}} \frac{1}{e ^{h c / λk T} - 1}$

Planck's Law describes the complete spectral distribution.
It resolves the issues of classical physics.

$E_{λ} \approx \frac{8 π k T}{λ ^{4}}$

Rayleigh-Jeans Law is the classical approximation.
Valid for long wavelengths ( $λ ≫ \frac{h c}{k T}$ ).

$λ_{IR} ≫ \frac{h c}{k T}$

Earth's infrared radiation falls in the long-wavelength tail.
It is correctly approximated by the Rayleigh-Jeans law.

00:00 / 00:00

The Sigma Insight: Heat Transfer

Solution Diagram

The Earth as a Cosmic Radiator

Imagine the Earth basking in the Sun's glow. It absorbs this immense solar energy and warms up to an average temperature of about

300 K

Like any object with a temperature above absolute zero, the Earth must radiate this heat back into the cold void of space to maintain thermal equilibrium.

But what kind of light does the Earth emit?

The Quest for the Perfect Curve

If we use Wien's Displacement Law,

λ_{m} T = b

, we find that for a temperature of

300 K

, the peak of the emitted radiation lies squarely in the infrared region.

We cannot see this light with our naked eyes, but we can certainly feel it as heat.

To describe the exact shape of this radiation curve across all wavelengths, physicists rely on Planck's Law of Radiation.

E_{λ} = \frac{8 π h c}{λ ^{5}} \frac{1}{e ^{h c / λk T} - 1}

Planck's Law perfectly maps out the energy distribution, forming a beautiful, continuous bell-like curve that matches experimental data flawlessly.

The Classical Approximation

Rayleigh-Jeans Law

However, before the dawn of quantum mechanics, classical physicists attempted to model this radiation using the Rayleigh-Jeans Law.

E_{λ} \approx \frac{8 π k T}{λ ^{4}}

This classical formula predicted that an object would emit infinite energy at short wavelengths—a catastrophic failure known as the Ultraviolet Catastrophe.

Despite this massive flaw at short wavelengths, the Rayleigh-Jeans law actually provides a remarkably accurate approximation for the long-wavelength tail of the spectrum.

Why This Answer?

Because the Earth radiates primarily in the far-infrared region, which corresponds to these longer wavelengths, its spectral behavior in this specific regime aligns beautifully with the classical prediction.

Therefore, in the context of this historical problem, the radiation spectrum of the Earth in the infrared region is correctly associated with the Rayleigh-Jeans law.

It is a fascinating glimpse into how classical approximations still hold immense value in specific physical limits!

Common Traps & FAQs

Isn't Planck's Law the correct answer for all blackbody radiation?

Yes, absolutely! Planck's Law is the exact quantum mechanical description for all wavelengths. However, historically and in the context of some specific classical physics problems, the Rayleigh-Jeans law is highlighted for its accuracy in the long-wavelength (infrared/microwave) limit.

What is the Ultraviolet Catastrophe?

It's the failure of the Rayleigh-Jeans law at short wavelengths (like ultraviolet). The classical formula predicted that an object would emit infinite energy at short wavelengths, which is physically impossible. Planck solved this by introducing quantized energy levels.

Why doesn't Wien's Law describe the whole spectrum?

Wien's Law is an approximation that works very well for short wavelengths but fails at long wavelengths. Wien's *Displacement* Law, however, is a specific rule derived from Planck's Law that tells us exactly where the peak of the spectrum is.

How hot is the Earth considered to be in these calculations?

For blackbody radiation calculations, the Earth's average surface temperature is taken to be roughly