Black-Body RadiationJEE Main

Concept

Every warm object glows. The Stefan–Boltzmann law says the total power radiated rockets up as the fourth power of temperature, while Wien's law says the colour of the peak shifts to shorter wavelengths as it gets hotter — which is why metal goes red, then white, then blue-hot.

Key formula

Derivation

A black body's emissive power integrates over all wavelengths to $\sigma T^4$ per unit area, with $\sigma = 5.67\times10^{-8}$. Multiplying by area and emissivity gives the total power.

The Planck spectrum peaks where $\lambda_{max}T = b$ is constant, so doubling the temperature halves the peak wavelength — the radiation gets bluer.

Scenarios to explore

• Black-Body Radiation — Stefan–Boltzmann power and Wien's peak wavelength.

Real-world applications

• Estimating stellar temperatures from their colour.
• Infrared thermometers and thermal imaging.
• Incandescent bulb and furnace design.

JEE exam tips

• $P \propto T^4$: doubling T multiplies power by 16.
• $\lambda_{max}T$ = constant — hotter means bluer.
• Net exchange with surroundings: $P = \varepsilon\sigma A(T^4 - T_0^4)$.

Common mistakes

• Using $T$ in °C instead of kelvin.
• Forgetting the fourth power — a small T change is a big power change.
• Mixing up Wien's $\lambda_{max} \propto 1/T$ direction.

Exam traps to avoid

• Temperature must be absolute (kelvin) in both laws.
• Emissivity scales power but never changes the peak wavelength.