Q.According to Stefan's law of radiation, a black body radiates energy $\sigma T^4$ from its unit surface area every second where $T$ is the surface temperature of the black body and $\sigma = 5.67 \times 10^{-8}$ W/m$^2$K$^4$ is known as Stefan's constant. A nuclear weapon may be thought of as a ball of radius 0.5 m. When detonated, it reaches temperature of $10^6$ K and can be treated as a black body.

Thermal Radiation Properties

Imagine holding your hand near a hot iron — you feel warmth without touching it. That warmth travels through empty space, not through air or metal. This is thermal radiation: energy emitted by any object purely because it has a temperature.

Unlike conduction (touching) or convection (fluid flow), radiation doesn't need a medium. It's electromagnetic waves — mostly infrared, but at high temperatures, visible light too. A glowing red-hot steel rod is radiating; so is your own body, though you can't see it.

The Core Idea: Every Object Radiates

Every object above absolute zero ($0\ \text{K}$) emits radiation. The amount and type depend on two things:

1. Temperature — hotter objects radiate more energy, and at shorter wavelengths.
2. Surface properties — some surfaces are good emitters, others are poor.

This leads to three key properties that describe how a real surface behaves compared to an ideal "perfect" radiator.

1. Emissivity ($\varepsilon$)

Intuition: A black matte surface feels hotter in sunlight than a shiny white one. Why? The black surface emits radiation more efficiently.

Definition: Emissivity is the ratio of radiation emitted by a real surface to the radiation emitted by an ideal blackbody at the same temperature.

• $\varepsilon = 1$ for a perfect blackbody (ideal emitter).
• $0 < \varepsilon < 1$ for all real surfaces.
• $\varepsilon$ depends on material, surface finish, and wavelength.

2. Absorptivity ($\alpha$)

Intuition: A black car roof gets hotter in summer than a white one. It absorbs more sunlight.

Definition: Absorptivity is the fraction of incident radiation that a surface absorbs.

• $\alpha = 1$ for a perfect blackbody (absorbs everything).
• $\alpha = 0$ for a perfect reflector.
• For opaque surfaces: $\alpha + \rho = 1$, where $\rho$ is reflectivity.

3. Reflectivity ($\rho$) and Transmissivity ($\tau$)

Intuition: A mirror reflects; a glass window transmits; a brick wall does neither.

Definition: For any surface, incident radiation is either absorbed, reflected, or transmitted:

• Reflectivity $\rho$: fraction reflected.
• Transmissivity $\tau$: fraction transmitted through.
• For opaque solids: $\tau = 0$, so $\alpha + \rho = 1$.

The Blackbody: The Ideal Reference

A blackbody is a theoretical surface that:

• Absorbs all incident radiation ($\alpha = 1$).
• Emits the maximum possible radiation at any temperature.
• Follows Planck's law, Stefan-Boltzmann law, and Wien's displacement law.

Real surfaces are compared to this ideal. The blackbody is a standard — like comparing a real engine to a Carnot engine.

Putting It Together: A Real Surface

A hot metal plate at $500\ \text{K}$ with $\varepsilon = 0.3$ emits:

$E = 0.3 \times 5.67 \times 10^{-8} \times (500)^4 \approx 1063\ \text{W/m}^2$ …