JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year APPLIED PHYSICS Previous Question Paper (July/August - 2021 - R18)

A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It does not reflect or transmit any radiation. When in thermal equilibrium at a specific temperature, it emits radiation known as black body radiation, which depends solely on the temperature of the body and not on its shape, size, or material composition.

Key characteristics include:

• It is a perfect absorber and a perfect emitter of radiation.
• The spectral distribution of the emitted radiation follows Planck's Law, which describes the intensity of radiation as a function of wavelength and temperature.
• As the temperature of the black body increases, the peak of the emission spectrum shifts toward shorter wavelengths (Wien's Displacement Law) and the total energy emitted increases (Stefan-Boltzmann Law).

An excellent practical example of a black body is a Lummer-Pringsheim cavity, which consists of an enclosure with a very small hole. Any radiation entering the hole undergoes multiple internal reflections, making it highly unlikely to escape, thus effectively trapping the radiation. The radiation emerging from the hole is a close approximation of ideal black body radiation.

Consider a particle of mass m confined to a one-dimensional box of length L, where the potential energy V(x) is zero inside the box (0 < x < L) and infinite outside. The time-independent Schrödinger wave equation is:

Inside the box (V=0), the equation simplifies to:

Let k² = 2mE/ħ². The general solution is ψ(x) = A sin(kx) + B cos(kx). Applying boundary conditions:

• At x = 0, ψ(0) = 0, which implies B = 0.
• At x = L, ψ(L) = 0, which implies A sin(kL) = 0. Since A ≠ 0, sin(kL) = 0, so kL = nπ, where n = 1, 2, 3...

Thus, k = nπ/L. Substituting k back into the energy expression:

The normalized wave function is found by setting the integral of |ψ|² over the box to 1: