Question

A black body radiates energy at the rate of $\mathbf{E}$ watt per metre $^{2}$ at a high temperature
T K. When the temperature is reduced to $(\mathrm{T} / 2) \mathrm{K}$, the radiant energy will be
(A) E/16
(B) E/4
(C) E/2
(D) 2E

Answer 1

Hint: We should know that the wavelength is the distance between two wave crests, which is the same as the distance between two troughs. The number of waves that pass-through a given point in one second is called the frequency, measured in units of cycles per second called Hertz. As the full spectrum of visible light travels through a prism, the wavelengths separate into the colours of the rainbow because each colour is a different wavelength. Violet has the shortest wavelength, at around 380 nanometres, and red has the longest wavelength, at around 700 nanometres. Gamma rays have the highest energies, the shortest wavelengths, and the highest frequencies. Radio waves, on the other hand, have the lowest energies, longest wavelengths, and lowest frequencies of any type of EM radiation.

Complete step by step answer
By Stefan-Boltzmann law, we know that the black body radiation of energy $\mathrm{E}$ is directly proportional to fourth power of $\mathrm{T}$ temperature of the black body.
This is equivalent to,
$\mathrm{E} \propto \mathrm{T}^{4}$
If the temperature of the body is reduced to $\dfrac{\mathrm{T}}{2},$ the energy of radiation will be $\dfrac{\mathrm{E}}{16}$.

Hence, Option A is the correct answer.

Note: We should know that black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature, called the Planck spectrum or Planck's law. As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. It occurs due to a process called thermal radiation. Thermal energy causes vibration of molecules or atoms, which in turn vibrates the charge distribution in the material, allowing radiation by the above mechanisms. That radiation, for a perfect absorber, follows the blackbody curve. The primary law governing blackbody radiation is the Planck Radiation Law, which governs the intensity of radiation emitted by unit surface area into a fixed direction from the blackbody as a function of wavelength for a fixed temperature. The mathematical function describing the shape is called the Planck function.
It should be known that blackbody radiation is a cornerstone in the study of quantum mechanics. This experiment is what led to the discovery of a field that would revolutionize physics and chemistry. Quantum mechanics gives a more complete understanding of the fundamental mechanisms at the sub-atomic level.