The energy radiated by a blackbody does indeed vary with both the temperature of the blackbody and the wavelength of the radiation. This statement is true.
According to Planck's law of blackbody radiation, the spectral radiance or intensity of radiation emitted by a blackbody at a particular wavelength is directly proportional to the temperature of the blackbody and the wavelength raised to the power of five. Mathematically, it can be expressed as:
B(λ, T) = (2hc²/λ⁵) * (1 / (exp(hc/λkT) - 1))
Where: B(λ, T) is the spectral radiance at a given wavelength λ and temperature T. h is Planck's constant, c is the speed of light, k is Boltzmann's constant, λ is the wavelength.
As per this equation, higher temperatures result in a greater amount of energy being radiated across all wavelengths. Additionally, the distribution of energy across different wavelengths changes with temperature. At higher temperatures, the peak of the blackbody radiation shifts towards shorter wavelengths (higher frequencies), indicating an increase in the proportion of high-energy photons. This phenomenon is known as Wien's displacement law.
Conversely, at lower temperatures, the total energy radiated by the blackbody decreases, and the peak of the radiation spectrum shifts towards longer wavelengths (lower frequencies).
In summary, the energy radiated by a blackbody is directly influenced by both the temperature of the blackbody and the wavelength of the radiation, as described by Planck's law of blackbody radiation and Wien's displacement law.