The relationship between intensity, frequency, and temperature in blackbody radiation is described by Planck's law, which states that the spectral radiance (or intensity) of blackbody radiation is proportional to both the frequency and the temperature of the object.
The equation for Planck's law is given by:
B(ν, T) = (2hν³ / c²) * (1 / (exp(hν / kT) - 1))
In this equation, B(ν, T) represents the spectral radiance (intensity) at a given frequency ν and temperature T. The other variables are constants: h is Planck's constant, c is the speed of light, and k is the Boltzmann constant.
The intensity of blackbody radiation is indeed proportional to temperature, as the term (1 / (exp(hν / kT) - 1)) in the equation depends on the temperature T. As the temperature increases, the exponential term in the denominator decreases, leading to an increase in the intensity. This relationship indicates that hotter objects emit more intense radiation across the entire spectrum of frequencies.
On the other hand, the frequency ν is not directly proportional to temperature. The relationship between frequency and temperature arises from the fact that the peak wavelength of the blackbody radiation spectrum, called the peak or dominant wavelength, is inversely proportional to temperature. This relationship is expressed by Wien's displacement law:
λ_max = b / T
In this equation, λ_max represents the peak wavelength, T is the temperature, and b is the Wien displacement constant.
As temperature increases, the peak wavelength decreases, which corresponds to an increase in frequency. However, it is important to note that the entire spectrum of blackbody radiation covers a range of frequencies, and the relationship between frequency and temperature is not a simple proportional one.
In summary, the intensity of blackbody radiation is proportional to temperature, whereas the relationship between frequency and temperature is described by Wien's displacement law, which determines the dominant wavelength or peak of the radiation spectrum.