- #1
Alettix
- 177
- 11
Hello!
I have a little trouble with understanding Planck's law of radiation, and wondered if you could help me with it. :)
The equation is:
## \frac{dI}{d\lambda} = \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} ## (1)
where T is the temperature, k Boltzmann's constant, h Planck's constant, ##\lambda## the wavelenght, c the speed of light in vacuum and I the radiated intensity per unit area.
What confuses me is the differential form on the left-hand side of the equation. As I have understood, the total intensity radiated in a small wavelenght interval from ## \lambda## to ##\lambda + \Delta\lambda## is given by:
## I = \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} \Delta\lambda## (2)
and the total intensity of all wavelenghts:
## I = \int_{0}^{\infty} \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} d\lambda = \sigma T^4 ## (3)
which is just the Stefan-Boltzmann law.
What I wonder about is how we can find the radiated intensity for one wavelenght only. That would mean ##\Delta\lambda \rightarrow 0 ## which would give ## I \rightarrow 0 ## with (2) if I am not wrong. But this doesn't seem logical, thus it would mean that no single wavelenght radiates any energy, wouldn't it? When trying to find information, I came across writings which said that the radiation intensity at one wavelenght is simply given by:
## I = \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} ## (4)
But to me this seems wrong as well. If ##\Delta\lambda## is small (which it is assumed to be), we get that (4)>(2) . This means that a wavelenght interval is radiating less energy than one single wavelenght in that interval, which just can't be right, can it?
I would be really glad if you could help me sort this out. :)
Thank you!
I have a little trouble with understanding Planck's law of radiation, and wondered if you could help me with it. :)
The equation is:
## \frac{dI}{d\lambda} = \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} ## (1)
where T is the temperature, k Boltzmann's constant, h Planck's constant, ##\lambda## the wavelenght, c the speed of light in vacuum and I the radiated intensity per unit area.
What confuses me is the differential form on the left-hand side of the equation. As I have understood, the total intensity radiated in a small wavelenght interval from ## \lambda## to ##\lambda + \Delta\lambda## is given by:
## I = \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} \Delta\lambda## (2)
and the total intensity of all wavelenghts:
## I = \int_{0}^{\infty} \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} d\lambda = \sigma T^4 ## (3)
which is just the Stefan-Boltzmann law.
What I wonder about is how we can find the radiated intensity for one wavelenght only. That would mean ##\Delta\lambda \rightarrow 0 ## which would give ## I \rightarrow 0 ## with (2) if I am not wrong. But this doesn't seem logical, thus it would mean that no single wavelenght radiates any energy, wouldn't it? When trying to find information, I came across writings which said that the radiation intensity at one wavelenght is simply given by:
## I = \frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)} ## (4)
But to me this seems wrong as well. If ##\Delta\lambda## is small (which it is assumed to be), we get that (4)>(2) . This means that a wavelenght interval is radiating less energy than one single wavelenght in that interval, which just can't be right, can it?
I would be really glad if you could help me sort this out. :)
Thank you!