Wavelength for highest radiation per unit wavelength

Like this: $$\rho_T(\lambda)d\lambda=\frac{8\pi{hc}}{\lambda^5}\frac{d\lambda}{e^{{hc}/{\lambda{kT}}}-1}$$. If you want it to appear on a separate line, use triple '#' signs. Like this:$$\rho_T(\lambda)d\lambda=\frac{8\pi{hc}}{\lambda^5}\frac{d\lambda}{e^{{hc}/{\lambda{kT}}}-1}$$In summary, the equation $$\rho_T(\lambda)d\lambda=\frac{8\pi{hc}}{\lambda^5}\frac{d\lambda}{e^{{hc}/{\lambda
  • #1
176
0

Homework Statement


At what wavelength does a cavity at 6000 degrees Kelvin radiate most per unit wavelength?


Homework Equations


$$\rho_T(\lambda)d\lambda=\frac{8\pi{hc}}{\lambda^5}\frac{d\lambda}{e^{{hc}/{\lambda{kT}}}-1}$$


The Attempt at a Solution


I'm pretty new to this whole topic, so don't judge me if I'm totally off...
First off can't the $$d\lambda$$'s cancel each other out? I'm not sure why the book I am using writes this equation like they did if they could cancel out that easily, but anyways I got rid of them because it seems to just make this more confusing for me otherwise.
So I figured if it wants the wavelength for highest radiation (per unit wavelength), I should take the derivative of $$\rho_T(\lambda)d\lambda$$ with respect to the wavelength, and set that expression equal to zero.
However now I am at an impasse, seeing as how solving for lambda would probably be really difficult, and I am not even sure if I am working in the correct direction. Any help would be appreciated much. Thanks
 
Physics news on Phys.org
  • #2
skate_nerd said:

Homework Statement


At what wavelength does a cavity at 6000 degrees Kelvin radiate most per unit wavelength?

Homework Equations


$$\rho_T(\lambda)d\lambda=\frac{8\pi{hc}}{\lambda^5}\frac{d\lambda}{e^{{hc}/{\lambda{kT}}}-1}$$

The Attempt at a Solution


I'm pretty new to this whole topic, so don't judge me if I'm totally off...
First off can't the $$d\lambda$$'s cancel each other out? I'm not sure why the book I am using writes this equation like they did if they could cancel out that easily, but anyways I got rid of them because it seems to just make this more confusing for me otherwise.
So I figured if it wants the wavelength for highest radiation (per unit wavelength), I should take the derivative of $$\rho_T(\lambda)d\lambda$$ with respect to the wavelength, and set that expression equal to zero.
However now I am at an impasse, seeing as how solving for lambda would probably be really difficult, and I am not even sure if I am working in the correct direction. Any help would be appreciated much. Thanks

You are working in the correct direction. Differentiate ##\rho_T(\lambda)## and set it equal to zero. It is hard to solve. But if you set ##x=\frac{hc}{\lambda{kT}}## and express the whole expression in terms of the dimensionless variable x you'll get a single equation for x you can solve numerically. See Wien's Displacement Law.
 
  • Like
Likes 1 person
  • #3
I actually did all that a bit ago and solved to get a wavelength of 4798 angstroms, aka blue light. Sounds about right. Then about 20 seconds ago I just realized I could have avoided that whole mess and simply used Wien's Displacement Law. Got me the same exact answer. -_- oh well
 
  • #4
By the way, I am not new to LaTeX but how do you write an equation on this site without using the double dollar signs? Obviously "\(___\)" doesn't work...
 
Last edited:
  • #5
skate_nerd said:
By the way, I am not new to LaTeX but how do you write an equation on this site without using the double dollar signs? Obviously "\(___\)" doesn't work...

Use double '#' signs if you want the math to stay on the same line.
 
  • Like
Likes 1 person

What is the definition of wavelength for highest radiation per unit wavelength?

The wavelength for highest radiation per unit wavelength, also known as the peak wavelength, is the specific wavelength at which the amount of radiation emitted or absorbed by a substance is at its highest point.

Why is the wavelength for highest radiation per unit wavelength important?

The wavelength for highest radiation per unit wavelength is important because it helps determine the behavior and properties of electromagnetic radiation, which is essential for many scientific applications such as spectroscopy and imaging.

How is the wavelength for highest radiation per unit wavelength calculated?

The wavelength for highest radiation per unit wavelength can be calculated using the formula λmax = b/T, where λmax is the peak wavelength, b is Wien's displacement constant (2.898 x 10^-3 m K), and T is the temperature of the substance in Kelvin.

What factors affect the wavelength for highest radiation per unit wavelength?

The wavelength for highest radiation per unit wavelength is primarily affected by the temperature of the substance. Other factors that can influence the peak wavelength include the atomic or molecular structure of the substance, the type of radiation, and the surrounding environment.

How does the wavelength for highest radiation per unit wavelength relate to other types of wavelengths?

The wavelength for highest radiation per unit wavelength is just one of many different types of wavelengths in the electromagnetic spectrum. It falls within the infrared region, which has longer wavelengths than visible light but shorter wavelengths than microwaves. The peak wavelength can also be used to calculate other types of wavelengths, such as the average wavelength or the effective wavelength.

Suggested for: Wavelength for highest radiation per unit wavelength

Back
Top