Wavelength for highest radiation per unit wavelength

[skate_nerd]

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

 

[Dick]

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.

 

[skate_nerd]

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

 

[skate_nerd]

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...

 

[Dick]

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.