How Is Lambda Max Derived from Planck's Law?

[georgeh]

I have a problem that states
Show that the wavelength Lamba_max=(2892 micro meters*K)/T
hint: set the dS/DLamba=0.
i have no idea how to do this.

 

[Hootenanny]

You need to find the maximum of a function. The maximum of a function is given when f'(x) = 0. So taking Planck's radiation formula for energy density per unit wavelength;

$$S_{\lambda} = \frac{8\pi hc}{\lambda^{5}} \cdot \frac{1}{e^{\frac{hc}{\lambda kT}} - 1}$$

The constant term $8\pi hc$ does not effect the position of the peak and can therefore be ignored. Thus the derivative becomes;

$$\frac{d}{d\lambda} = \frac{1}{\lambda^5}\cdot \frac{1}{e^{\frac{hc}{\lambda kT}} - 1} \;\; d\lambda = 0$$

Once you have found the derivative all that remains is to solve the equation. As far as I know there exists no analytical solution to the equation and it must be solved numerically. However, if I have time I may venture into the maths forums and inquire as to whether an analytical solution exists.

HINT: Solve numerically for $\lambda_{max}T$ first.

~H

 

[kyle8921]

First, let's break down Planck's spectral radiation law. It states that the spectral radiance of a black body at a given wavelength (λ) and temperature (T) can be calculated using the following equation:

B(λ,T) = (2hc^2/λ^5) * (1/(e^(hc/λkT) - 1))

Where:
- B(λ,T) is the spectral radiance
- h is Planck's constant
- c is the speed of light
- k is the Boltzmann constant

To find the maximum wavelength, we need to find the wavelength (λ) that maximizes the spectral radiance (B(λ,T)). This can be done by setting the derivative of the spectral radiance with respect to wavelength (dS/dλ) equal to 0 and solving for λ.

So, let's start by taking the derivative of the spectral radiance:

dS/dλ = (2hc^2/λ^5) * (1/(e^(hc/λkT) - 1)) - (10hc^2/λ^6) * (1/(e^(hc/λkT) - 1)^2) * (e^(hc/λkT))

Next, we can simplify this equation by multiplying both terms by λ^6 and rearranging:

dS/dλ = (2hc^2/λ^4) * (1/(e^(hc/λkT) - 1)) - (10hc^2/λ^4) * (1/(e^(hc/λkT) - 1)^2) * (e^(hc/λkT))

Now, let's set this derivative equal to 0 and solve for λ:

dS/dλ = 0
(2hc^2/λ^4) * (1/(e^(hc/λkT) - 1)) - (10hc^2/λ^4) * (1/(e^(hc/λkT) - 1)^2) * (e^(hc/λkT)) = 0
2/(e^(hc/λkT) - 1) - 10/(e^(hc/λkT) - 1)^2 * (e^(hc/λkT)) = 0
2 - 10e^(hc/λkT)