- #1

Alcubierre

- 80

- 0

## Homework Statement

The energy density of electromagnetic radiation at wavelength λ from a black body at temperature [itex]T[/itex] (degrees Kelvin) is given by Planck's law of black body radiation:

[itex]f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)}[/itex]

where [itex]h[/itex] is Planck's constant, [itex]c[/itex] is the speed of light, and [itex]k[/itex] is Boltzmann's constant. To find the wavelength of peak emissions, maximize [itex]f(\lambda)[/itex] by minimizing [itex]g(\lambda) = λ^{5}(e^{hc/λkT} - 1)[/itex]. Use a Taylor polynomial for [itex]e^{x}[/itex] with n = 7 to expand the expression in parentheses and find the critical number of the resulting function. (Hint: Use [itex]\frac{hc}{k}[/itex] = 0.014.) Compare this to Wien's law:

[itex]\lambda _{max} = \frac{0.014}{T}[/itex]. Wien's law is accurate for small λ. Discuss the flaw in our use of Maclaurin series.

## Homework Equations

[itex]e^{x} = \sum_{n = 0}^{∞} \frac{x^{n}}{n!}[/itex]

## The Attempt at a Solution

I have no idea where to begin. I started with setting x to [itex]\frac{0.014}{\lambda T}[/itex] and expanding the series to the 7th term but I don't know the direction to go.