(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the number of electrons whose velocity's x-component is between [x,x+dx] is given by

[tex]

dN = \frac{4\pi V m^2 k_B T}{h^3} ln [exp(\frac{E_F -mv_x^2/2}{k_B T})+1]dv_x

[/tex]

2. Relevant equations

The Fermi-Dirac Distribution function:

[tex]

\frac{dn}{dE}=\frac{4\pi V \sqrt{2m^3}}{h^3}\frac{\sqrt{E}}{exp \left((E-E_F)/ k_B T\right) +1}

[/tex]

Where E is kinetic energy,

[tex]

E=\frac{1}{2} m v^2.

[/tex]

3. The attempt at a solution

First, I used the definition of kinetic energy above to rewrite the distribution as a function of velocity. I got

[tex]

dn=\frac{V m^{3/2}}{h^3}\frac{1}{exp((\frac{1}{2} m v^2 -E_F)/k_B T) +1} 4 \pi v^2 dv

[/tex]

Now, we know that

[tex]

4 \pi v^2 dv = dv_x dv_y dv_z.

[/tex]

Now I should be able to get the number of particles with v_{x}in the given range by integratin out dv_{y}and dv_{z}. In the assignment it is suggested that I write

[tex]

t^2=v_y^2+v_z^2

[/tex]

and then I can use the formula

[tex]

\int_0^{\infty} (ae^x+1)^{-1}dx=ln(1+\frac{1}{a}).

[/tex]

But I don't know how to do that. What is the differential element dv_{y}dv_{z}written with dt?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: A question regarding the number of fermions with a certain velocity component

**Physics Forums | Science Articles, Homework Help, Discussion**