- #1
BRN
- 108
- 10
Hi guys! I have a problem with this exercise:
1. Homework Statement
The stars called white dwarfs may have inside them a density in order of 1011 kg m-3. For semplicity, we assume:
I'm at T=0, but electrons are relatevistic, so P=-∂U/∂V.
Internal energy is:
U=∫0εFg(ε)ε dε
where:
states density g(ε)=(2me3/2V√ε)/(√2π2ħ3)
ε=ħck ; k2=p2/ħ2=me2v2/ħ2=2εme/ħ2 ⇒ k=√(2εme/ħ2)
Then:
U=∫0εF (2me3/2V√ε)/(√2π2ħ3) √(2εme/ħ2) ħc dε =
= (me3/2V c εF3/2 √(εFme))/(π2ħ2)
So:
P=-∂U/∂V=-(me3/2 c εF3/2 √(εFme))/(π2ħ2)
Fermi energy is:
εF=ħ2/(2me)(3π2N/V)2/3
Then:
P=-3/4ħ2c(3π2)1/3(N/V)4/3
This solution is wrong. The correct solution is:
P=(ħ c(3π2)1/3(N/V)4/3)/4
Could someone tell me where I'm wrong?
1. Homework Statement
The stars called white dwarfs may have inside them a density in order of 1011 kg m-3. For semplicity, we assume:
- these stars are made with non interacting protons and electrons at the same quantity and with uniform density;
- Inside, the electrons are relatevistic with energy ε=c|p|=cħ|k|;
- Temperature is nothing.
The Attempt at a Solution
I'm at T=0, but electrons are relatevistic, so P=-∂U/∂V.
Internal energy is:
U=∫0εFg(ε)ε dε
where:
states density g(ε)=(2me3/2V√ε)/(√2π2ħ3)
ε=ħck ; k2=p2/ħ2=me2v2/ħ2=2εme/ħ2 ⇒ k=√(2εme/ħ2)
Then:
U=∫0εF (2me3/2V√ε)/(√2π2ħ3) √(2εme/ħ2) ħc dε =
= (me3/2V c εF3/2 √(εFme))/(π2ħ2)
So:
P=-∂U/∂V=-(me3/2 c εF3/2 √(εFme))/(π2ħ2)
Fermi energy is:
εF=ħ2/(2me)(3π2N/V)2/3
Then:
P=-3/4ħ2c(3π2)1/3(N/V)4/3
This solution is wrong. The correct solution is:
P=(ħ c(3π2)1/3(N/V)4/3)/4
Could someone tell me where I'm wrong?