Electron Boost (Homework 8-323)
The energy of an electron is twice its rest mass (rest energy). a. Find the electron’s boost factor $γ$ and the velocity factor $β$. b. Find the electron’s velocity factor β for the case where its momentum is its rest mass (times c).
a)
$E^2=p^2 c^2+m_0^2 c^4$ with $E=2m_0 c^2$
$4m_0^2 c^4=p^2 c^2+m_0^2 c^4$
or $3m_0^2 c^4=p^2 c^2$ with $p=γm_0 v$
$3m_0^2 c^4=(m_0^2 v^2)/(1-β^2 )*c^2$
or $3c^2=v^2/(1-β^2 )$
or $3/β^2 =1/(1-β^2 )$ or $3-3β^2=β^2$
$β^2=3/4=0.75$ and $γ=1/\sqrt{1-β^2 }=1/\sqrt{(1-0.75)}=2$
b) If
$p=m_0 c$ this is $γm_0 v=m_0 c$ or $1/\sqrt{(1-β^2 )}=1/β$ or $β^2=1-β^2$ or $β^2=1/2$
$β=1/√2=0.707$ and $γ=1/\sqrt{(1-1/2)}=1/√0.5=√2=1.41$