Question

Consider the fusion in helium plasma. Find the temperature at which the average thermal energy 1.5 kT equals the Coulomb potential energy at 2 fm.

Answer 1

Given:
Average thermal energy, E = 1.5 kT
Point of coulomb potential energy = 2 fm
Potential energy is given by
U = $\frac{K{q}_1{q}_2}{r}$ .....(1)
Here, K = $\frac{1}{4{\mathrm{\pi \epsilon }}_0}$ = 9$\times$10⁹

Charge, $q_1=q_2=2\times 1.6\times {10}^{-19}\mathrm{C}$

Average kinetic energy $\left(E\right)$ is given by
$E=\frac{3}{2}kT$ .....(2)
Here, k = Boltzman constant
T = Temperature

Equating equation (1) and (2), we get
$\frac{k{q}_1{q}_2}{r}=\frac{3}{2}kT$
$\Rightarrow T=\frac{2K{q}_1{q}_2}{3kr}$
= $\frac{2\times 9\times {10}^9\times 4\times {\left(1.6\times {10}^{-19}\right)}^2}{3\times 1.38\times {10}^{-23}\times 2\times {10}^{-15}}$
= 2.23$\times$10¹⁰ K