Question

The deuteron is bound by nuclear forces just as $H$-atom is made up of $p$ and $e$ bound by electrostatic forces. If we consider the force between neutron and proton in deuteron as given in the form of a Coulomb potential but with an effective charge $e$'
$F=\frac{1}{4\pi\varepsilon_{0}} \frac{e'^{2}}{r}$
Estimate the value of (e'/e) given that the binding energy of a deuteron is $2.2\, MeV$

• A. $1.89$
• B. $9.24$
• C. $3.64$
• D. $7.62$

Answer 1

Answer: (C)

We know that binding energy of hydrogen atom in ground state,
$E=\frac{me^{4}}{8\varepsilon_{0}^{2}h^{2}}=13.6\,eV \ldots\left(i\right)$
Replacing $e$ by e' and m by m', reduced mass of neutron - proton,
$m'=\frac{M\times M}{M+M}=\frac{M}{2}=\frac{1836m}{2}=918m \ldots\left(ii\right)$
($M$ = mass of neutron/proton)
$\therefore $ Binding energy, $E'=\frac{918 me'^{4}}{8\varepsilon_{0}^{2}h^{2}}=2.2\,MeV$
Dividing $\left(ii\right)$ by $\left(i\right)$
$918\left(\frac{e'}{e}\right)^{4}=\frac{2.2\times10^{6}}{13.6}$
$\frac{e'}{e}=\left(176.21\right)^{1/ 4}=3.64$