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 $e$'
$F=\frac{1}{4\pi {\epsilon }_0}\frac{e^{\prime 2}}{r}$
Estimate the value of (e'/e) given that the binding energy of a deuteron is $2.2MeV$

• 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,
$E=\frac{m{e}^4}{8{\epsilon }_0^2{h}^2}=13.6eV\dots \left(i\right)$
Replacing $e$ by e' and m by m', reduced mass of neutron - proton,
$m^{\prime }=\frac{M\times M}{M+M}=\frac{M}{2}=\frac{1836m}{2}=918m\dots \left(ii\right)$
($M$ = mass of neutron/proton)
$\therefore$ Binding energy, $E^{\prime }=\frac{918m{e}^{\prime 4}}{8{\epsilon }_0^2{h}^2}=2.2MeV$
Dividing $\left(ii\right)$ by $\left(i\right)$
$918{\left(\frac{e^{\prime }}{e}\right)}^4=\frac{2.2\times 10^6}{13.6}$
$\frac{e^{\prime }}{e}={\left(176.21\right)}^{1/4}=3.64$