Q.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 $e'$: $F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{e'^2}{r}$, estimate the value of $(e'/e)$ given that the binding energy of a deuteron is $2.2\ \text{MeV}$.

Treat the neutron-proton pair like a hydrogen-like (Bohr) bound state, but with the electron charge $e$ replaced by an effective charge $e'$ and the electron mass replaced by the reduced mass of the two nucleons.

Bohr scaling: the ground-state binding energy of a hydrogen-like system is $E \propto \mu\,(\text{charge})^4$, where $\mu$ is the reduced mass.

Reduced mass of the deuteron: $\mu = \dfrac{m_p m_n}{m_p + m_n} \approx \dfrac{m_p}{2} = \dfrac{1836}{2}\,m_e = 918\,m_e.$

Scale from hydrogen ($E_H = 13.6\ \text{eV}$, electron mass $m_e$, charge $e$): …

The analogy

In the hydrogen atom an electron (mass $m_e$, charge $e$) is Coulomb-bound to the proton with ground-state binding energy $E_H = 13.6\ \text{eV}$. The stem tells us to treat the neutron-proton force in the deuteron as a Coulomb-type interaction with an effective charge $e'$. So the deuteron is a hydrogen-like system with two changes: the charge $e \to e'$, and the light electron mass is replaced by the reduced mass of the two heavy nucleons.

Step 1 — Bohr energy scaling

For any hydrogen-like bound state the ground-state energy is

so $E \propto \mu\,(\text{charge})^4$, where $\mu$ is the reduced mass of the bound pair.

Step 2 — Reduced mass of the deuteron

The neutron and proton have nearly equal masses $\approx m_p$, so

Step 3 — Scale the binding energy

Replacing $m_e \to \mu$ and $e \to e'$, …

Method: Scaling a Bound-State Energy Between Two "Hydrogen-Like" Systems (Bohr-Model Scaling)

This method applies whenever a problem asks you to model an unfamiliar two-body bound system (here, the deuteron) as if it were a hydrogen-like atom with a different charge and/or different constituent masses, and to find an unknown parameter (like an effective charge) by matching binding energies.

Steps

Step 1: Recognise the two-body Coulomb-bound-state analogy

Any two particles held together purely by an inverse-square attractive force behave like a hydrogen-like atom. The Bohr-model ground-state binding energy for such a system depends only on the reduced mass of the pair and the fourth power of the interaction charge:

Step 2: Compute the reduced mass of the actual bound system

For two particles of mass $m_1$ and $m_2$, the reduced mass is

Use the real masses of the particles in the problem (here, the neutron and proton) — not the electron mass used in ordinary hydrogen.

Step 3: Set up the scaling ratio against the known hydrogen-atom result

Since you know hydrogen's own ground-state binding energy $E_H = 13.6\ \text{eV}$ (mass $m_e$, charge $e$), write the ratio of the new system's energy to hydrogen's:

$\frac{E_{\text{new}}}{E_H} = \left(\frac{\mu_{\text{new}}}{m_e}\right)\left(\frac{\text{charge}_{\text{new}}}{e}\right)^4$ …