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 forces between neutron and proton in deuteron as given in the form of a Coulomb force but with an effective charge e': `F=1/(4piepsilon_(0)) (e'^(2))/r^(2)`
estimate the value of `(e'//e)` given that the following binding energy of a deuteron is 2.2MeV.

Options

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

Explanation

To estimate the value of \( \frac{e'}{e} \) given the binding energy of a deuteron, we can follow these steps:

### Step 1: Understand the Binding Energy
The binding energy of a deuteron is given as 2.2 MeV. We need to convert this energy into electron volts (eV) for our calculations:
\[ E_{binding} = 2.2 \text{ MeV} = 2.2 \times 10^6 \text{ eV} \]

### Step 2: Use the Binding Energy Formula
For a hydrogen atom, the binding energy in the ground state is given by:
\[ E = \frac{m_e e^4}{8 \pi \epsilon_0^2 h^2} \]
where \( m_e \) is the mass of the electron, \( e \) is the charge of the electron, \( \epsilon_0 \) is the permittivity of free space, and \( h \) is Planck's constant. The binding energy for hydrogen is known to be approximately 13.6 eV.

### Step 3: Write the Binding Energy for Deuteron
For the deuteron, which consists of a proton and a neutron, we can express the binding energy in a similar manner but with the effective charge \( e' \):
\[ E_{binding} = \frac{m' e'^4}{8 \pi \epsilon_0^2 h^2} \]
where \( m' \) is the reduced mass of the proton-neutron system.

### Step 4: Calculate the Reduced Mass
The reduced mass \( m' \) of the deuteron can be approximated as:
\[ m' \approx \frac{m_p m_n}{m_p + m_n} \]
Given that the mass of the proton \( m_p \) and the neutron \( m_n \) are nearly equal, we can simplify this to:
\[ m' \approx \frac{m_p^2}{2m_p} = \frac{m_p}{2} \]

### Step 5: Relate the Two Binding Energies
Now, we can set up the ratio of the binding energies:
\[ \frac{E_{binding, deuteron}}{E_{binding, hydrogen}} = \frac{m' e'^4}{m_e e^4} \]
Substituting the known values:
\[ \frac{2.2 \times 10^6 \text{ eV}}{13.6 \text{ eV}} = \frac{\frac{m_p}{2} e'^4}{m_e e^4} \]

### Step 6: Solve for \( \frac{e'}{e} \)
Rearranging gives:
\[ \frac{e'^4}{e^4} = \frac{2.2 \times 10^6}{13.6} \cdot \frac{2 m_e}{m_p} \]
Calculating the left side:
\[ \frac{2.2 \times 10^6}{13.6} \approx 161.76 \]
Substituting \( m_p \approx 1836 m_e \):
\[ \frac{e'^4}{e^4} = 161.76 \cdot \frac{2}{1836} \]
Calculating:
\[ \frac{e'^4}{e^4} \approx 0.176 \]
Taking the fourth root:
\[ \frac{e'}{e} = (0.176)^{1/4} \approx 0.664 \]

### Final Result
Thus, the estimated value of \( \frac{e'}{e} \) is approximately:
\[ \frac{e'}{e} \approx 0.664 \]