If in Bohr's atomic model, it is assumed that force between electron and proton varies inversely as `r^(4)` , energy of the system will be proportional to

Options

A. `n^(2)`
B. `n^(4)`
C. `n^(6)`
D. `n^(8)`

Answer

To solve the problem, we need to analyze the situation where the force between the electron and proton varies inversely as \( r^4 \). We will derive the relationship for the energy of the system based on this assumption.

### Step-by-Step Solution:

1. **Understanding the Force**:
The force \( F \) between the electron and proton is given by:
\[ F = \frac{B}{r^4} \]
where \( B \) is a proportionality constant and \( r \) is the distance between the electron and proton.

2. **Centripetal Force**:
The centripetal force required to keep the electron in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

3. **Equating Forces**:
For circular motion, the centripetal force is provided by the electrostatic force:
\[ \frac{mv^2}{r} = \frac{B}{r^4} \]
Rearranging gives:
\[ mv^2 = \frac{B}{r^3} \]

4. **Angular Momentum Conservation**:
According to Bohr's model, the angular momentum \( L \) of the electron is quantized:
\[ L = mvr = \frac{nh}{2\pi} \]
where \( n \) is a quantum number and \( h \) is Planck's constant.

5. **Squaring the Angular Momentum Equation**:
Squaring both sides:
\[ (mvr)^2 = \left(\frac{nh}{2\pi}\right)^2 \]
Expanding gives:
\[ m^2v^2r^2 = \frac{n^2h^2}{4\pi^2} \]

6. **Substituting for \( mv^2 \)**:
From step 3, we have \( mv^2 = \frac{B}{r^3} \). Substituting this into the squared angular momentum equation:
\[ m \left(\frac{B}{r^3}\right) r^2 = \frac{n^2h^2}{4\pi^2} \]
Simplifying gives:
\[ \frac{mB}{r} = \frac{n^2h^2}{4\pi^2} \]

7. **Finding \( r \)**:
Rearranging for \( r \):
\[ r = \frac{4\pi^2mB}{n^2h^2} \]

8. **Kinetic Energy**:
The kinetic energy \( K \) of the electron is given by:
\[ K = \frac{1}{2}mv^2 \]
Substituting \( mv^2 = \frac{B}{r^3} \):
\[ K = \frac{1}{2} \cdot \frac{B}{r^3} \]

9. **Substituting for \( r \)**:
Substitute \( r \) from step 7 into the kinetic energy equation:
\[ K = \frac{1}{2} \cdot \frac{B}{\left(\frac{4\pi^2mB}{n^2h^2}\right)^3} \]
Simplifying this will show that \( K \) is proportional to \( n^6 \).

10. **Total Energy**:
The total energy \( E \) of the system is the sum of kinetic and potential energy. However, since we are asked for the proportionality, we can conclude that:
\[ E \propto n^6 \]

### Final Answer:
The energy of the system will be proportional to \( n^6 \).