In a hydrogen-like species, the net force acting on a revolving electron in an orbit as given by the Bohr model is proportional to [n `rarr` principal quantum numbe, Z `rarr` atomic number]

Options

A. `(Z^(3))/(n^(4))`
B. `(n^(3))/(Z^(2))`
C. `(Z^(2))/(n^(3))`
D. `(n^(4))/(Z^(3))`

Answer

To solve the problem, we need to analyze the forces acting on an electron in a hydrogen-like atom according to the Bohr model.

### Step-by-Step Solution:

1. **Understanding the Forces**:
In a hydrogen-like atom, the electron revolves around the nucleus due to the electrostatic force of attraction between the positively charged nucleus (with charge +Ze) and the negatively charged electron (with charge -e). The net force acting on the electron is the centripetal force required to keep it in circular motion.

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

3. **Electrostatic Force**:
The electrostatic force \( F_e \) between the nucleus and the electron is given by Coulomb's law:
\[ F_e = \frac{Ze^2}{r^2} \]
where \( Z \) is the atomic number and \( e \) is the charge of the electron.

4. **Setting Forces Equal**:
For the electron to be in a stable orbit, the centripetal force must equal the electrostatic force:
\[ \frac{mv^2}{r} = \frac{Ze^2}{r^2} \]

5. **Finding Velocity**:
Rearranging the equation gives:
\[ mv^2 = \frac{Ze^2}{r} \]
From the Bohr model, the velocity \( v \) of the electron in the nth orbit is given by:
\[ v = \frac{Z e^2}{4 \pi \epsilon_0 h n} \]
where \( h \) is Planck's constant and \( \epsilon_0 \) is the permittivity of free space.

6. **Finding Radius**:
The radius \( r \) of the nth orbit is given by:
\[ r = \frac{n^2 h^2}{4 \pi^2 m e^2 Z} \]

7. **Substituting Values**:
Now substituting the expressions for \( v \) and \( r \) back into the centripetal force equation, we can find the relationship between the net force and the quantum numbers:
\[ F \propto \frac{(Z^2 e^4)}{(4 \pi \epsilon_0)^2} \cdot \frac{1}{n^4} \]

8. **Final Proportionality**:
Therefore, the net force acting on the electron is proportional to:
\[ F \propto \frac{Z^2}{n^4} \]

### Conclusion:
The net force acting on a revolving electron in a hydrogen-like atom is proportional to \( Z^2/n^4 \).