In Bohr model of hydrogen atom, the force on the electron depends on the principal quantum number (n) as

Options

A. independent of n
B. `Fprop(1)/(n^(5))`
C. `Fprop(1)/(n^(4))`
D. `Fprop(1)/(n^(3))`

Explanation

To solve the problem of how the force on the electron in the Bohr model of the hydrogen atom depends on the principal quantum number \( n \), we can follow these steps:

### Step 1: Understand the Forces Acting on the Electron
In the Bohr model, the electron revolves around the nucleus in circular orbits. The force acting on the electron is the centripetal force, which can be expressed as:
\[ F = \frac{mv^2}{R} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( R \) is the radius of the orbit.

### Step 2: Determine the Velocity and Radius
According to the Bohr model:
- The velocity \( v \) of the electron is given by:
\[ v \propto \frac{Z}{n} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)) and \( n \) is the principal quantum number.

- The radius \( R \) of the electron's orbit is given by:
\[ R \propto \frac{n^2}{Z} \]

### Step 3: Substitute Velocity and Radius into the Force Equation
Now, substituting the expressions for \( v \) and \( R \) into the centripetal force equation:
\[ F = \frac{m \left(\frac{Z}{n}\right)^2}{\frac{n^2}{Z}} \]

### Step 4: Simplify the Force Expression
Now, simplify the expression for the force:
\[ F = \frac{m \cdot \frac{Z^2}{n^2}}{\frac{n^2}{Z}} = m \cdot \frac{Z^2}{n^2} \cdot \frac{Z}{n^2} = m \cdot \frac{Z^3}{n^4} \]

### Step 5: Analyze the Dependence on Principal Quantum Number \( n \)
From the simplified expression, we can see that the force \( F \) is proportional to:
\[ F \propto \frac{Z^3}{n^4} \]
This indicates that the force on the electron is inversely proportional to the fourth power of the principal quantum number \( n \).

### Conclusion
Thus, we conclude that in the Bohr model of the hydrogen atom, the force on the electron depends on the principal quantum number \( n \) as:
\[ F \propto \frac{1}{n^4} \]