The force acting on the electron in a hydrogen atom depends on the principal quantum number as

Options

A. `Fprop1//n^(2)`
B. `Fprop1//n^(4)`
C. `Fprop1//n^(5)`
D. Does not depend on n+

Answer

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

### Step 1: Understand the Force Acting on the Electron
The force acting on the electron in a hydrogen atom can be described using the centripetal force formula:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the electron's orbit.

### Step 2: Relate Velocity to Principal Quantum Number
According to Bohr's model of the hydrogen atom, the velocity \( v \) of the electron is inversely proportional to the principal quantum number \( n \):
\[ v \propto \frac{1}{n} \]
This means that as \( n \) increases, the velocity \( v \) decreases.

### Step 3: Relate Radius to Principal Quantum Number
Bohr's model also tells us that the radius \( r \) of the electron's orbit is directly proportional to the square of the principal quantum number \( n \):
\[ r \propto n^2 \]
This indicates that as \( n \) increases, the radius \( r \) increases as well.

### Step 4: Substitute Relationships into the Force Equation
Now, substituting these relationships into the force equation:
1. Substitute \( v \) in terms of \( n \):
\[ v^2 \propto \left(\frac{1}{n}\right)^2 = \frac{1}{n^2} \]
2. Substitute \( r \) in terms of \( n \):
\[ r \propto n^2 \]

### Step 5: Combine the Relationships
Now, substituting \( v^2 \) and \( r \) into the force equation:
\[ F \propto \frac{m \cdot \frac{1}{n^2}}{n^2} = \frac{m}{n^4} \]
Thus, we find that the force \( F \) is inversely proportional to the fourth power of the principal quantum number \( n \):
\[ F \propto \frac{1}{n^4} \]

### Conclusion
Therefore, the force acting on the electron in a hydrogen atom depends on the principal quantum number \( n \) as:
\[ F \propto \frac{1}{n^4} \]

### Final Answer
The correct option is that the force is inversely proportional to \( n^4 \).

---