In the Bohr's model of hydrogen-like atom the force between the nucleus and the electron is modified as `F=(e^2)/(4 pi epsi_0)((1)/(r^2)+(beta)/(r^3))`, where B is a constant. For this atom, the radius of the nth orbit in terms of the Bohr radius `(a_0 = ( epsi_0 h^2)/(m pi e^2))` is :

Options

A. `r_n =a_0 n- beta`
B. `r_n =a_0 n^2 + beta`
C. `r_n = a_0 n^2 - beta`
D. `r_n =a_0 n+beta`

Explanation

To solve the problem, we need to find the radius of the nth orbit in a hydrogen-like atom using the modified force equation provided. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Force Equation
The force between the nucleus and the electron is given as:
\[ F = \frac{e^2}{4 \pi \epsilon_0} \left(\frac{1}{r^2} + \frac{\beta}{r^3}\right) \]
This force must equal the centripetal force required to keep the electron in a circular orbit.

### Step 2: Set Up the Centripetal Force Equation
The centripetal force \( F_c \) acting on the electron 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.

### Step 3: Equate the Forces
Set the modified force equal to the centripetal force:
\[ \frac{mv^2}{r} = \frac{e^2}{4 \pi \epsilon_0} \left(\frac{1}{r^2} + \frac{\beta}{r^3}\right) \]

### Step 4: Use Bohr's Quantization Condition
According to Bohr's model, the angular momentum is quantized:
\[ L = mvr = n\frac{h}{2\pi} \]
From this, we can express the velocity \( v \) as:
\[ v = \frac{n h}{2 \pi m r} \]

### Step 5: Substitute Velocity into the Force Equation
Substituting \( v \) into the centripetal force equation gives:
\[ \frac{m \left(\frac{n h}{2 \pi m r}\right)^2}{r} = \frac{e^2}{4 \pi \epsilon_0} \left(\frac{1}{r^2} + \frac{\beta}{r^3}\right) \]
This simplifies to:
\[ \frac{n^2 h^2}{4 \pi^2 m r^3} = \frac{e^2}{4 \pi \epsilon_0} \left(\frac{1}{r^2} + \frac{\beta}{r^3}\right) \]

### Step 6: Rearrange the Equation
Cross-multiplying gives:
\[ n^2 h^2 = \frac{e^2}{\epsilon_0} \left(r + \beta\right) \]

### Step 7: Express in Terms of Bohr Radius
The Bohr radius \( a_0 \) is defined as:
\[ a_0 = \frac{\epsilon_0 h^2}{m \pi e^2} \]
We can substitute this into our equation to express \( r \) in terms of \( a_0 \).

### Step 8: Final Expression for Radius
After simplification, we find:
\[ r = a_0 n^2 - \beta \]

### Conclusion
Thus, the radius of the nth orbit in terms of the Bohr radius \( a_0 \) is:
\[ r_n = a_0 n^2 - \beta \]