suppose an electron is attracted towards the origin by a force `k//r`, where k is a constant and r is the distance of the electron form the origin. By applying bohr model to this system, the radius of `n^(th)` orbit of the electron is found to be `r_n` and

Options

A. E
B. 2E
C. 3E
D. 4E

Explanation

To find the radius of the nth orbit of an electron attracted towards the origin by a force of the form \( \frac{k}{r} \), we can apply the Bohr model. Here’s a step-by-step solution:

### Step 1: Understand the Force Acting on the Electron
The force acting on the electron is given by:
\[ F = \frac{k}{r} \]
where \( k \) is a constant and \( r \) is the distance of the electron from the origin.

### Step 2: Relate the Force to Centripetal Motion
Since the electron is moving in a circular orbit, the centripetal force required to keep the electron in its circular path is provided by the attractive force. Therefore, we can write:
\[ \frac{mv^2}{r} = \frac{k}{r} \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 3: Simplify the Equation
Multiplying both sides by \( r \) gives:
\[ mv^2 = k \]
From this, we can express the velocity \( v \) as:
\[ v = \sqrt{\frac{k}{m}} \]

### Step 4: Apply Bohr's Quantization Condition
According to Bohr's model, the angular momentum \( L \) of the electron is quantized:
\[ L = mvr = \frac{nh}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 5: Substitute for Velocity
Substituting \( v \) from Step 3 into the angular momentum equation gives:
\[ m \left(\sqrt{\frac{k}{m}}\right) r = \frac{nh}{2\pi} \]
This simplifies to:
\[ \sqrt{mk} \cdot r = \frac{nh}{2\pi} \]

### Step 6: Solve for the Radius \( r \)
Rearranging the equation to solve for \( r \):
\[ r = \frac{nh}{2\pi \sqrt{mk}} \]

### Step 7: Identify the Relationship
From the final expression, we can see that the radius \( r \) of the nth orbit is directly proportional to \( n \):
\[ r_n = \frac{nh}{2\pi \sqrt{mk}} \]

### Conclusion
Thus, the radius of the nth orbit of the electron in this system is given by:
\[ r_n = \frac{nh}{2\pi \sqrt{mk}} \]