The correct relation between scattering angle (`theta`), impact parameter (b) and distance of closest approach (D) is

Options

A. `sin theta = Db`
B. `tan"" (theta )/(2) = (D )/(2b)`
C. `(cos theta )/(b ) =D`
D. `cot"" (theta )/(2) =(b )/( 2D)`

Explanation

To find the correct relation between the scattering angle (\( \theta \)), impact parameter (\( b \)), and distance of closest approach (\( D \)), we can follow these steps:

### Step 1: Understand the Concept of Distance of Closest Approach
The distance of closest approach (\( D \)) is defined as the minimum distance between the alpha particle and the nucleus when the alpha particle is deflected due to the electrostatic force. At this point, the kinetic energy of the alpha particle is equal to the potential energy due to the electrostatic interaction with the nucleus.

### Step 2: Set Up the Energy Relation
The kinetic energy (\( KE \)) of the alpha particle can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the alpha particle and \( v \) is its velocity.

The potential energy (\( PE \)) at the distance of closest approach can be expressed using Coulomb's law:
\[ PE = \frac{k \cdot Z_1 Z_2 e^2}{D} \]
where \( k \) is Coulomb's constant, \( Z_1 \) and \( Z_2 \) are the atomic numbers of the interacting particles, and \( e \) is the charge of an electron.

### Step 3: Equate Kinetic and Potential Energy
At the distance of closest approach, we set the kinetic energy equal to the potential energy:
\[ \frac{1}{2} mv^2 = \frac{k \cdot Z_1 Z_2 e^2}{D} \]

### Step 4: Define the Impact Parameter
The impact parameter (\( b \)) is the perpendicular distance from the center of the nucleus to the line along which the alpha particle would travel if it were not deflected.

### Step 5: Relate Scattering Angle to Impact Parameter
The scattering angle (\( \theta \)) is related to the impact parameter and the distance of closest approach. The relationship can be derived from the geometry of the scattering process. The relationship is given by:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{D}{b} \]

### Step 6: Derive the Final Relation
From the above relationship, we can rearrange it to express \( D \) in terms of \( b \) and \( \theta \):
\[ D = b \tan\left(\frac{\theta}{2}\right) \]

### Step 7: Final Relation
Using the small angle approximation for \( \tan\left(\frac{\theta}{2}\right) \approx \frac{\theta}{2} \) when \( \theta \) is small, we can express the relationship as:
\[ D \approx b \cdot \frac{\theta}{2} \]
This can be rearranged to give:
\[ \frac{\theta}{2} = \frac{D}{b} \]

### Conclusion
Thus, the correct relation between the scattering angle (\( \theta \)), impact parameter (\( b \)), and distance of closest approach (\( D \)) is:
\[ \frac{\theta}{2} = \frac{D}{b} \]