In mu-particle scattering experiment, the expression of distance of closest approach (`r_0`)is (symbols have their usual meanings)

Options

A. `1/(4πepsilon_0) × (Ze^2)/E`
B. `1/(4πepsilon_0) × (2Ze^2)/E`
C. `1/(4πepsilon_0) × (Z^2e^2)`
D. `1/(4πepsilon_0)×(Z^2e^4)/E`

Answer

To solve the problem regarding the distance of closest approach (`r_0`) in a mu-particle scattering experiment, we can derive the expression step by step.

### Step 1: Understand the Concept
The distance of closest approach refers to the minimum distance between the incoming particle (in this case, a muon) and the target nucleus during a scattering event. This distance can be derived from the principles of electrostatics and conservation of energy.

### Step 2: Use the Relevant Formula
For a charged particle (like a muon) approaching a nucleus, the distance of closest approach can be derived from the balance of kinetic energy and potential energy. The potential energy at the distance of closest approach is given by Coulomb's law:

\[ U = \frac{k \cdot Z \cdot e^2}{r_0} \]

where:
- \( U \) is the potential energy,
- \( k \) is Coulomb's constant,
- \( Z \) is the atomic number of the nucleus,
- \( e \) is the charge of the muon,
- \( r_0 \) is the distance of closest approach.

### Step 3: Set Up the Energy Conservation Equation
The kinetic energy of the muon before the interaction is given by:

\[ K.E. = \frac{1}{2} m v^2 \]

where:
- \( m \) is the mass of the muon,
- \( v \) is the velocity of the muon.

At the distance of closest approach, all the kinetic energy is converted into potential energy:

\[ \frac{1}{2} m v^2 = \frac{k \cdot Z \cdot e^2}{r_0} \]

### Step 4: Rearranging the Equation
Rearranging the above equation to solve for \( r_0 \):

\[ r_0 = \frac{2k \cdot Z \cdot e^2}{m v^2} \]

### Step 5: Final Expression
Thus, the expression for the distance of closest approach \( r_0 \) in a mu-particle scattering experiment is:

\[ r_0 = \frac{2k \cdot Z \cdot e^2}{m v^2} \]