In a Rutherford scattering experiment when a projectile of change `Z_(1)` and mass `M_(1)` approaches s target nucleus of change `Z_(2)` and mass `M_(2)`, te distance of closed approach is `r_(0)`. The energy of the projectile is

Options

A. Directly proportional to mass `M_(1)`
B. Directly proportional to `M_(1) xx M_(2)`
C. Directly proportional to `Z_(1)Z_(2)`
D. Inversely proportional to `Z_(1)`

Explanation

To solve the problem regarding the energy of a projectile in a Rutherford scattering experiment, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a projectile with charge \( Z_1 \) and mass \( M_1 \).
- It approaches a target nucleus with charge \( Z_2 \) and mass \( M_2 \).
- The distance of closest approach is denoted as \( r_0 \).

2. **Energy Conservation Principle**:
- In the Rutherford scattering experiment, we can use the principle of conservation of energy. The kinetic energy of the projectile at a distance far away from the nucleus will be converted into electrostatic potential energy at the closest approach.

3. **Kinetic Energy of the Projectile**:
- The initial kinetic energy (KE) of the projectile when it is far away from the nucleus is given by:
\[ KE = \frac{1}{2} M_1 v^2 \]
- Here, \( v \) is the velocity of the projectile.

4. **Electrostatic Potential Energy at Closest Approach**:
- The electrostatic potential energy (PE) between two charges at a distance \( r_0 \) is given by:
\[ PE = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Z_1 Z_2}{r_0} \]
- Here, \( \epsilon_0 \) is the permittivity of free space.

5. **Setting Up the Energy Conservation Equation**:
- At the closest approach, all the kinetic energy is converted into potential energy:
\[ \frac{1}{2} M_1 v^2 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Z_1 Z_2}{r_0} \]

6. **Solving for the Energy of the Projectile**:
- Rearranging the equation to express the energy of the projectile:
\[ E = \frac{1}{2} M_1 v^2 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Z_1 Z_2}{r_0} \]
- Thus, the energy \( E \) of the projectile is:
\[ E = \frac{Z_1 Z_2}{4 \pi \epsilon_0 r_0} \]

### Final Expression:
The energy of the projectile in a Rutherford scattering experiment is given by:
\[ E = \frac{Z_1 Z_2}{4 \pi \epsilon_0 r_0} \]