In a Rutherford scattering experiment when a projectile of charge `Z_(1)` and mass `M_(1)` approaches s target nucleus of charge `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 of `M_(1) ** M_(2)`
C. Directly proportional of `z_(1)z_(2)`
D. Inversely proportional to `z_(1)`

Answer

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

### Step 1: Understand the scenario
In a Rutherford scattering experiment, a charged projectile (with charge \( Z_1 \) and mass \( M_1 \)) approaches a target nucleus (with charge \( Z_2 \) and mass \( M_2 \)). The distance of closest approach is denoted as \( r_0 \).

### Step 2: Identify energy conversion
At the distance of closest approach \( r_0 \), the kinetic energy of the projectile is converted into electrostatic potential energy. This is a key point in understanding the relationship between the kinetic energy and the potential energy.

### Step 3: Write the expression for kinetic energy
The kinetic energy (KE) of the projectile before it approaches the nucleus can be expressed as:
\[ KE = \frac{1}{2} M_1 v^2 \]
where \( v \) is the velocity of the projectile.

### Step 4: Write the expression for electrostatic potential energy
The electrostatic potential energy (PE) at the distance of closest approach \( r_0 \) can be expressed using Coulomb's law:
\[ PE = \frac{k \cdot |Z_1| \cdot |Z_2|}{r_0} \]
where \( k \) is Coulomb's constant.

### Step 5: Set kinetic energy equal to potential energy
At the point of closest approach, the kinetic energy is equal to the potential energy:
\[ \frac{1}{2} M_1 v^2 = \frac{k \cdot |Z_1| \cdot |Z_2|}{r_0} \]

### Step 6: Solve for the energy of the projectile
From the above equation, we can express the kinetic energy of the projectile in terms of the other variables:
\[ KE = \frac{k \cdot |Z_1| \cdot |Z_2|}{2r_0} \]
Thus, the energy of the projectile is directly proportional to the product of the charges \( Z_1 \) and \( Z_2 \) and inversely proportional to the distance of closest approach \( r_0 \).

### Final Expression
The energy of the projectile can be summarized as:
\[ E = \frac{k \cdot |Z_1| \cdot |Z_2|}{2r_0} \]