Question

In a Rutherford scattering experiment when a projectile of charge $Z _{1}$ and mass $M _{1}$ approaches a target nucleus of charge $Z _{2}$ and mass $M _{2}$, the distance of closest approach is $r _{0}$. The energy of the projectile is :

• A. Directly proportional to $z_1 z_2$
• B. Inversely proportional to $z_1$
• C. Directly proportional to mass $M_1$
• D. Directly proportional to $M_1 \times M_2$

Answer 1

The Correct Option is A

To solve this problem, we need to understand the concept of the closest approach in Rutherford scattering experiments. The closest approach r_0 is the minimum distance between a projectile (charged particle) and a target nucleus during a scattering event. It can be understood using the principles of conservation of energy.

The total energy of the system before scattering is essentially kinetic energy, due to the incoming projectile, and potential energy at the closest approach. The formula for electrostatic potential energy U at the distance of closest approach is given by:

U = \dfrac{k \cdot Z_1 \cdot Z_2 \cdot e^2}{r_0}

Where:

• k is Coulomb's constant.
• Z_1 and Z_2 are the atomic numbers (or charge numbers) of the projectile and target nucleus respectively.
• e is the elementary charge.
• r_0 is the distance of closest approach.

The energy of the projectile is primarily kinetic and therefore at the point of closest approach, it is completely converted into potential energy. Thus, the initial kinetic energy K of the projectile is equal to U, so:

K = \dfrac{k \cdot Z_1 \cdot Z_2 \cdot e^2}{r_0}

This shows that the kinetic energy is directly proportional to the product Z_1 Z_2.

Analyzing the given options:

• Directly proportional to z_1 z_2: This option is correct as we derived that the energy is proportional to the product of the charges.
• Inversely proportional to z_1: Incorrect, since proportionality is direct.
• Directly proportional to mass M_1: Incorrect, as the mass does not appear directly in the derived formula for the energy.
• Directly proportional to M_1 \times M_2: Incorrect, as mass is not part of the proportionality in the given context.

Therefore, the correct answer is that the energy of the projectile is directly proportional to Z_1 Z_2.