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In a Rutherford scattering experiment when a projectile of charge Z₁ and mass M₁ approaches a target nucleus of charge Z₂ and mass M₂ the distance of the closest approach is r₀. What is the energy of the projectile?

A

Directly proportional to M₁ × M₂

B

Directly proportional to Z₁Z₂

C

Inversely proportional to Z₁

D

Directly proportional to the mass M₁

Step-by-Step Solution

At the distance of closest approach (r0r_0), the initial kinetic energy (KK) of the projectile is completely converted into electrostatic potential energy (UU). The formula for potential energy between two charges q1q_1 (Z1eZ_1e) and q2q_2 (Z2eZ_2e) separated by a distance r0r_0 is: U=14πϵ0q1q2r0=14πϵ0(Z1e)(Z2e)r0U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_0} = \frac{1}{4\pi\epsilon_0} \frac{(Z_1e)(Z_2e)}{r_0}. Since K=UK = U, the energy of the projectile is proportional to the product of the charges (Z1Z2Z_1 Z_2). Therefore, EnergyZ1Z2Energy \propto Z_1 Z_2.

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