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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 mass $M_1$
B
Directly proportional to $M_1\times M_2$
C
Directly proportional to $Z_1Z_2$
D
Inversely proportional to $Z_1$
Detailed Solution
At the distance of closest approach $r_0$ the projectile is momentarily at rest, so all its kinetic energy has been converted into electrostatic potential energy.
$E = \frac{1}{4\pi\varepsilon_0}\frac{(Z_1e)(Z_2e)}{r_0}$
$E = \frac{Z_1Z_2e^2}{4\pi\varepsilon_0r_0}$
For a given $r_0$, the energy does not involve the masses and is directly proportional to the product $Z_1Z_2$.
Hence the energy of the projectile is directly proportional to $Z_1Z_2$.
$E = \frac{1}{4\pi\varepsilon_0}\frac{(Z_1e)(Z_2e)}{r_0}$
$E = \frac{Z_1Z_2e^2}{4\pi\varepsilon_0r_0}$
For a given $r_0$, the energy does not involve the masses and is directly proportional to the product $Z_1Z_2$.
Hence the energy of the projectile is directly proportional to $Z_1Z_2$.