The "distance of closest approach" signifies the minimum separation between an incoming proton and a nucleus, driven by electrostatic repulsion. This point is reached when the proton's kinetic energy is fully converted into potential energy.
Energy conservation is applied:
\[ \text{Initial kinetic energy} = \text{Final potential energy} \]
The potential energy \( U \) at the closest distance \( r \) is determined by Coulomb's law:
\[ U = \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r} \]
Where:
Equating initial kinetic energy to final potential energy:
\[ K.E. = U \]
The proton's kinetic energy \( K.E. \) is:
\[ K.E. = 3.95 \, \text{MeV} = 3.95 \times 10^6 \times 1.6 \times 10^{-13} \, \text{J} \]
Equating energies:
\[ 3.95 \times 10^6 \times 1.6 \times 10^{-13} = \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{r} \]
Solving for \( r \):
\[ r = \frac{1}{4 \pi \epsilon_0} \frac{Z e^2}{K.E.} \]
Substituting values:
\[ r = \frac{(9 \times 10^9) \times (79) \times (1.6 \times 10^{-19})^2}{3.95 \times 10^6 \times 1.6 \times 10^{-13}} \, \text{m} \]
The calculated distance of closest approach is:
\[ r = 28.8 \times 10^{-15} \, \text{m} = 28.8 \, \text{fm} \]
The distance of closest approach is \( 28.8 \times 10^{-15} \, \text{m} \).