I Shooting an electron past a positive nucleus (trajectory)

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When an electron is shot horizontally past a proton, the trajectory can be analyzed as a two-body problem, though it simplifies to a one-body problem with the proton fixed. The distance of closest approach can be derived using conservation of energy and angular momentum principles. The relationship between scattering angle and trajectory can be expressed using hyperbolic parameters, specifically the formulas L² = γb²/a and E = γ/2a, where γ = e²/m. For further understanding, references to Rutherford scattering provide additional insights into the behavior of charged particles in such interactions. This analysis highlights the importance of classical mechanics in understanding particle trajectories in electromagnetic fields.
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An electron is shot horizontally. There is a proton located somewhere else, but not in the horizontal path of the electron. Is there a distance of closest approach, and how do you derive it? A physical explanation would be appreciated too. Feel free to use any variables.
An electron is shot horizontally. There is a proton located somewhere else, but not in the horizontal path of the electron. Is there a distance of closest approach, and how do you derive it? A physical explanation would be appreciated too.
 
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It's straightforward as a one-body problem but a little more subtle as a two-body problem. For the one-body (fixed proton) scenario it's enough to conserve energy and angular momentum about the proton. If you want the scattering angle/trajectory then you can relate these to the hyperbolic parameters ##a## and ##b## with the standard formulae ##L^2 = \gamma b^2 / a## and ##E = \gamma / 2a## [where ##\gamma = e^2/m##]; the proton is at the far focus.
 
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You could search for Rutherford scattering. There's quite a bit online about this.
 
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