For a hydrogen atom, an electron (e1, with charge -e and mass m) is in a circula

Question

For a hydrogen atom, an electron (e1, with charge -e and mass m) is in a circular orbit of radius r_1 around a proton (p_1, with charge +e and mass M) which is at rest and fixed in position. The Coulomb force (for our purposes) is what keeps the electron in a circular orbit.

A) Show that e_1’s kinetic energy is equal to minus one-half its potential energy (T = -½U) and calculate the total energy in terms of the potential. (10 points)

From very far away a second electron (e_2, with charge -e and mass m) approaches the hydrogen atom with a known kinetic energy T_2. e_2 collides with the hydrogen atom in an in-elastic collision causing
e_1 to be knocked free and e_2 to be captured in a circular orbit of radius r_2
B) Find the total energy of the three particles (hint: there should be two kinetic energy terms and three potential energy terms) (5 points)
C) Find the total energy of the system well before the collision (when e_2 is far away) (5 points)
D) Find the total energy of the system well after the collision (when e_1 is far away) (5 points)
E) Find the kinetic energy of e_1 when it is far away from the collision in terms of T_2, r_1, and r_2. (10 points)

Explanation / Answer

A) the electron e1 is going round in an orbit of radius r1

around the nucleus there is a Coulmb field E = ke/r , due to the charge on the proton(e)

the culomb force is equal to the centrifugal force of the orbit

ke2/r12 = mv2/r1

KE of the electron K = mv2/2 = ke2/2r1

Potential energy of the electron in the culomb field U =(E*e) = -ke2/r1

K = -U/2

Total energy of the atom = K+U = -U/2 +U = -U/2

B) kinetic energy of the far-away elctron before collison = T2

The incoming electron occupies oribit of radius r2

kinetic energy of the new-electorn in the oribit r2

                    K2 = ke2/2r2

potential energy U2 = -ke2/r2

Total energy of the electron in the orbit r2 E = K2 + U2 = -ke2/r2

When the electron e1 is knocked out from the atom its potential energy =0

Kienetic energy = T2 - (ke2/r2 - ke2/r1 )

The atom is taken from a state of -ke2/r1 to -ke2/r1 , Part of the energy of the incoming electron is used to take the atom from lower energy state to the higher one and the remaining energy is given to the electron as KE.

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