Fundamental Constants: e = 1.60 Times 10^-19 C - elementary charge K_e = 8.99 Ti

Question

Fundamental Constants: e = 1.60 Times 10^-19 C - elementary charge K_e = 8.99 Times 10^9 N middot m^2/C^2 - Coulomb constant G = 6.67 Times 10^-11 m^3/(kg middot s^2) - gravitational constant m_3 = 9.11 Times 10^-31 kg - electron mass m_p = 1.67 Times 10^-27 kg - proton mass Consider a "classic" model of the hydrogen atom, an electron spins around a proton in a prefect circular orbit of radius r_o = 5.29 Times 10^-11 m (Bohr radius). The electrostatic force F_e between the negatively charged electron and the positively charged proton according to the Third Newton's Law must be equal in magnitude to the centrifugal force F_c = m_3 v^2/r, where m_e is the electron's mass, and upsilon is the electron's speed. Calculate the period of rotation T of the electron (the time that takes for the electron to travel one complete circle of radius r_o). Calculate the magnitude of the electrostatic force F_e between the electron and the proton. Also, since the electron and the proton have mass, they are attracted to each other due to the gravitational force F_g. Calculate the magnitude of the gravitational force F_g. Based on the results, make a conclusion, if the electron anti the proton are held in a hydrogen atom primarily due to gravitational or electrostatic interaction. Calculate the electric potential phi(r_o) on the electron's orbit in a hydrogen atom. Calculate the potential energy E_p(r_o), the kinetic energy E_k(r_o), and the total energy E_tot (r_o) of the electron in a hydrogen atom.

Explanation / Answer

Ke2/ro2 = mv2/ro

v = sqrt(ke2/mro)

T = 2pi ro/sqrt(ke2/mro) = sqrt[ 4*pi2*ro3*m/ke2)

b) Fe = ke2/ro2

  Fg = Gmemp/ro2

here Fe >> Fg

so electrostatic forces are dominating.

[put the values in the results to get the numeral answers]

c) potential = ke/ro

   potential energy = -ke2/ro

kinetic energy = ke2/2ro

total energy = -ke2/2ro

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