Bohr Model of Hydrogen Atom?

Question

Consider an electron with charge -e and mass m orbiting in a circle around a hydrogen nucleus (a single proton) with charge +e. In the classical model, the electron orbits around the nucleus, being held in orbit by the electromagnetic interaction between itself and the protons in the nucleus, much like planets orbit around the sun, being held in orbit by their gravitational interaction. When the electron is in a circular orbit, it must meet the condition for circular motion: The magnitude of the net force toward the center, Fe, is equal to mv2/r. Given these two pieces of information, deduce the velocity v, of the electron as it orbits around the nucleus.

Express your answer in terms of e, m, r, and E0, the permittivity of free space.

Answer 1

Equate teh two forces:

Fe = Fc

Fe = -q1*q2/(4pi*r^2)   Fc = mv^2/r

e^2/(4pi*E0*r^2) = mv^2/r

Solve for v:

v^2 = e^2/(4pi*E0*r*m)

v = e/sqrt(4pi*E0*r*m)

Answer 2

If mass of electron   =   m

electron velocity   =   v

radius of orbit   =   r

charge on electron, and proton    =   e

permittivity of free space   =   eo

Force required to make electron move in a circle

                           F     =     m * v^2 / r

Electrostatic attracting force between proton and neutron:-

                           F     =    (charge1)  *  (charge 2) /  4 * pi * eo * r^2

The two forces are the same, so

                   m * v^2 / r     =     e * e / 4 * pi * eo * r

so,     m * v^2     =     e * e / 4 * pi * eo

    

                v^2     =     e^2 / 4 * pi * m * eo * r

    velocity of electron:-

                                     v     =     e / SQR(4*pi*m*eo*r)