Notes to a video lecture on http://www.unizor.com
Orbiting Electron
Let's analyze the dynamics of an electron rotating around a nucleus of a hydrogen atom on a circular orbit.
Considering the strength of electric forces significantly exceeds the strength of gravitational forces, we will ignore the gravity between a nucleus and an electron.
Assume, the mass of an electron is m (it's a known constant), its negative electric charge is −e (also a known constant) and a radius of its orbit is r (variable).
An atom of hydrogen has only one negatively charged electron. Therefore, to maintain electric neutrality, its nucleus has to have positive electric charge equal in magnitude to a charge of an electron, that is, e.
The Coulomb's Law states that the magnitude of the force of attraction between a nucleus of an atom of hydrogen and its electron equals to
F = k·e·e/r² = k·e²/r²
where k is a Coulomb's constant.
On the other hand, according to Rotational Kinematics, that same force gives an electron a centripetal acceleration
a = v²/r
where v is a linear speed of an electron circulating around a nucleus.
Applying the Newton's Second Law
F = m·a,
we obtain an equation that connects radius of an orbit, linear speed of an electron, its charge and mass:
F = m·a = m·v²/r = k·e²/r²
or
m·v²·r = k·e²
The direct consequence of this equation is an expression for a kinetic energy of an electron, as a function of a radius of its orbit:
Ekin = m·v²/2 = k·e²/(2·r)
Potential energy of a negative charge e in the centrally symmetrical electric field of a nucleus carrying a positive charge e of the same magnitude (that is, work needed to bring a charge −e from infinity to a distance r from a central charge +e) is
Epot = −k·e²/r
Total energy of an electron is, therefore,
E = Ekin + Epot = −k·e²/(2·r)
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