Orbiting Electrons: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks of Matter
Channel: Zor Shekhtman on UNIZOR Education
Date: 2023-01-20

UNIZOR.COM - Physics through a Looking Glass of Mathematician
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)