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Bohr Atom
----------
Niels Bohr introduced the first quantized model of the atom in 1913, in an attempt to
overcome a major shortcoming of Rutherford's classical model. In classical
electrodynamics, a charge moving in a circle should radiate electromagnetic
radiation*. If that charge were to be an electron orbiting a nucleus, the radiation
would cause it to lose energy and spiral down into the nucleus. Bohr solved this
paradox with explicit reference to Planck's work: an electron in a Bohr atom could
only have certain defined energies E_{n}. Once the electron reached the lowest energy
level (n = 1), it could not get any closer to the nucleus (lower energy). This approach
also allowed Bohr to account for the Rydberg formula, an empirical description of the
atomic spectrum of hydrogen, and to account for the value of the Rydberg constant in
terms of other fundamental constants.
Bohr also introduced the quantity h/2π, now known as the reduced Planck constant, as
the quantum of angular momentum. At first, Bohr thought that this was the angular
momentum of each electron in an atom: this proved incorrect and, despite developments
by Sommerfeld and others, an accurate description of the electron angular momentum proved
beyond the Bohr model. The correct quantization rules for electrons - in which the energy
reduces to the Bohr-model equation in the case of the hydrogen atom - were given by
Heisenberg's matrix mechanics in 1925 and the Schrodinger wave equation in 1926.
* Classically, any charged particle that is accelerated will emit electromagnetic
radiation. This can be shown by solving Maxwell's equations using retarded potentials.
However, J.J. Thomson had a simpler approach to prove this. Consider:
E_{r} = q/r^{2} (electric field is electric force per unit charge by definition)
/ /
/ /
/θ /
.q->.q
Δvt
From geometry, the ratio of strengths E_{p}/E_{r} is given by:
/^
/ |E_{p} = Δvtsinθ
/ |
/θ |
---> E_{r}
E_{p}/E_{r} = Δvtsinθ/cΔt where E_{p} is field perpendicular to radial field
and θ is the angle between the acceleration vector and the line from the charge
to the observer.
E_{p} = E_{r}Δvtsinθ/cΔt
= (qt/r^{2}c)sinθ(Δv/Δt) where r is the distance from the charge.
= (qsinθ/c^{2}r)(Δv/Δt) since t = r/c
From the observer's point of view, only the visible acceleration perpendicular to the line
of sight contributes to the radiated electric field; the invisible component of acceleration
parallel to the line of sight does not radiate - what you see is what you get.
For a non-relativistic circular orbit, the acceleration is just the centripetal
acceleration, v^{2}/r.