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Physics II, 4/19/99

Energy Levels of the Bohr Atom; Revisiting Induced Charge


Energy Levels of the Bohr Atom

Setting the expression for the Coulomb force equal to the expression for the centripetal force and applying the condition of quantization on the angular momentum, we again obtain the expressions for allowed orbital radii rn and orbital velocities vn, where n = 1, 2, 3, .... We obtain the expressions for the potential and kinetic energies -k qe / rn and .5 m vn^2 and for the total energy of the orbiting electron. We find that the total energy is En = - 1 / n^2 * [ `me k^2 qe^4 / `hBar^2 ], which is approximately equal to - 1 / n^2 * 13.6 eV. We represent these energy levels by a schematic diagram.

Electrostatic Induction Revisited

We consider the charging of the ends of an initially uncharged aluminum rod when a charged PVC rod is brought near the center of the aluminum rod, and the resultant 'jumping away' of an initially uncharged conductor when it touches one of these ends.  These phenomena are easily understood in terms of the free migration of charge over the surface of a conductor.


Energy Levels of the Bohr Atom

As seen in the preceding class, when an electron with charge -qe orbits a proton with charge +qe at a distance r, its velocity v is related to r by equating the Coulomb's Law force and a centripetal force on the electron.

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The velocity v associated with the allowed orbital radius rn = n^2 [ `hBar^2 / ( k qe^2 `me) ] is vn = k qe^2 / ( n `hBar ).

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The total energy is thus equal in magnitude to the kinetic energy, but due to the greater magnitude of the (negative) potential energy the total energy is negative.

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Video Clip #01

The picture at the top of the figure below depicts an electron being accelerated from rest at the negative plate of a capacitor to the positive plate. The potential difference between the plates is 1 volt, so the electron to choose a kinetic energy of 1 eV. We note that the final velocity of this electron with be approximately 10^6 m/s.

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Video Clip #02

Electrostatic Induction Revisited

We observed experimentally that when a charged PVC rod is brought near the denter of an aluminum rod, the ends of the aluminum rod have the same effect on charged objects as would the PVC rod.

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It is also observed that when an uncharged conductor is brought into contact with the charged end of the aluminum rod, the conductor will tend to quickly 'jump' away from the aluminum rod.

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Video Clip #03

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