In the Bohr model

PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Modern Physics · Modern Physics

Problem

In the Bohr model, electrons occupy discrete circular orbits about a nucleus of charge

+Ze

. The standing-wave constraint requires an integer number

n

of de Broglie wavelengths to fit exactly around the circumference of the orbit. (a) Starting from first principles, derive an expression for the allowed radii

r_{n}

of the electron orbits in terms of the quantum number

n

, the nuclear charge number

Z

, the electron mass

m_{e}

, the elementary charge

e

, Planck’s constant

h

, and the vacuum permittivity

\varepsilon_{0}

. Show all reasoning clearly. (b) Using your result from part (a), calculate the radius of the

n=3

orbit for the singly-ionized helium ion (He

^{+}

). Use

a_{0}=5.29\times10^{-11}\,\text{m}

for numerical evaluation. (c) Determine the electron’s speed in this

n=3

orbit of He

^{+}

, expressing the answer in metres per second. (d) Compute the ratio of this speed to the speed of light in vacuum (

c=3.00\times10^{8}\,\text{m\,s}^{-1}

) and comment on whether relativistic effects are expected to be significant for the electron in this state.

Related theory

Browse the problem library