Physics quiz question: fine structure constant units

Physics 205A Quiz 1, fall semester 2016
Cuesta College, San Luis Obispo, CA

The Rydberg constant R_∞ (measured in units of m^–1) is a physical constant used in spectroscopy[*]:

R_∞ = α²·m_e·c/(2·h).

The electron mass m_e is measured in units of kg, the speed of light c is measured in units of m·s^–1, and the Planck constant h is measured in units of kg·m²·s^–1. The units for the constant α are:
(A) kg^1/2·m^3/2.
(B) m²·s^–1.
(C) kg·m·s.
(D) (No units, as it is a dimensionless quantity.)

[*] wki.pe/Rydberg_constant#Alternative_expressions.

Correct answer (highlight to unhide): (D)

The constant α can be expressed as:

α = sqrt(2·h·R_∞/(m_e·c)),

and replacing the quantities in the equation with their kg, m and s units, the units of α are:

α = sqrt((kg·m²·s^–1)·(m^–1)/((kg)·(m·s^–1))),

α = sqrt((kg·m·s^–1)/(kg·m·s^–1)),

α = sqrt(1);

thus making α a dimensionless quantity.

Sections 70854, 70855, 73320
Exam code: quiz01bIR6
(A) : 11 students
(B) : 14 students
(C) : 6 students
(D) : 26 students

Success level: 46%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.50