Discussion
Which one of the following is a dimensionless constant?
Answer: D
The dimensionless constant is the fine structure constant.
\[ \begin{aligned} \alpha &= \frac{e^{2}}{4\pi\epsilon_{0}\hbar c} \\ &\approx \frac{1}{137} \end{aligned} \]
The units of \(e^{2}/4\pi\epsilon_{0}\) (energy times length) cancel exactly with those of \(\hbar c\), so \(\alpha\) is a pure number and has the same value in every system of units.
The permittivity of free space has units F/m, the permeability of free space H/m (or N/A2), and the Bohr magneton, \(e\hbar/2m_{e}\), J/T. All three carry dimensions.
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