Dimension of Rydberg Constant

Theorem

The Rydberg constant has the dimension $\mathsf {L^{-1} }$.

Proof

By definition, the Rydberg constant is:

$R_\infty = \dfrac {m_\E \E^4} {8 \varepsilon_0^2 h^3 c}$

where:

$m_\E$ denotes the electron rest mass

$\E$ denotes the elementary charge

$\varepsilon_0$ denotes the vacuum permittivity

$h$ denotes Planck's constant

$c$ denotes the speed of light.

We have:

\(\ds m_\E\)

\(\text {has dimension}\)

\(\ds \mathsf M\)

Definition of Mass of Electron

\(\ds \E\)

\(\text {has dimension}\)

\(\ds \mathsf {I T}\)

Definition of Elementary Charge

\(\ds \varepsilon_0\)

\(\text {has dimension}\)

\(\ds \mathsf {M^{-1} L^{-3} T^4 I^2}\)

Definition of Vacuum Permittivity

\(\ds h\)

\(\text {has dimension}\)

\(\ds \mathsf {M L^2 T^{-1} }\)

Definition of Planck's Constant

\(\ds c\)

\(\text {has dimension}\)

\(\ds \mathsf {L T^{-1} }\)

Definition of Speed of Light

\(\ds \leadsto \ \ \)

\(\ds R_\infty\)

\(\text {has dimension}\)

\(\ds \dfrac {\mathsf M \cdot \paren {\mathsf {I T} }^4} {\paren {\mathsf {M^{-1} L^{-3} T^4 I^2} }^2 \cdot \paren {\mathsf {M L^2 T^{-1} } }^3 \cdot \mathsf {L T^{-1} } }\)

\(\ds \)

\(=\)

\(\ds \dfrac {\mathsf {M I^4 T^4 } } {\mathsf {M^{-2} L^{-6} T^8 I^4} \cdot \mathsf {M^3 L^6 T^{-3} } \cdot \mathsf {L T^{-1} } }\)

\(\ds \)

\(=\)

\(\ds \dfrac {\mathsf {M I^4 T^4 } } {\mathsf {M L T^4 I^4} }\)

\(\ds \)

\(=\)

\(\ds \mathsf {L^{-1} }\)

$\blacksquare$