Dimension of Rydberg Constant
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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 | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds R_\infty\) | \(\text {has dimension}\) | \(\ds \dfrac {\mathsf M \cdot \paren {\mathsf {I T} }^4} {\paren {\mathsf {M^{-1} L^{-3} T^4 I^2} }^2 \cdot \paren {\mathsf {M L^2 T^{-1} } }^3 \cdot \mathsf {L T^{-1} } }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\mathsf {M I^4 T^4 } } {\mathsf {M^{-2} L^{-6} T^8 I^4} \cdot \mathsf {M^3 L^6 T^{-3} } \cdot \mathsf {L T^{-1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\mathsf {M I^4 T^4 } } {\mathsf {M L T^4 I^4} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathsf {L^{-1} }\) |
$\blacksquare$