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Asked in GSEB Board March 2022 · Units and dimensions
Given: the combination (me⁴)/(8ε₀²h³c), with m and e the electron's mass and charge.
Idea: this is the Rydberg constant of the Bohr model, so it should come out as a reciprocal length. Work it out factor by factor.
Numerator: M¹×(T¹A¹)⁴=M¹T⁴A⁴; the 8 is a pure number.
Denominator: [ε₀²]=M⁻²L⁻⁶T⁸A⁴, [h³]=M³L⁶T⁻³ and [c]=L¹T⁻¹, whose product is M¹L¹T⁴A⁴.
Dividing: (M¹T⁴A⁴)/(M¹L¹T⁴A⁴)=M⁰L⁻¹T⁰ -- everything cancels but one power of length, and that inverted.
So the dimensions are M⁰L⁻¹T⁰, exactly as the unit m⁻¹ of the Rydberg constant requires.
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