 Solution

ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/section-a/4/solution

In SI base dimensions,

[e] = I T, [ϵ_{0}] = I^{2} T^{4} M^{- 1} L^{- 3}, [m] = M, [h] = M L^{2} T^{- 1} .

Equating current, time, mass and length powers respectively yields

α + 2 β = 0, α + 4 β - δ = 0, - β + γ + δ = 0, - 3 β + 2 δ = 1.

Thus

α = - 2, β = 1, γ = - 1, δ = 2, a_{0} \propto \frac{ϵ _{0} h ^{2}}{m e ^{2}} .

Dimensional analysis cannot determine the dimensionless proportionality constant.

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