Dimensional analysis - quantum and classical lengthscale ratio

[kapitan90]

Homework Statement

I was supposed to use dimensional analysis to approximate the length scale (in classical and quantum mechanics). The results I got(same as those in the answer sheet): classicaly$$l_c≈\frac{e^2}{4πm_ec^2ε_0}≈2.8*10^-15m$$ In quantum mechanics$$l_q≈\frac{h}{m_ec}≈2.4*10^-12m$$

Homework Equations

The next question is: how would your analysis in case of $$l_q$$have to change if $$l_q≈l_c?$$

The Attempt at a Solution

I have the answer to question b), but I don't think I understand it. The answer they give is:
'If$$l_q≈l_c$$ it would not be possible to estimate l_q without considering electromagnetic effects and write $$l_q=f(\frac{l_c}{l_q})$$ In fact ratio l_c/l_q is is small (apart from a factor of 2π it is just the fine structure constant).'

Could anyone please explain or comment on the given answer?

 

[kapitan90]

Could anyone explain why the electromagnetic effects need to be considered and how it influences the ratio?