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Present limit on the electron electric dipole moment ($e$EDM) is based on theelectron spin precession measurement. We propose an alternative approach -observation of the $\mathcal{P}$,$\mathcal{T}$-odd Faraday effect in anexternal electric field on atoms and molecules using cavity-enhancedpolarimetric scheme in combination with molecular (atomic) beam crossing thecavity. Our calculations of the effective electric fields and theoreticalsimulation of the proposed experiment on Tl and Pb atoms, PbF, YbF, ThO, andYbOH show that the present limit on the $e$EDM can be improved by 6-7 orders ofmagnitude.

$\mathcal{P}$,$\mathcal{T}$-odd Faraday rotation on atoms and molecules  in intra-cavity absorption spectroscopy as an alternative way to search for  the $\mathcal{P}$,$\mathcal{T}$-odd effects in nature