  Calculations of the cross sections of inelastic electron scattering $(e,e')$
on a nucleus in the multi-keV energy range strongly depend on the description
of the screening of the nuclear Coulomb potential as well as on the deformation
of the wave functions of the incoming and outgoing electron in the vicinity of
the nucleus. These cross sections are evaluated at values lower than $10^{-30}$
cm$^2$ which vary by several orders of magnitude according to the models.
Experimental measurements would be required to constrain the models but it is a
real challenge to measure such low cross sections. In this study, we
demonstrate that inelastic electron scattering is the main nuclear excitation
mechanism in a $^{181}$Ta target irradiated with a new intense 10 - 30 keV
electron beam produced with a biased laser-plasma. Calculations show that
through the detection of conversion electrons, it should be possible to measure
the nuclear excitation yields. The effect of electron beam heating and of
plasma deposition on the tantalum target are quantified, thus allowing the
dimensioning of a possible experimental configuration to study $(e,e')$
processes in this range of energy for the first time.
