  Systematic differences in the the proton's charge radius, as determined by
ordinary atoms and muonic atoms, have caused a resurgence of interest in
elastic lepton scattering measurements. The proton's charge radius, defined as
the slope of the charge form factor at Q$^2$=0, does not depend on the probe.
Any difference in the apparent size of the proton, when determined from
ordinary versus muonic hydrogen, could point to new physics or need for the
higher order corrections. While recent measurements seem to now be in
agreement, there is to date no high precision elastic scattering data with both
electrons and positrons. A high precision proton radius measurement could be
performed in Hall B at Jefferson Lab with a positron beam and the calorimeter
based setup of the PRad experiment. This measurement could also be extended to
deuterons where a similar discrepancy has been observed between the muonic and
electronic determination of deuteron charge radius. A new, high precision
measurement with positrons, when viewed alongside electron scattering
measurements and the forthcoming MUSE muon scattering measurement, could help
provide new insights into the origins of the proton radius puzzle, and also
provide new experimental constraints on radiative correction calculations.
