  The detection of weak forces is a central problem in physics and engineering,
ranging in importance from fundamental pursuits such as precision tests of
gravity, gravitational-wave detection, and searches for dark matter, to
applications such as force microscopy. These pursuits require a low-mass
mechanical force transducer with a high quality factor, whose motion can be
measured in a quantum-noise-limited manner. Here we study the ultimate example
of such a transducer: a single trapped electron. We propose and analyze in
detail a new scheme for high-sensitivity continuous force detection using a
single trapped electron whose motion is coupled to a microwave cavity field via
image currents induced in an antenna. We derive the fundamental and technical
limits to the sensitivity of this scheme and show that despite the disparity in
size between that of a single electron and the wavelength of the microwave
field, it is possible to continuously monitor the charge's zero-point motion
and use it as a force detector with a sensitivity as low as $6\times 10^{-27}\,
\mathrm{N}/ \sqrt{\mathrm{Hz}}$ in the gigahertz regime. This sensitivity
improves on the state-of-the-art by four orders of magnitude and thus paves the
way to novel precision experiments.
