The physical analysis of a uniformly accelerating point charge provides a rich problem to explore in advanced courses in electrodynamics and relativity since it brings together fundamental concepts in relation to electromagnetic radiation, Einstein's equivalence principle and the inertial mass of field energy in ways that reveal subtleties in each of these concepts. By first exploring the heuristic value of Maxwell's and Faraday's idea that the electromagnetic field is "like" a stressed material medium, it is shown in this paper that the problem also provides an interesting application of the electromagnetic stress-energy-momentum tensor and that an analysis using this tensor provides clear physical insight into this highly subtle and contentious problem and the so-called "4/3 problem" of classical electromagnetic theory. In particular, it is shown that the stress force on a uniformly accelerating, uniformly charged spherical shell due to its own field is simply the (relativistic) inertial mass of the charge's electrostatic field times the acceleration. Since the inertial mass of the electromagnetic field forms part of the observed rest mass of a charged particle, it is argued that the results are therefore consistent with the Lorentz-Abraham-Dirac equation of motion for an accelerating point charge, which implies that for uniform acceleration, the work done by the force acting on the charge only goes into increasing the kinetic energy of the charge, none goes into the creation of radiation. (Contains 4 figures.)