The dielectric breakdown of solids is a problem of great practical and theoretical interest. It is the electrical analog of the fracture of solids under applied loads. In the case of fracture, the reigning theory for linear elastic materials is linear elastic fracture mechanics. This paper develops the analogous theory, linear dielectric-breakdown electrostatics, based on a Griffith-like energy-balance calculation applied to a single conducting crack in an isotropic dielectric medium. Results include the development of the critical field-intensity factor, ${K}_{c}^{e}$, and the introduction of a contour-independent line integral, ${J}^{e}$, which is analogous to the J integral of linear and nonlinear elastic fracture mechanics. Some discussion of the relation between these results and recent lattice models of dielectric breakdown is given.