Planar wedge disclinations are angular mismatches at the nanoscale in crystal lattices, representing a violation of rotational symmetry. Alongside dislocations, they are observed in crystal plasticity and in Shape-Memory Alloys undergoing austenite-to-martensite phase transitions. We present a novel mathematical formulation for the elastic energy of finite, isotropic systems of planar wedge disclinations and edge dislocations [1], working within the planar strain regime and linearized kinematics. In this framework, disclinations and dislocations emerge as solutions to energy minimization problems under kinematic incompatibility constraints. Following the standard approach for 2D geometries in linear elasticity, we introduce the Airy potential to reformulate the mechanical equilibrium problem. We develop a new variational formulation suited to incompatible elasticity in non-simply connected domains. Our main result establishes the exact energetic equivalence between a disclination dipole and an edge dislocation under appropriate energy rescaling. Using the core-radius regularization method of P. Cermelli and G. Leoni (SIAM SIMA 37, 2005), we prove that as the core radius tends to zero, the renormalized and rescaled energy of a disclination dipole converges to that of an edge dislocation. Finally, we discuss how this framework can be extended to model the dissipative evolution of disclination systems [2] and to study kinematically incompatible elastic plates. These results provide new insights into the mathematical modeling of crystalline defects and their role in the mechanics of complex materials.