This chapter focuses on the behavior of conductors in electrostatic equilibrium and introduces Poisson’s equation ( \(\mathbf {\nabla }^2 V = - \varrho /\epsilon _0\) ) as a powerful mathematical tool for determining the electric potential. It begins by establishing key properties of conductors in static electric fields: the electric field vanishes inside a conductor, all excess charge resides on its surface, and the electric field at the surface is always normal to it. These properties lead to Coulomb’s theorem, which quantifies the discontinuity of the electric field across a conductor’s surface, and the constant nature of the electric potential throughout the conductor and on its surface. The concept of electrostatic energy of a conductor and the electrostatic pressure exerted on its surface are also discussed. The chapter then introduces Poisson’s equation as the governing partial differential equation for the electric potential, which simplifies to Laplace’s equation ( \(\mathbf {\nabla }^2 V =0\) ) in charge-free regions. It emphasizes that solving this equation, subject to appropriate boundary conditions, is the primary method for determining the electric potential and, consequently, the electric field in complex scenarios where charge distributions are not known a priori. Two main types of boundary conditions are detailed: Dirichlet’s boundary condition (specifying potential values on the boundary) and Neumann’s boundary condition (specifying the normal component of the electric field on the boundary). Advanced techniques for solving Poisson’s and Laplace’s equations are presented, including the method of images for problems involving conductors with simple geometries (like infinite grounded planes) and expansion in Legendre polynomials for problems with azimuthal symmetry. Finally, the chapter introduces the concept of Green’s functions as a formal solution to Poisson’s equation with given boundary conditions, providing a general framework for addressing complex electrostatic problems.