We study Maxwell's equation as a theory for smooth k-forms on globally hyperbolic spacetimes with timelike boundary as defined by Aké, Flores and Sanchez. In particular we start by investigating on these backgrounds the D'Alembert - de Rham wave operator and we highlight the boundary conditions which yield a Green's formula. Subsequently, we characterize the space of solutions of the associated initial and boundary value problem under the assumption that advanced and retarded Green operators do exist. This hypothesis is proven to be verified by a large class of boundary conditions using the method of boundary triples and under the additional assumption that the underlying spacetime is ultrastatic. Subsequently we focus on the Maxwell operator. First we construct the boundary conditions which entail a Green's formula for such operator and then we highlight two distinguished cases, dubbed
δd-tangential and δd-normal boundary conditions. Associated to these we introduce two different notions of gauge equivalence and we prove that in both cases, every equivalence class admits a coclosed representative. We use this property and the analysis of the operator □ to construct and to classify the space of gauge equivalence classes of solutions of the Maxwell's equations with the prescribed boundary conditions.