In material distribution-based topology optimization, we place material inside a design domain to extremize an objective function. The optimization problem is solved using a gradient-based algorithm. An efficient way to compute the gradients is to use the adjoint method. This study performs the sensitivity analysis of a coupled plasmonic problem using the adjoint method. More precisely, a TE-polarized Helmholtz equation is coupled to a Poisson equation. The sensitivity analysis of the coupled plasmonic problem poses some challenges stemming from the complex solution of the plasmonic problem. Therefore, we first consider a model problem whose structure is similar to the main problem in some ways but is simpler to study. After examining the model problem, we perform the sensitivity analysis of the coupled plasmonic problem, highlighting key differences between the two problems.