In this paper we present a fully discrete $A$-$ϕ$ finite element method to solve Maxwell's equations with a nonlinear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of the electric field $E$ and the magnetic field $H$ obeys a power-law nonlinearity of the type $H × n = n × (|E × n|^{α-1}E × n)$ with $α ∈ (0,1]$. We prove the existence and uniqueness of the solutions of the proposed $A$-$ϕ$ scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.