The first author introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,\omega_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(-R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on the number of such trajectories.