  On a compact surface endowed with any $\Spinc$ structure, we give a formula
involving the Energy-Momentum tensor in terms of geometric quantities. A new
proof of a B\"{a}r-type inequality for the eigenvalues of the Dirac operator is
given. The round sphere $\mathbb{S}^2$ with its canonical $\Spinc$ structure
satisfies the limiting case. Finally, we give a spinorial characterization of
immersed surfaces in $\mathbb{S}^2\times \mathbb{R}$ by solutions of the
generalized Killing spinor equation associated with the induced $\Spinc$
structure on $\mathbb{S}^2\times \mathbb{R}$
