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We prove boundedness and polynomial decay statements for solutions to thespin $\pm1$ Teukolsky-type equation projected to the $\ell=1$ sphericalharmonic on Reissner-Nordstr\"om spacetime. The equation is verified by agauge-invariant quantity which we identify and which involves theelectromagnetic and curvature tensor. This gives a first description inphysical space of gauge-invariant quantities transporting the electromagneticradiation in perturbations of a charged black hole.  The proof is based on the use of derived quantities, introduced in previousworks on linear stability of Schwarzschild by Dafermos-Holzegel-Rodnianski. Thederived quantity verifies a Fackerell-Ipser-type equation, with right hand sidevanishing at the $\ell=1$ spherical harmonics. The boundedness and decay forthe projection to the $\ell\geq 2$ spherical harmonics are implied by theboundedness and decay for the Teukolsky system of spin $\pm2$ obtained in ourprevious work.  The spin $\pm1$ Teukolsky-type equation is verified by the curvature andelectromagnetic components of a gravitational and electromagnetic perturbationof the Reissner-Nordstr\"om spacetime. Consequently, together with theestimates obtained in our previous work, these bounds allow to prove the fulllinear stability of Reissner-Nordstr\"om metric for small charge to coupledgravitational and electromagnetic perturbations.

Boundedness and decay for the Teukolsky equation of spin $\pm1$ on Reissner-Nordstr\"om spacetime: the $\ell=1$ spherical mode