The symplectic cohomology of magnetic cotangent bundles

We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced in \cite{Groman2015}. As an application, we deduce that if $N$ is a closed manifold and $\sigma$ is a magnetic form that is not weakly exact, then the ${\pi }_1$-sensitive Hofer-Zehnder capacity of any compact set in the magnetic cotangent bundle determined by $\sigma$ is finite.