Change in Magnetic Flux: Eddy Currents vs Moving Wire

The flux of the magnetic field through any surface whose border lies inside the rod is defined as $$int_Svec{B}cdothat{n}dS$$ where the integral is calculated over the whole surface and $hat{n}$ is the versor perpendicular to the surface (point by point). For example if you consider a rectanguar border and the rectangle as the surface, the flux is simply given by $$int_Svec{B}cdothat{n}dS=vec{B}cdothat{n}cdot A$$ where $A$ is the area of the rectangle and $vec{B}cdothat{n}$ is the same in any point of the rectangle because the field is uniform. For this particular surface the flux is constant. Without actually calculating it but just by looking at the flux definition we can conclude that this is true for any surface with any border that moves with the rod because the field is uniform and if the rod moves perpendicularly to the field, the angle between $vec{B}$ and $hat{n}$ does not change over time.