Plotting a vector field with polar coordinates on a ring

Assuming I have a vector components: A_r(r,theta) and A_t(r,theta) so A=(A_r,A_t)
I would like to show the Vector A on a ring. for some reason when I’m using VectorPlot it has to be in cartesian coordinates. So I have define A_x and A_y from A(r,t) and tried to use VectorPlot yet it shows nothing.
the line:

VectorPlot({ar(r, t), at(r, t)}, {r, R1, R2}, {t, 0, 2pi})

shows a vector field, yet it is not on a ring.
when I’m trying to use the cartesian coordinates:

VectorPlot({FullSimplify(ax(x, y)), FullSimplify(ay(x, y))}, {x, -R2, R2}, {y, -R2, R2}, RegionFunction -> Function({x, y}, R1^2 < x^2 + y^2 < R2^2))

It shows nothing.
I have made the transformation by



For some reasons it does not work,
Any help?