Consider a geometric system consisting of two surfaces

Consider a geometric system consisting of two surfaces (perfectly two-dimensional) cylindrical, coaxial and infinite, extended towards the z axis. The inner and outer cylindrical surfaces have, respectively, radii A and B.

On each of the surfaces, a certain amount of charge per unit length (in the direction of the z axis) is evenly distributed so as to remain fixed on the surfaces. On the internal and external surfaces, the loads per unit of length are, respectively, λ_A and λ_B.

Both surfaces are free to print rotations, around the z axis. Assume they have been set to rotate such that the angular velocities of the inner and outer surfaces are constant, necessarily non-zero and are given, respectively, by ω_A and ω_B.

In the space between the two cylindrical surfaces, a test point particle, of charge q and mass m, is placed such that its distance from the z axis is given by s. The point particle is set in motion, with speed exclusively in the direction azimuthal, constant and magnitude v.

Obtain, exclusively in terms of the parameters provided, the necessary condition so that the point particle remains in uniform circular motion between the two cylindrical surfaces.