# Critical Helical Convergence

Consider the following setup which consists of a semi-infinite solenoid (in which a constant current $$I$$ is flowing )connected end to end and coaxially to a double cylindrical set up. The double cylindrical set up consists of two coaxial fixed cylinders of radius r with a very small radii difference. In between the gap of the two cylinders a particle is given a certain horizontal velocity $$V_{1_{0}}$$ as well as some tangential velocity $$V_{2_{0}}=1kms^{-1}$$ at time t=0. The number of turns density is n. What is the maximum horizontal velocity in $$km/s$$ given so that the particle returns to its original position?

## Details and assumptions:

• The cylinder is also infinitely long
• There is no friction or no collisions between the cylinder and particle.
• $$r=0.001m$$
• $$I=10^{7}A$$
• $$n=10^{3}m^-1$$
• $$\frac{q}{m}=10^{3}Ckg^{-1}$$
• $$\mu_{o}=4\pi10^{-7}m kg s^{-2} A^{-2}$$
×