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.
  • \(r=0.001m\)
  • \(I=10^{7}A\)
  • \(n=10^{3}m^-1 \)
  • \(\frac{q}{m}=10^{3}Ckg^{-1}\)
  • \(\mu_{o}=4\pi10^{-7}m kg s^{-2} A^{-2}\)

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