Consider a plasma-phase electrothermal thruster like a helicon, shown below. Fuel inlet Quartz tube Pyrex tube Magnetic Field Lines Magnets RF Copper Plasma Helical Sheath power Antenna These thrusters operate on lighter gases, and Argon (MM = 40 g/mol) is a typical propellant. The applied axial magnetic field has a field strength of B = 0.05 T. For the sake of this problem, you may assume that the Pyrex tube has a length of 0.25 m, a radius of 2.5 cm, and that the plasma is fully ionized within the entire volume of the Pyrex tube. You may assume that the ions are moving axially at 10,000 m/s, and their velocity orthogonal to the magnetic field is equal to their thermal speed at T = 500K. Given a plasma density of n = 5 x 1020 m, and a plasma temperature of T = 1.5 eV, determine the following quantities: e A. The electron gyro radius (Le) B. The ion gyro radius (r;); assume the velocity of the ions perpendicular to the magnetic field is the ion thermal velocity C. The Coulomb collision cross section (Qei); assume Ln(1) = 8 D. The electron collision frequency (Vei) (assume fully ionized n 0 m and that electrons don't interact with each other). E. The mean free path for collisions for electrons (e) F. If electrons are moving down the channel with their thermal velocity, how many interactions (collisions) will the electrons undergo with ions? Assume Ce >> Vion. Dimensional arguments: Based on the above calculations answer the following questions: G. At this magnetic field strength, what is the smallest tube diameter that won't impede electron cyclotron motion along the tube centerline? H. At this magnetic field strength, what is the smallest tube diameter that won't impede ion cyclotron motion along the tube centerline? I. J. How long must the tube be in order to ensure that the number of e-l collisions is fewer than 10 collisions (number of collisions is a possible way to characterize plasma thermal relaxation)? What is the maximum plasma (electron) residence time in the tube if we want to design the thruster so that electrons undergo fewer than 10 collisions?