Answer this question please:

(30 pts.) Let us evaluate an arc discharge plasma column (ions and electrons) that is conducting a discharge current of 30 A in a vacuum chamber. Assume that the plasma column has a radius of Ry, the current density is uniform in this column and the current density is zero outside of the plasma column, i.e., j, = j for r Hy. (a) (b) () (8 pts.) Derive the induced magnetic field as a function of the radius, sketeh the profile. (4 pts.) If the plasma pressure (p = p; +p.) is initially constant, in which direction will the Lorentz force act? (8 pts.) After some time, the arc plasma is at equilibrium. Assume that the plasma column has a radius of R, the current density can be written as, j. = jp (7o is constant) for r R. Derive the plasma pressure, p, as a function of the radius, and sketch the profile. Consider that the Lorentz force balances the pressure gradient in the radial direction inside the arc column at equilibrium. (10 pts.) Assuming that the plasma is fully ionized in the arc column and p = 100 Torr at the center of the arc column, estimate the radius and the current density of the arc column