Plasma Confinement in Cylindrical Column · RMPH 2016 #3
Physics · electromagnetism
Problem
A cylindrical column of plasma having the radius R and the length L is generated in a vacuum chamber. The plasma appears as result of ionization of a gas, such that the concentrations of electrons ne(r) and ions ni(r) are equal at every point ni(r)=ne(r)=n(r); the common value n(r) depends only on the distance between the point and the axis of symmetry of the plasma cylinder. It is assumed that the plasma is in a stationary state, and therefore all its macroscopic characteristics are independent on time. It may be admitted that the temperature T of plasma is the same at every point of the column and that at these temperature the parameters describing plasma abide the perfect gas law. The electric charge of electron is −e and the ions are monovalent, carrying an electric charge +e. Consider as known the magnetic permeability of plasma μ and Boltzmann's constant kB. Between the electrodes at the ends of the plasma column passes through the plasma an electrical current characterized by a constant density j(r)=j.
Consider an elementary portion of the hollow cylinder having the radiuses r and r+dr as in joined figure. Elementary portion has a height equal to the unit. In the annulus (circular crown) representing the cross section of the plasma column the elementary portion covers the angle dϕ.
[Figure: Elementary hollow cylindrical portion of plasma with inner radius r, outer radius r+dr, height 1, and angular span dϕ]
3.i. Write the expressions of the forces acting on this elementary volume of plasma (Fp due to pressure p(r) in the column of plasma, Fe due to the interaction with electrical charges carriers that there are in plasma, Fm due to the interaction with magnetic field of electrical current flowing through plasma). Write the equation describing the equilibrium of the considered elementary portion of plasma. (1.20p)
3.ii. Deduce the expression of pressure p(r) in a point of the column of plasma. Express the answer as function of r, j, R and magnetic permeability μ. (1.00p)
3.iii. Determine the expression of the number of particles N carrying electrical charge in the column of plasma as function of L, I, T, μ and kB. (0.80p)