Question

You have a great summer research position in a research laboratory working on generating power from fusion. You are designing a device that confines a hot gas of positively charged ions, called plasma, in a very long cylinder with a radius of 2.0 cm. The charge density of the plasma in the cylinder is 6.0 × 10−5 C/m3. Positively charged Tritium ions are to be injected into the plasma perpendicular to the axis of the cylinder in a direction toward the center of the cylinder. Tritium is an isotope of Hydrogen with one proton and two neutrons. Your job is to determine the speed that a Tritium ion should have when it enters the plasma cylinder so that its velocity is zero when it reaches the axis of the cylinder.

Answer 1

The plasma cylinder is a uniformly charged cylinder with charge density = 6.0e-5 C/cu.m

consider a cylindrical Guassian surface of length l at a distance r from the center

E is the electric field

charge enclosed inside the Guassian surface Q_enc = r² l

Flux through the surface = E2rl

using Guass law

E2rl = Q_enc / = r² l/

Electric field at r   E(r) = r/

Force on a charge q   F = Eq

work done voer a distance r by the aprticle dW = Fdr = q/ rdr

By the time the aprticle reaches the center the KE of the particle =0

initial KE = mv²/2 , this must be the work done for bringing the particle to the center

   W =

         = (6.0e-5*1.6e-19/ 8.54e-12) 0.02²/2 ( charge on tritium = 1.60e-19 C)

          = 2.25e-16

= mv²/2

mass of trituim = 5.01 e-27 kg

v² = 2*2.25e-16/5.01e-27

speed of tritium at entry v = 3.0e-6 m/s