Problem/Project Set 1

1. Create a log-log plot of ne vs kTe with ne from 106 to 1025 m−3, and kT , from 0.01 to105 eV. Draw lines of constant λD and ND. On this graph, place the following points(n in m−3, kT in eV):

(a) Typical fusion reactor: n = 1021, kT = 10000

(b) Typical fusion experiment: n = 1019, kT = 3500

(c) Ionosphere: n = 1011, kT = 0.05

(d) Typical glow discharge: n = 1015, kT = 2

(e) Typical flame: n = 1014, kT = 0.1

(f) Typical Cs plasma: n = 1017, kT = 0.2

(g) Interplanetary space: n = 106, kT = 0.01

Verify that these are all plasma

2. Calculate the pressure, in atmospheres, exerted by a thermonuclear plasma on its con-tainer. Assume kTe = kTi = 20keV, n = 1021 m−3 and p = nkT , where T = Te + Ti.

3. For the case of an infinite, transparent grid charged to a potential φ, show that theDebye shielding distance is given by

λ−2d =

nee2

ε0

(1

kTe+

1

kTi

)

(hint: extend the example in class to include the ions)

4. An alterantive derivation of λD will give further insight to its meaning. Consider twoinfinite, parallel planes at x = ±d, set at potential φ = 0. The space between them isuniformly filled by a gas of density n of particles of charge q.

(a) Using Poisson’s equation, show that the potential distribution between the platesis

φ =nq

2ε0(d2 − x2)

(b) Show that for d > λD the energy needed to transport a particle from a plate to themidplane is greater than the average kinetic energy of the particles.