Fundamentals of Plasma Physics

1.13 Assignments 29

that the cathode potential is zero, in which case the anode potential isV 0. Assuming
that electrons leave the cathode with zero velocity, show that the electron velocity as
a function of position is given by

v(x) =

√

2 eV(x)

me

.

Show that the above two equations, plus Poisson’s equation, can be combined to give
the following differential equation for the potential

d^2 V

dx^2

−λV−^1 /^2 = 0

whereλ=ǫ− 01 |J|

√

me/ 2 e. By multiplying this equation with the integrating factor
dV/dxand using the space charge limited boundary condition thatE= 0atx= 0,
solve forV(x).By rearranging the expression forV(x)show that the space charge
limited current is

J=

4

9

ǫ 0

√

2 e

me

V^3 /^2

d^2

.

This is called the Child-Langmuir space charge limited current. For reference the
temperature limited current is given by the Richardson-Dushman law,

J=AT^2 e−φ^0 /κT

where coefficientAand thework functionφ 0 are properties of the cathode mate-
rial, whileTis the cathode temperature. Thus, the actual cathode current will be
whichever is the smaller of the above two expressions. Show there is a close relation-
ship between the physics underlying the Child-Langmuir law and Debye shielding
(hint-characterize the electron velocity as being a thermal velocity and its energy as
being a thermal energy, show that the inter-electrode spacing correspondsto ?). Sup-
pose that a cathode was operating in the space charge limited regime and thatsome
positively charged ions were placed in the inter-electrode region. What would happen
to the space charge-would it be possible to draw more or less current from thecath-
ode? Suppose the entire inter-electrode region were filled with plasma with electron
temperatureTe.What would be the appropriate value ofdand how much current could
be drawn from the cathode (assuming it were sufficiently hot)? Does this giveyou any
ideas on why high current switch tubes (called ignitrons) use plasma to conduct the
current?