Fundamentals of Plasma Physics

28 Chapter 1. Basic concepts

cathode anode

space charge

d

V

− 

electrons emitted

from cathode surface

Figure 1.5: Electron cloud accelerated from cathode to anode encounters space charge of
previously emitted electrons.

This configuration can operate in two regimes: (i) thetemperature limitedregime

where the current is determined by the thermionic emission capability of the cathode,

and (ii) thespace charge limitedregime, where the current is determined by a buildup

of electron density in the region between cathode and anode (inter-electrode region).

Let us now discuss this space charge limited regime: If the current is small then the

number of electrons required to carry the current is small and so the inter-electrode

region is nearly vacuum in which case the electric field in this region will be nearly

uniform and be given byE=V/dwhereVis the anode-cathode potential difference

anddis the anode cathode separation. This electric field will accelerate the electrons

from anode to cathode. However, if the current is large, there will be a significant

electron density in the inter-electrode region. This space charge will create a localized

depression in the potential (since electrons have negative charge). Theresult is that

the electric field will be reduced in the region near the cathode. If the space charge is

sufficiently large, the electric field at the cathode vanishes. In this situation attempting

to increase the current by increasing the number of electrons ejected by the cathode

will not succeed because an increase in current (which will give an increase in space

charge) will produce a repulsive electric field which will prevent the additional elec-

trons from leaving the cathode. Let us now calculate the space charge limited current

and relate it to our discussion on Debye shielding. The current density in thissystem

is

J=−n(x)ev(x) =a negative constant

Since potential is undefined with respect to a constant, let us choose this constant so