Solving Poisson's Equation: Child-Langmuir Law

As a potential difference is applied to two plates, electrons are "boiled" off of the hot cathode (V=0 plate) at essentially vanishing small speed and are then attracted towards the anode. A cloud of electrons (space charge) builds up as current flows in the space. How does the current depend on the applied potential difference? Does it satisfy Ohm's Law? Figure 1



QUESTIONS

(Score is number right minus number wrong.)

The electric field created by the positive potential difference created by the plates alone would accelerate electrons
right-to-left.
left-to-right.
Because of the electrons in the space between the electrodes (space charge), the electric field created by the electrodes alone is
weakened.
strengthened.
The system will seek a "steady state" such that the electric field near the cathode will neither act to enhance nor diminish the space charge, i.e., the electric field near the cathode will vanish.
True
False
For large plates and small separation, we can probably ignore fringe effects near the edges of the plates.
True
False
Between the plates, neither the electron speed v(x), nor the potential V(x) nor the space charge density are likely to depend on position x.
True
False
At steady state, the current I between the plates can be expected to be constant.
True
False
The current I between the plates is given by
Equation (4a).
Equation (4b).
Equation (4c).
Equation (4d).
Equation (4e).
Equation (4f).
Although the electrons accelerate, the total energy W of the electrons should be constant in the space between the electrodes (barring collisions).
True
False
When steady state is achieved, the energy of the electrons at x=0 should essentially vanish.
True
False
The speed of the electrons, v(x) is given by
Equation (4a).
Equation (4b).
Equation (4c).
Equation (4d).
Equation (4e).
Equation (4f).
For electrons, -q is a
positive number.
negative number.
Poisson's equation takes the form,
Equation (5a).
Equation (5b).
The coefficient beta in Poisson's equations (Equation (5)) is given by
Equation (4a).
Equation (4b).
Equation (4c).
Equation (4d).
Equation (4e).
Equation (4f).
Poisson's equation is a nonlinear differential equation. Straightforward separation of variables doesn't work. See the transmogrification of Equation (6).

Equations (6a)-(6d) are all correct.
True
False
The constant of integration in Equation (6d) vanishes because both V and E=dV/dx vanish at
Equation (7a).
Equation (7b).
Equation (7c).
Separation of variables can be applied directly to Equation (6d). The constant of integration again vanishes and the potential is given by
Equation (8a)
Equation (8b)
Equation (8c)
Equation (8d)
Equation (8e)
Since V is known at x=d and since the constant beta harbors a factor of I, we conclude that the relationship between current and potential difference is given by
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
Equation (9e)
EQUATIONS

Equations


Equations









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