The discussion revolves around a vacuum diode consisting of two parallel plates, where electrons are emitted from a cathode and accelerated towards an anode. The problem involves deriving Poisson's equation for the electric potential between the plates and exploring the relationships between charge density, electric field, and velocity of electrons.
The discussion is active, with participants providing insights and guidance on how to approach the problem. There are multiple interpretations being explored, particularly regarding the relationships between charge density, electric field, and potential. Some participants are seeking clarification on the definitions and mathematical relationships involved.
Participants note that the problem constraints include the assumption of steady state and the independence of current with respect to position. There is also an emphasis on deriving relationships without explicit values for certain parameters, such as capacitance.
gabbagabbahey said:Yes, you now have an ODE for V(x) involving only V(x), its derivatives, and some constants; so there is your answer to part (d). For part (e) you are asked to solve this ODE for V(x), to make it easier to work with I recommend you collect all your constants into one, say,
\kappa \equiv \frac{-I}{\epsilon_0 A} \sqrt{\frac{m}{2q}}
\Rightarrow \frac{d^2V}{dx^2}= \kappa V^{-1/2}
Now, try to solve this ODE.