The discussion centers on solving a conductor problem involving a vacuum diode with two parallel plates, where electrons are emitted from a cathode and accelerated towards an anode. The participants derive Poisson's equation, {\nabla}^2V=\frac{\rho}{{\epsilon}_0}, and explore the relationship between charge density, current, and electric field. They establish that in steady state, the current is independent of position, leading to a differential equation for the potential V(x) that incorporates constants and the charge density. The final equation derived is \frac{d^2V}{dx^2}= \kappa V^{-1/2}, where \kappa is a constant defined as \kappa \equiv \frac{-I}{\epsilon_0 A} \sqrt{\frac{m}{2q}}.
Students and professionals in electrical engineering, physicists studying electromagnetism, and anyone interested in the behavior of vacuum diodes and electrostatic fields.
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.