[HW4 Q2.2/2.3]
Summary
Initial boundary conditions / Implementation of \dfrac{d\psi (x=h/2)}{dx}=0.
Details
Q2.2: I solved the Poisson-Boltzmann equation with initial boundary conditions for \psi (x=0)=\psi(0) and \dfrac{d\psi (x=0)}{dx} being my guessing parameter and it works well. I then looked for a value of \dfrac{d\psi (x=0)}{dx} giving a solution satisfying \dfrac{d\psi (x=h/2)}{dx}=0, which must be valid in the middle between the two plates. The solution I obtained satisfies equation 6 as well.
Q2.3: My result is not realistic (\psi=40mV over whole gap). It seems that my solution doesnt work for realistic parameter values.
Is the implementation of the boundary condition \dfrac{d\psi (x=h/2)}{dx}=0 correct? I hesitated with this part cause we are looking for a solution to satisfy eq. 6. However, I couldnt find another way to implement both conditions simultaneously.