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Two parallel plates are kept in such a way that one is earthed and the other one is held at a potential $V_1$. They have a positive charge density $\rho$ in the volume between the inner faces of the plates. The distance between the parallel plates is $d$. What will be the distribution of potential inside the plates?

First of all, I tried to take a Gaussian surface in the middle of the two plates. At a distance $x$ from the centre, the electric field I found out was $E(r)=4\pi\rho x$. Now, I can use $\text{grad} V=-E$ and $\text{div} E = 4\pi\rho$.

But I am confused what will be the direction of the electric field between the plates if there is a positive charge everywhere? Does this mean that the electric field at the centre would be $0$?

And if I want to take the plate with $V=0$ (i.e. the plate which is earthed) as my $x=0$ point and the other plate at $V=V_1$ as $x=d$, how can I find the electric field then?

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The electric potential (or electric field) at points between the plates is the superposition of the potential (or field) due to the plates and that due to the uniform distribution of charge.

By symmetry the electric field along the mid-line of the charge distribution is zero, but the electric field along this line due to the plates is non-zero. The electric field due to the charge distribution points away from the mid-line (in both directions) and increases with distance from it, whereas the electric field due to the plates is the same at all points, pointing from the +ve plate to the -ve plate.

Assuming the plates are flat and much larger than the separation between them, the electric field is perpendicular to the plates at all points. So this is a 1D (linear) problem.

Levitopher's answer to Field between the plates of a parallel plate capacitor using Gauss's Law might be useful.

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  • $\begingroup$ If we take a Gaussian surface whose centre coincides with the centre line of the plates, then I used Φ2E.dS=4πq=4π(ρ*ds*2x). What’s wrong in this equation? $\endgroup$
    – Sid
    Commented Nov 6, 2017 at 23:20
  • $\begingroup$ Apologies, I did not realise you are using Gaussian Units. $\endgroup$
    – sammy gerbil
    Commented Nov 7, 2017 at 13:11

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