Consider two parallel infinite metal plates, one at $z=0$ and the other at $z=a$, maintained at potentials $\phi=0$ and $\phi=\phi_0$ respectively. The potential in between the plates and also on the plates is given by the expression $$\phi(z)=\phi_0\frac{z}{a}.$$ If we use Poisson's equation $d^2\phi/dz^2=-\rho/\epsilon_0$, the expression written above gives $\rho=0$ (which is charge density in between the plates). But I have a feeling that this formula also contains the information that the charges on the plates are nonzero. How can we find the surface charge densities $\sigma$ on the plates?

Consider two parallel infinite metal plates, one at $z=0$ and the other at $z=a$, maintained at potentials $\phi=0$ and $\phi=\phi_0$ respectively. The potential in between the plates and also on the plates is given by the expression $$\phi(z)=\phi_0\frac{z}{a}.$$ The electric field between the plates is then given by $$\vec{E}=-\frac{d\phi}{dz}\hat{z}=-\frac{\phi_0}{a}\hat{z}.$$ If we use Poisson's equation in the next step $d^2\phi/dz^2=-\rho/\epsilon_0$, the expression written above gives $\rho=0$ (which is charge density in between the plates). But I have a feeling that this formula also contains the information that the charges on the plates are nonzero. How can we find the surface charge densities $\sigma$ on the plates?