Given the potential between two infinite parallel plates, how to find charge densities on the plates?

Question

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?

Answer 1

Not sure if you need to solve Poisson's equation for this. Couldn't you simply take the gradient of the potential to give you the field?

$$\mathbf{E} = -\frac{\partial{\phi}}{{\partial z}}\, \hat{z} = -\frac{\phi_0}{a}\, \hat{z}.$$

Then, you can use the expression for the field between the plates to solve for $\sigma$, the charge density on the positive plate:

$$\mathbf{E} = \frac{\sigma}{\epsilon_0}\, \hat{z} = -\frac{\phi_0}{a} \, \hat{z};$$ so, $$|\sigma| = \frac{\epsilon_0 \, \phi_0}{a}.$$