The relation between zero curl and divergence of E and B and their respective values

Question

Take the following scenarios: A 2 parallel plates capacitor, the electric field between them is non zero, but the charge density is zero. A solenoid, the magnetic field inside it is non zero, but the current density is. In both cases the curl and divergence of E and B is zero, because the charge density is zero, and the current density is zero, but if the curl and divergence of a vector field is zero, doesn't this mean that the vector field is zero? Where am i going wrong here? I thought that a vector field was determined by its curl, divergence and boundary conditions.

Answer 1

There is an electric field between the plates because there is a potential difference between the capacitor plates. Namely, one face of the capacitor has charge $+Q$, and the other face has charge $-Q$. The voltage will be given by

$$ V(x) = \frac{Q}{C} = -E \ x . $$

The way to show this is to first use Gauss's law with the "pillbox". Each plate gives a contribution of

$$ E_{plate} = \frac{\sigma}{2\epsilon_0} $$

so that when you add them you get that

$$ E_{total} = \frac{\sigma}{\epsilon_0} $$

where $\sigma = \frac{Q}{A}$ where $Q$ is the charge induced on the capacitor and $A$ is the area of the plate.

To relate this to your question of $\nabla \cdot\mathbf{E}$, note that this is just the differential form of Gauss's law. Namely we have that

$$\iint_{\partial V} \mathbf{E}\cdot d\mathbf{A} = \frac{1}{\epsilon_0} \iiint_V \nabla \cdot E \ dV = \frac{1}{\epsilon_0}\iiint_V \rho(\mathbf{r}) \ dV \equiv \frac{Q_{nec}}{\epsilon_0} $$

So while it is true that there is no charge density in between the plates, that doesn't cover the whole space. You need to consider the boundaries as well (namely the faces of the plates). In particular, for a parallel plate capacitor your density integral reduces to

$$ \iiint_V \rho(x) dV = \sigma A \int_{-\infty}^\infty dx\ \delta(x) + \delta(x-d)$$

if one plate is located at $\mathbf{x=0}$ and the other at $\mathbf{x =d}$.

I could go through the derivation of the electric field of a parallel plate capacitor, but these are easier to find than a parking space in Arkansas.