# Homework Help: Laplacian for hyperbolic plates

1. Dec 23, 2017

### Physgeek64

I don't see how you could sketch it without working out $\sigma$

Could we not have a non zero volume charge though? To satisfy the conservation of charge

2. Dec 23, 2017

### TSny

Sketch E-field lines in the region of the origin. Look for symmetry. This should help you see any symmetry of the surface charge density.

Usually, it is assumed that the susceptibility $\chi_e$ is constant inside the material (i.e., does not depend on position). From $\bf D = \epsilon_0 \left( 1 + \chi_e \right) \bf E$, deduce that $\nabla \cdot \bf D$ $= \epsilon_0 \left( 1 + \chi_e \right)\nabla \cdot \bf E$. But we know $\nabla \cdot \bf D$ $= 0$ (assuming no free volume charge density $\rho_f$). Hence, $\nabla\cdot \bf E$ $= 0$. And this implies zero total volume charge density $\rho_f + \rho_b$. Hence, what can you conclude about any bound charge density $\rho_b$?