# Dielectric Boundary Condition Question

1. Jun 9, 2013

### Apteronotus

Hi,

I have a question regarding the boundary condition present for a dielectric immersed in a static field. I hope one of you physics guru's can shed some light on this.

Suppose we have a dielectric in space subjected to some external static electric field.

I have read (without explanation) that at the boundary of the dielectric the potential $\Phi$ satisfies

$k\frac{\partial \Phi}{\partial n_i} = \frac{\partial \Phi}{\partial n_e}$

where $\frac{\partial}{\partial n}$ represent the derivatives along the outward unit normal just interior, $i$, and just exterior, $e$, of the dielectric and $k$ is the dielectric constant.

can anyone shed some light on why this is so?

Last edited: Jun 9, 2013
2. Jun 9, 2013

### MisterX

The equation relates the normal component of the electric fields on either side of the boundary
$\frac{\partial \Phi}{\partial x} = -E_x$

The boundary condition is

$\epsilon_1 E^\perp_1 - \epsilon_2 E^\perp_1 = \sigma_q$

where $\sigma_q$ is the charge density on the surface.

This can be shown by using Gauss's law with a "pillbox" surface.

http://www.scribd.com/doc/136393324/27/Boundary-conditions-for-perpendicular-field-components

This corresponds to your equation when $\frac{\epsilon_i}{\epsilon_e} = k$ and $\sigma_q = 0$

Last edited: Jun 9, 2013
3. Jun 9, 2013

### Apteronotus

MisterX you are amazing!
I am grateful. Thank you.