# Boundary conditions for static electric field

Consider a surface that carries surface charge density. In electrostatics, boundary conditions are studied by showing that there is a discontinuity in the normal component of the electric field across the charged surface and that there is no discontinuity in the horizontal component.

My question is aren't the electric fields at the surface always perpendicular to the surface? I mean where does the horizontal component come from if wherever we look locally, close enough to the surface, it looks like a flat infinite patch and there is no horizontal components!

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Comments to the question(v1): What electrical properties are you assuming about the ambient material on both sides of the surface? Is your question essentially this question? –  Qmechanic Mar 14 '12 at 20:56
@Qmechanic Just vacuum on both sides of the interface. It is discussed in Griffith's in CH2 before talking about conductors or dielectrics or any of that, just charged interface. No, my question is not that question. –  Revo Mar 15 '12 at 6:28

Assume that both the surface and the bulk are insulators with vacuum permittivity $\varepsilon_0$, so that the charges cannot redistribute themselves.

1. Consider first the electric field $$\vec{E}~=~\frac{\sigma}{2\varepsilon_0} \begin{pmatrix} {\rm sgn}(x)\\0\\0 \end{pmatrix}$$ associated with a uniformly charged capacitor plate at $x=0$ parallel to the $yz$ plane.

2. Consider next the electric field $$\vec{E}~=~\frac{\sigma}{2\varepsilon_0} \begin{pmatrix} 0\\{\rm sgn}(y)\\0 \end{pmatrix}$$ associated with a uniformly charged capacitor plate at $y=0$ parallel to the $xz$ plane.

3. Now construct a simple counterexample (where $\vec{E}$ is not perpendicular to the surface) by adding together the charge distributions in situation 1 and 2, cf. figure. Use superposition principle to determine $$\vec{E}~=~\frac{\sigma}{2\varepsilon_0} \begin{pmatrix} {\rm sgn}(x)\\{\rm sgn}(y)\\0 \end{pmatrix},$$

Figure:

        Capacitor plate
|
\ | /         E-field lines
\    \    \|/    /    /
\    \    |    /    /
\    \   |   /    /
-----------------------------  Capacitor plate
/    /   |   \    \
/    /    |    \    \
/    /    /|\    \    \
/ | \
|

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As an example, consider the boundary of a dielectric with an applied external field $\vec{E}$. The component $\vec{E_{\perp}}$ will induce a surface charge and accompanying discontinuity in $\vec{E}$ by polarizing the medium. The field component parallel to the surface $\vec{E_{\parallel}}$ will be continuous across the surface and is not necessarily zero inside.