[Physics] Boundary conditions for static electric field

boundary conditionselectric-fieldselectrostatics

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!

Best Answer

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
     /    /   |   \    \
    /    /    |    \    \ 
   /    /    /|\    \    \
            / | \
              |
Related Question