How Are Electrostatic Boundary Conditions Derived?

[ak416]

Im having trouble following how this is derived: The normal component of the electric field is discontinuous by an amount sigma/epsilon_0 at any boundary (when you cross a continuous surface charge). They talk about taking a little box so that the surface integral E dot da = 1/epsilon_0 * sigma * A (where A is area parallel to surface charge) and making its width perpendicular to the surface charge very small. Somehow they get that this implies E_perpendicualAbove -E_perpendicularBelow = 1/epsilon_0 * sigma. How's this? And also, they go on saying that in cases like the surface of a uniformly charged solid sphere this doesn't apply because there is no surface charge, but I don't get this...what about the edge of the sphere, its still charged. So please any clarification will help, as i have a test tomorrow.

 

[Galileo]

It's just an application of Gauss' law applied to a small pillbox straddling the surface. Looking only at the perpendicular component of E, the sides contribute nothing to flux. The flux through the pillbox is (E_\perp^{above} -E_{\perp}^{below})A. The enclosed charge is sigma*A, so by Gauss' law you get the result.

In a continuous volume charge distribution the E field is always continuous.

 

[Meir Achuz]

How's this? And also, they go on saying that in cases like the surface of a uniformly charged solid sphere this doesn't apply because there is no surface charge, but I don't get this...what about the edge of the sphere, its still charged. So please any clarification will help, as i have a test tomorrow.

The Gaussian pillbox has infintesimal thickness \Delta t, so the charge enclose at the surface of a uniformly charged sphere goes to zero
with \Delta t.