# boundary condition of electrical field

The boundary condition of electrical field is well known.

$$E^{\perp} _{above} -E^{\perp} _{below} = \frac{\sigma}{\epsilon_0}$$

Griffiths wrote that where there is no surface charge $E^{\perp}$ is continuous, as for instance at the surface of a uniformly charged solid sphere.

Let's think of electrical field of solid sphere which carries a uniformly volume charge density $\rho$. It's obviously true that $E$ is continuous by Gauss theorem, but I think it has surface charge density $\sigma$ too. Why doesn't it have $\sigma$ in this situation?

One way to see this is the following. In your uniform sphere consider a shell with small thickness. What is the amount of charge in this shell? If this shell has radius $r$ it is $\Delta Q=4\pi r^2 \Delta r \rho$ (volume times density). If you want to calculate the surface density you should divide for the total area of the surface and send the thickness to zero. It gives $\Delta r \rho \,\to\,0$ as $\Delta r \,\to\,0$. Put it simply, you have a volume charge density therefore the total charge of an object goes to zero as the volume goes to zero which is what happens in any surface (by definition it has volume zero).
Suppose now you want to solve another problem in which, besides a charge density sphere, you will also put as a condition of the problem a surface charge $\sigma$ and want to determine the electric field. In this case in applying Gauss theorem there is a discontinuity on the total charge enclosed by a sphere as it crosses this surface: $\Delta Q(R+\delta r)-\Delta Q(R-\delta r) = \sigma 4\pi R^2$. But again, this only happens if surface density is directly imposed and not because you have a volume charge density.
• I understand that surface charge density is zero, but I'm confused in understanding $\Delta Q(R+\delta r)-\Delta Q(R-\delta r) = \sigma 4\pi R^2$. You said that $\sigma$ will be zero so I think $\Delta Q(R+\delta r)-\Delta Q(R-\delta r)$ should also be zero. Am I right? – cokecokecoke Feb 4 '16 at 14:43
• Yes! That is correct. I did not explain well what I meant. What I meant is that if you consider the problem of "an uniformly charged sphere with charge density $\rho$" then $\Delta Q (R+\delta r) -\Delta Q (R+\delta r) =0$. If you consider instead other problem of "an uniformly charged sphere with charge density $\rho$ PLUS (as an additional feature) a surface charge $\sigma$" then $\Delta Q (R+\delta r) -\Delta Q (R+\delta r) =\sigma 4 \pi R^2$ which will justify a discontinuity on the electric field as you well pointed out. If this answers your question I edit the answer. Feel free to ask – BartLima Feb 4 '16 at 14:59