Gauss surface

Fig. 25. A Gauss surface for the Gouy-Chapman theory the broken line indicates the boundaries of the Gauss surface.

Refer now to the arguments that led to equation 4.18. It is recalled that the Gauss surface was constructed to include all of the charge in the diffuse double-layer. If, however, we draw the surface in a way similar to that of Section B, 2, but so that it only includes part of the charge we find that... 

Let us comment on the expression derived, hi this expression the point charge Q is comprised of a Gauss surface S. It was taken in spherical form, but it can also be a irregnlar shaped surface. On the right-hand side the charge Q can be seen which we encompassed by the Gauss surface (Figure 4.4). 

If the charge is outside, the flux through the Gauss surface is equal to zero (Figure 4.5) since part of the flux is negative and the other part is positive (because of various orientations of a normal n to the surface relative to vector E). This is the case for all those charges which lie outside the Gauss surface they can be excluded from consideration. 

Figure 4.4 The Gauss theorem Q is a charge, 5 is a closed Gauss surface, aK is an elementary area and n is a unit vector to dS.

Figure 4.5 The zero total flux of the electric field strength for a particular case when an electric charge is outside the Gauss surface.

Figure 4.6 The electric field of a point charge. Point A is a point of observation, the dashed line represent a Gauss surface.

Calculation of the field outside the sphere is easier the Gauss surface encompasses the whole sphere. Then... 

Choose accordingly a closed Gauss surface in the form of a coaxial cylinder of length I (cyl) and end surfaces (end) perpendicular to the cylinder axes (Figure 4.9). Then the integral over the Gauss surface will be separated into three parts ... 

According to the Gauss theorem, this expression is equal to the charge inside the Gauss surface, i.e.. 

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