In this topic we describe a general relationship between the net electric flux through a closed surface (often called a gaussian surface) and the charge enclosed by the surface. This relationship, known as Gauss’s law, is of fundamental importance in the study of electric fields.
Let us again consider a positive point charge q located at the center of a sphere of radius r, as shown in Figure 24.6. From Equation 23.4 we know that the magnitude of the electric field everywhere on the surface of the sphere is E = keq / r2. As noted in Example 24.1, the field lines are directed radially outward and hence perpendicular to the surface at every point on the surface. That is, at each surface point, E is parallel to the vector ΔAi representing a local element of area ΔAi surrounding the surface point. Therefore,
and from Equation 24.4 we find that the net flux through the gaussian surface is
where we have moved E outside of the integral because, by symmetry, E is constant
over the surface and given by E = keq / r2. Furthermore, because the surface is
spherical, ∮ dA = A = 4πr2. Hence, the net flux through the gaussian surface is
we can write this equation in the ke = 1/(4πε0), form
We can verify that this expression for the net flux gives the same result as Example 24.1: ΦE = (1.00 × 10-6 C)/(8.85 ×10-12 C2/N.m2) = 1.13 × 105 N.m2/C.
Note from Equation 24.5 that the net flux through the spherical surface is proportional to the charge inside. The flux is independent of the radius r because the area of the spherical surface is proportional to r2, whereas the electric field is proportional to 1/r2. Thus, in the product of area and electric field, the dependence on r cancels.
Now consider several closed surfaces surrounding a charge q, as shown in Figure 24.7. Surface S1 is spherical, but surfaces S2 and S3 are not. From Equation 24.5, the flux that passes through S1 has the value q/ε0 . As we discussed in the previous section, flux is proportional to the number of electric field lines passing through a surface. The construction shown in Figure 24.7 shows that the number of lines through S1 is equal to the number of lines through the nonspherical surfaces S2 and S3 . Therefore, we conclude that the net flux through any closed surface is independent of the shape of that surface. The net flux through any closed surface surrounding a point charge q is given by q/ε0.
Now consider a point charge located outside a closed surface of arbitrary shape, as shown in Figure 24.8. As you can see from this construction, any electric field line that enters the surface leaves the surface at another point. The number of electric field lines entering the surface equals the number leaving the surface. Therefore, we conclude that the net electric flux through a closed surface that surrounds no charge is zero. If we apply this result to Example 24.2, we can easily see that the net flux through the cube is zero because there is no charge inside the cube.
Let us extend these arguments to two generalized cases: (1) that of many point charges and (2) that of a continuous distribution of charge. We once again use the superposition principle, which states that the electric field due to many charges is the vector sum of the electric fields produced by the individual charges. Therefore, we can express the flux through any closed surface as
where E is the total electric field at any point on the surface produced by the vector addition of the electric fields at that point due to the individual charges.
Consider the system of charges shown in Figure 24.9. The surface S surrounds only one charge, q1 ; hence, the net flux through S is q1/ε0 . The flux through S due to charges q2 and q3 outside it is zero because each electric field line that enters S at one point leaves it at another. The surface S surrounds charges q2 and q3 ; hence, the net flux through it is (q2 + q3)/ε0. Finally, the net flux through surface S ʹʹ is zero because there is no charge inside this surface. That is, all the electric
field lines that enter S ʹʹ at one point leave at another.
Gauss’s law, which is a generalization of what we have just described, states that the net flux through any closed surface is
where qin represents the net charge inside the surface and E represents the electric field at any point on the surface.
A formal proof of Gauss’s law is presented in later topic. When using Equation 24.6, you should note that although the charge qin is the net charge inside the gaussian surface, E represents the total electric field, which includes contributions from charges both inside and outside the surface.
In principle, Gauss’s law can be solved for E to determine the electric field due to a system of charges or a continuous distribution of charge. In practice, however, this type of solution is applicable only in a limited number of highly symmetric situations. As we shall see in the next section, Gauss’s law can be used to evaluate the electric field for charge distributions that have spherical, cylindrical, or planar symmetry. If one chooses the gaussian surface surrounding the charge distribution carefully, the integral in Equation 24.6 can be simplified. You should also note that a gaussian surface is a mathematical construction and need not coincide with any real physical surface.