6.3 Applying gauss’s law  (Page 3/13)

The magnitude of the electric field $\overrightarrow{\text{E}}$ must be the same everywhere on a spherical Gaussian surface concentric with the distribution. For a spherical surface of radius r ,

Using gauss’s law

According to Gauss’s law, the flux through a closed surface is equal to the total charge enclosed within the closed surface divided by the permittivity of vacuum ${\epsilon }_0$ . Let $q_{\text{enc}}$ be the total charge enclosed inside the distance r from the origin, which is the space inside the Gaussian spherical surface of radius r . This gives the following relation for Gauss’s law:

Hence, the electric field at point P that is a distance r from the center of a spherically symmetrical charge distribution has the following magnitude and direction:

Direction: radial from O to P or from P to O .

The direction of the field at point P depends on whether the charge in the sphere is positive or negative. For a net positive charge enclosed within the Gaussian surface, the direction is from O to P , and for a net negative charge, the direction is from P to O . This is all we need for a point charge, and you will notice that the result above is identical to that for a point charge. However, Gauss’s law becomes truly useful in cases where the charge occupies a finite volume.

Computing enclosed charge

The more interesting case is when a spherical charge distribution occupies a volume, and asking what the electric field inside the charge distribution is thus becomes relevant. In this case, the charge enclosed depends on the distance r of the field point relative to the radius of the charge distribution R , such as that shown in [link] .

If point P is located outside the charge distribution—that is, if $r\ge R$ —then the Gaussian surface containing P encloses all charges in the sphere. In this case, $q_{\text{enc}}$ equals the total charge in the sphere. On the other hand, if point P is within the spherical charge distribution, that is, if $r<R,$ then the Gaussian surface encloses a smaller sphere than the sphere of charge distribution. In this case, $q_{\text{enc}}$ is less than the total charge present in the sphere. Referring to [link] , we can write $q_{\text{enc}}$ as

The field at a point outside the charge distribution is also called ${\overrightarrow{\text{E}}}_{\text{out}}$ , and the field at a point inside the charge distribution is called ${\overrightarrow{\text{E}}}_{\text{in}}.$ Focusing on the two types of field points, either inside or outside the charge distribution, we can now write the magnitude of the electric field as