Gauss' Law

Qualitatively Gauss' Law is often stated as, "the net number of flux lines out of any closed surface containing a charge is proportional to the net charge inside the surface" or

But what is meant by the term electric flux, Φ? These are electric field lines penetrating a surface. If a field line enters the surface its value can be thought of as -1 while a field line existing the surface can be thought of as +1. The unit for flux is a Nm²/C.

Isolated positive charges are sometimes called sources since all of their field lines begin within the surface while isolated negative charges are sometimes called sinks since all of their field lines enter the surface.

positive isolated charge "+16" flux lines arrows point out		negative isolated charge "-16" flux lines arrows point in

If the surface encloses a mixture of charges, the number of flux lines is equal to the net, or sum, of the field lines entering and/or exiting the surface.

8 field lines leave and 6 field lines enter
(notice that 8 field lines are always inside the surface and are not counted)

net flux = +2 telling us that the positive charge is larger

Remember that the relative strength of an electric field can be represented pictorially as a proportional number of field lines. If one charge has twice the magnitude of another, it would have two times as many field lines.

Dot Product

When calculating the magnitude of the electric flux passing through a surface, the formula is

where

• E is the magnitude of the electric field,
• A is the cross-sectional area of the plane, and
• θ is the angle measured between the electric field lines and the normal to the area (which is called the area vector).

Remember that dot products produce scalar answers. So your results will no longer have a direction, only magnitude. That is, you will not be asked to find the components of the number of flux lines

In the top diagram the angle between the field lines equals 0º, so Φ = EA cos(0) = EA, its maximum value.

In the middle diagram the angle between the field lines equals 45º, so Φ = EA cos(45) = E(0.707A).

In the final diagram the angle between the field lines equals 90º, so Φ = EA cos(90) = 0.

To use Gauss' Law to calculate the electric field in a region, we choose a convenient Gaussian surface whose "edges or sides" lie either perpendicular or parallel to the field lines emanating from the charged object. We will only be responsible for highly symmetrical objects: point charges, charged wires/rods/cylinders, and sheets/disks of charge. We will begin our study with point charges.

Point Charges

Notice that by taking our Gaussian surface to also be a sphere, the field lines will always pass perpendicular to its surface [θ = 90º and cos(90º) = 1] so we can write Gauss' Law as

Charged Plane

Suppose we now look at a uniformly charged plane; that is, a surface