Gauss Law For The Electric Field

Gauss Law, Previous Concepts

Before stating Gauss' Law, let us first discuss the flux of a vector field. This is a concept of great usefulness in many physical problems.

Consider a surface S placed in a region where there is a vector field V. One can divide the surface into very small surfaces of areas dS₁, dS₂, dS₃,... and draw at each of them a unit vector u₁, u₂, u₃,... perpendicular to the surface at that point. Notice we are associating a vector to a surface.

If the surface is closed, the vectors u_N are oriented in the outward direction. Let ß₁, ß₂, ß₃,... be the angles between the normal vectors u₁, u₂, u₃,... and the field vectors V₁, V₂, V₃,... at each point on the surface. Then, by definition, the flux Ø of the vector field V through the surface S is

Ø = V₁dS₁cosß₁ + V₂dS₂cosß₂ + V₃dS₃cosß₃ +...
= V₁· u₁dS₁ + V₂· u₂dS₂ + V₃· u₃dS₃ +...

Where the integral extends over all the surface S. This is known as a surface integral. The name flux given to the integral is due to its application in the study of fluid flow. For instance, let´s calculate the water flow through a rectangular frame of sides a and b immersed in a river whose waters have velocity v.

The angle between the vector velocity and the vector perpendicular to the surface is ß.

Then the flux is

If v is in m/s and a and b in meters, this water flow is given in m³/s.

GAUSS' LAW.

Let us compute the flux of the electric field E produced by a point charge q through a spherical surface of radius r concentric with the charge. The electric field on each point of the spherical surface is

The unit vector normal to the surface coincides with the unit vector u_r along the radial direction. Therefore, the angle between E and the normal unit vector u_r is zero, so its cosine is 1.

The electric flux Ø_E is

and independent of the radius of the surface.

It can be demonstrated that this result does not depend on the shape of the closed surface S, the surface S can be an arbitrary closed surface S. If there are several charges q₁, q₂, q₃,... inside the arbitrary surface S the total electric flux will be the sum of the fluxes produced by each charge.

The Gauss' Law is then stated as:

The electric flux through a closed surface surrounding charges q₁, q₂, q₃,... is

Where q = q₁ + q₂ + q₃ +... is the total charge inside the closed surface.

Gauss' Law Exercises