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 uN 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
= 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.
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 ur along the radial direction. Therefore, the angle between E and the normal unit vector ur is zero, so its cosine is 1.
The electric flux ØE 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.
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