**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 dS _{1}, dS_{2}, dS_{3},... and draw
at each of them a unit vector u_{1}, u_{2}, u_{3},...
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 ß_{1}, ß_{2}, ß_{3},...
be the angles between the normal vectors u_{1}, u_{2}, u_{3},...
and the field vectors V_{1}, V_{2}, V_{3},... at each
point on the surface. Then, by definition, the flux Ø of the vector field
V through the surface S is **

Ø
= V_{1}dS_{1}cosß_{1} + V_{2}dS_{2}cosß_{2}
+ V_{3}dS_{3}cosß_{3} +...

= V_{1}·
u_{1}dS_{1} + V_{2}· u_{2}dS_{2} +
V_{3}· u_{3}dS_{3} +...

** 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 m ^{3}/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 q _{1},
q_{2}, q_{3},... 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 q _{1}, q_{2}, q_{3},... is **

**Where q = q _{1} + q_{2} + q_{3} +... is the total
charge inside the closed surface.**

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