Gauss's Law

Gauss's Law Do you remember our introduction to Conservation of Momentum? Newton's Third Law -- "action and reaction" -- led us to the idea that the total momentum of a system remains a constant. Conservation of Momentum and Newton's Third Law are essentially restatements of the same idea. In a similiar manner we will start with

Coulomb's Lawand developGauss' Law; they are restatements of the same idea.Consider the flux through a

spherical surfacedue to the electric field from apoint chargeq at the center of the sphere:We know the magnitude of the electric field is given by

E = k q / r ^{2}and we know that it points radially outward. We also know that "radially outward" means it passes through the sphere

perpendicularto the area. Therefore, we can writeThis is just the flux through a small area. When we sum that up -- or take the integral of it -- over the whole sphere, we have

for the electric field E is constant for constant radius; E = k q/r

^{2}. The integral of dA over the sphere's surface is 4 r2. That means the total flux isWhen we first looked at Coulomb's Law, F = k Qq/r

^{2}or E = k q/r^{2}, it was pointed out that this constant k in Coulomb's Law is sometimes written ask = 1/4 or Coulomb's Law can be written as

E = (1/4 ) q / r ^{2}This means the flux through our spherical surface due to a point charge q can be written as

Gauss's Lawis an extension or a generalization of this.Gauss's Lawis that the [net] electric flux throughanyclosed surface is equal to the charge inside that surface divided by this constant epsilon ().Consider several different surfaces surrounding a point charge:

Remember, we can also think of the electric flux as the

numberof electric field lines passing through a surface. In the diagram above, the number of lines passing through each of the different surfaces is clearly the same. That is, thefluxthrough each of these different surfaces must bethe same. The charge inside each of these surfaces is thesame-- just the charge q -- so the total or net flux must also be thesame.Consider a surface with

no chargeinside it. Thenetflux is what's important. Flux ispositivefor electric fieldsEpointingoutof the surface and flux isnegativefor electric fieldsEpointingintothe surface.

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(c) Doug Davis, 2002; all rights reserved