Gauss' Law is not just an alternative way to state the relationship between

electric charge and

electric field (the other being

Coulomb's Law). In fact, Coloumb's Law can be derived directly from Gauss' Law--the two are truly the same law, stated in different terms.

Note: in my notation, variables in bold are vector quantities. ε_{0} is the permitivity of free space and is 8.85 x 10^{-12} Farads/meter or coloumbs-squared/Newtons-meters-squared. All integrals are to be understood as loop integrals, over the entire Gaussian surface.

Take a point charge +q, around which is envisioned a Gaussian concentric spherical surface of radius r. Divide this surface into differential areas dA. By definition, the

area vector for each area dA is

**dA** with magnitude equal to the area and direction perpendicular to the surface (directed outward from the interior of the sphere). From the

symmetry of the sphere, we also know that at any point on the sphere, the

electric field vector **E** is also perpendicular to dA and directed outward, and so is parallel to

**dA**. Therefore the

angle between

**E** and

**dA** is zero.

Gauss' Law is usually stated:

ε_{0}∫**E**•**dA** = q_{enclosed}

Since

**E** and

**dA** are parallel, their dot product is simply EdA. The enclosed charge is simply q (since we made our sphere around this charge), and is a constant. E is also a constant, because it varies only radially for a point charge, and we have restricted the radius to the radius r of our sphere. So Gauss' Law can be rewritten:

ε_{0}E∫dA = q

The

loop integral of the differntial areas is just the area of the sphere, 4πr

^{2}. Substituting, we have:

ε_{0}E(4πr^{2}) = q.

This is:

E = q/(4πε_{0}r^{2})

which is Coloumb's Law! :-)

Btw, if anyone knows how to make loop integrals in

HTML, I'd love to know!