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. ε0 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:
ε0∫E•dA = qenclosed
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:
ε0E∫dA = q
The
loop integral of the differntial areas is just the area of the sphere, 4πr
2. Substituting, we have:
ε0E(4πr2) = q.
This is:
E = q/(4πε0r2)
which is Coloumb's Law! :-)
Btw, if anyone knows how to make loop integrals in
HTML, I'd love to know!