' 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
Gauss' Law is usually stated:
ε0∫E•dA = qenclosed
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, 4πr2
. Substituting, we have:
ε0E(4πr2) = q.
E = q/(4πε0r2)
which is Coloumb's Law! :-)
Btw, if anyone knows how to make loop integrals in HTML
, I'd love to know!