Gauss’s Law Electric Flux Gauss’s Law Examples

Gauss’s Law What’s in the box ??

Gauss’s Law What’s in the box ?? Net # of lines going out is related to the Net charge enclosed + + +

Recall electric flux: -Scalar quantity - related to number of filed lines

For a closed surface S, the element of area vectors point outward from the surface. The surface may be a real surface or just an ‘imaginary’ mathematical surface of our choosing.

Gauss’s Law For a closed surface S, take the outward direction as the direction for ; then… Net outward flux through S

Or… The closed “Gaussian surface” S is arbitrary (and usually exists only in your imagination).

Example: point charge + r q E S: sphere, radius r, centered on the charge. dA E E Find the magnitude of E on the sphere.

Solution:

Example: Find the net outward flux for each surface + centered + outside + near edge

Example: +Q -2Q +3Q S1S1 S2S2 S3S3 S 1, S 2, and S 3 represent closed 3-dimensional surfaces. Find the net outward electric flux through each surface.

+Q -2Q +3Q S1S1 S2S2 S3S3 Field Lines:

Example: Uniformly Charged Sphere Total charge Q P R r Find: at P for… i) r > R and ii) r < R

P R r E(r) r > R

Solution part for r>R:

Solution part for r<R:

In General… For any uniform distribution of charge Q within a sphere of radius R:

Gauss’s Law What if the charge distribution is not uniform? Can we still find the electric field? Just rewrite Gauss’s Law to look like this:

Summary Gauss’s Law: The net outward electric flux through any closed surface is proportional to the net charge enclosed. Gauss’s Law is a consequence of Coulomb’s Law Example: for spherically symmetric charges, it is easy to calculate E using Gauss’s law