1. Oregon State University PH 213, Class #8
Gauss’s Law
“The net electric flux through any closed surface is directly proportional to the net charge contained within that surface.”
Qenclosed = e0(FE)
4/19/17
Oregon State University PH 213, Class #8

2. Oregon State University PH 213, Class #8
Gauss’s Law is always true, but often it’s no more convenient to use than Coulomb-based E-field integrals… In fact, the only time Gauss’ Law is really handy is when its integration is trivial—and that’s when |E| and cosq are constants over entire portions of the Gaussian surface:
Now as a counter-example, consider: Is there a Gaussian surface you could define that would allow you to use Gauss’s law to easily determine the electric field outside a uniformly charged cube?
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

3. Oregon State University PH 213, Class #8
So, when is Gauss’s Law useful? When we already know the shape/form of the E-field (but not necessarily its magnitude)—and that form is symmetric and/or uniform, so that we can avoid nasty surface integrals by selecting certain simple Gaussian surfaces. There are three such situations:
A spherical surface (for spherically symmetric charge distributions).
A cylindrical surface (for very long lines/cylinders of cylindrically symmetric charge distributions).
A rectangular “box” surface (for very large planes of rectangular symmetric charge distributions).
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

4. Oregon State University PH 213, Class #8
4/19/17
Oregon State University PH 213, Class #8

5. Oregon State University PH 213, Class #8
Example: Suppose we want to determine the E-field at any point either inside or outside a solid sphere (centered at the origin) of known radius R, and known uniform charge distribution r.
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

6. Oregon State University PH 213, Class #8
Example: Suppose we want to determine the E-field at any point either inside or outside a long, solid cylinder (centered on the z-axis) of known radius R, and known uniform charge distribution r.
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

7. Oregon State University PH 213, Class #8
Example: Suppose we want to determine the E-field at any point either inside or outside a long, solid slab (centered on the xz-plane) of known thickness W, and known uniform charge distribution r.
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

8. Oregon State University PH 213, Class #8
Example: Suppose we want to determine the E-field at any point either inside or outside a solid sphere (centered at the origin) of known radius R, with this known volumetric charge distribution:
r = cr (c is a positive known constant—with what units?)
This is spherically symmetric (depends only on r, not on q or f).
(See After Class 8.)
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

9. Oregon State University PH 213, Class #8
Example: Suppose we want to determine the E-field at any point either inside or outside a long cylinder (parallel to, and centered on, the z-axis) of known radius R and with the following known volumetric charge distribution:
r = cr (c is a known positive constant—with what units?)
This is cylindrically symmetric (depends only on r, not on q or z).
(See After Class 8.)
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8

10. Oregon State University PH 213, Class #8
Example: Suppose we want to determine the E-field at any point either inside or outside a long slab of known thickness W (parallel to and centered on the x-z plane), with the following volumetric charge distribution:
r = c|y| (c is a known positive constant—with what units?)
This is rectangularly symmetric (depends only on y, not on x or z).
(See After Class 8.)
STT27.5
Answer: C
4/19/17
Oregon State University PH 213, Class #8