1. Cultural enlightenment time. Conductors in electrostatic equilibrium.
Today’s agenda:
Announcements.
Gauss’ Law Examples.
You must be able to use Gauss’ Law to calculate the electric field of a high-symmetry charge distribution.
Cultural enlightenment time.
You must be culturally enlightened by this lecture.
Conductors in electrostatic equilibrium.
You must be able to use Gauss’ law to draw conclusions about the behavior of charged particles on, and electric fields in, conductors in electrostatic equilibrium.

2. Always true, not always easy to apply.
Gauss’ Law
Last time we learned that
Gauss’ Law
Always true, not always easy to apply.
and used Gauss’ Law to calculate the electric field for spherically-symmetric charge distributions
Today we will calculate electric fields for charge distributions that are non-spherical but exhibit a high degree of symmetry, and then consider what Gauss’ Law has to say about conductors in electrostatic equilibrium.

3. To be worked at the blackboard in lecture.
Example: calculate the electric field outside a long cylinder of finite radius R with a uniform volume charge density  spread throughout the volume of the cylinder.
To be worked at the blackboard in lecture.
“Long” cylinder with “finite” radius means neglect end effects; i.e., treat cylinder as if it were infinitely long.
More details of the calculation shown here:

4. Example: calculate the electric field outside a long cylinder of finite radius R with a uniform volume charge density  spread throughout the volume of the cylinder.
Cylinder is looooooong. I’m just showing a bit of it here.
I don’t even want to think of trying to use dE=kdq/r2 for this.

5. Example: calculate the electric field outside a long cylinder of finite radius R with a uniform volume charge density  spread throughout the volume of the cylinder. 
>0 R E

6. Looking down the axis of the cylinder.dA
Looking down the axis of the cylinder.

7. dA 
>0 r R L E

8. Inside the charged cylinder, by symmetry must be radial.

9. E dA dA E

10. Also must be constant at any given r.dA E r L
Also must be constant at any given r.

11. dA E r L

12. dA E 
>0 r R L

13. dA E 
>0 r R L

14. Why does this vary as 1/r instead of 1/r2?dA E 
>0 r R L
For positive :
In general:
Why does this vary as 1/r instead of 1/r2?

15. For a solid cylinder…
Charge per volume is
Charge per length is So
And