1. Patterns of Fields in Space
Box versus open surface
…no clue…
Seem to be able to tell
if there are charges inside
Gauss’s law:
If we know the field distribution on closed surface
we can tell what is inside.
Define Electric Flux
Gauss’s Law
Apply it
Symmetry

2. Electric Flux: Direction of E
Need a way to quantify pattern of electric field on surface: electric flux
1. Direction
flux>0 : electric field comes out
flux<0 : electric field goes in +1 -1
Show how total flux becomes zero in the third picture if flux~cos(theta)
Relate flux to the angle between outward-going normal and E:
flux ~ cos()

3. Electric Flux: Magnitude of E
flux ~ E
flux ~ Ecos()
𝑓𝑙𝑢𝑥 ~ 𝐸 ∙ 𝑛
𝑓𝑙𝑢𝑥 ~ 𝐸 ∙ 𝑛 = 𝐸 𝑛 cos⁡(𝜃)

4. Electric Flux: Surface Area
flux through small area:
Definition of electric flux on a surface:

5. Electric Flux: Perpendicular Field or Area
Perpendicular area x y q

6. Adding up the Flux

7. Gauss’s Law Features: 1. Proportionality constant
2. Size and shape independence
3. Dependence on sum of charges inside
4. Charges outside contribute zero

8. 1. Gauss’s Law: Proportionality Constant
What if charge is negative?
Works at least for one charge and spherical surface

9. 2. Gauss’s Law: The Size of the Surface
universe would be
much different if
exponent was not exactly 2!

10. 3. Gauss’s Law: The Shape of the Surface
∆ 𝐴 1⊥ ∝ 𝑟 1 2
The flux through the inner sphere is the same as the flux through the outer.

11. 4. Gauss’s Law: Outside Charges–
Outside charges contribute 0 to total flux

12. 5. Gauss’s Law: Superposition

13. Gauss’s Law Features: 1. Proportionality constant
2. Size and shape independence
3. Independence on number of charges inside
4. Charges outside contribute zero
Gauss’s Law and Coulomb’s Law?
Can derive one from another
Last shown.
Gauss’s law is more universal:
works at relativistic speeds

14. Applications of Gauss’s Law
Knowing E can conclude what is inside
Knowing charges inside can conclude what is E

15. The Electric Field of a Large Plate
Symmetry:
Field must be perpendicular to surface
Eleft=Eright
Start here.
Could be a sheet of charge or a metal plate with charge Q/A on each side. Assumption: we are finding the field in a region far from the plate edges.

16. The Electric Field of a Uniform Spherical Shell of Charge
Symmetry:
Field should be radial
The same at every location on spherical surface 𝑟 𝑟
A. Outer Dashed Sphere:
B. Inner Dashed Sphere:
Finally!

17. The Electric Field of a Uniform Cube
Is Gauss’s law still valid?
Yes, it’s always valid.
Can we find E using Gauss’s law?
Without symmetry, Gauss’s law loses much of its power.

18. Gauss’s Law for Electric Dipole
No symmetry
Direction and Magnitude of E varies
Numerical
Solution
Numerical solution to cube with 100 charges per side …

19. Clicker Question What is the net electric flux through the box? 0 V∙m

20. Gauss’s Law: Properties of Metal
Can we have excess charge inside a metal that is in
static equilibrium?
Proof by contradiction: =0

21. Gauss’s Law: Hole in a Metal=0
Less formal: imagine solid piece of metal. remove some (hole) – there are no excess charges, no field – so nothing changes.
Is the metal itself as shown electrically neutral? No, apparently, it has a net + charge.

22. Gauss’s Law: Charges Inside a Hole=0
+5nC

23. Next Class is a Review