1. Patterns of Fields in Space
Chapter 22
Patterns of Fields in Space
Electric flux
Gauss’s law
Ampere’s law
Maxwell equations

2. Patterns of Fields in Space
What is in the box?
no charges?
vertical charged plate?
Gauss’s law:
If we know the field distribution on closed surface
we can tell what is inside.

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

4. Adding up the Flux

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

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

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

8. 3. Gauss’s Law: The Shape of the Surface
All elements of the outer surface
can be projected onto corresponding
areas on the inner sphere with the
same flux

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

10. 5. Gauss’s Law: Superposition

11. Gauss’s Law Is it a law or a theorem? Can derive one from another
Last shown.
Gauss’s law is more universal:
works at relativistic speeds

12. Clicker Question What is the net electric flux on the box? 0 V*m

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

14. 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.

15. The Electric Field of a Uniform Spherical Shell of Charge
Symmetry:
Field should be radial
The same at every location
on spherical surface
A. Outer sphere:
B. Inner sphere:

16. The Electric Field of a Uniform Cube
Is Gauss’s law still valid?
Can we find E using Gauss’s law?

17. Clicker Question What is the electric flux through the area A?
E = 100 V/m
q = 30o
DA = 2 m2
100 V*m
173 V*m
50 V*m
87 V*m c

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

19. Gauss’s Law: Hole in a Metal=0
What is electric field inside the hole? =
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.
No charges on the surface of an empty hole
E is zero inside a hole

20. Gauss’s Law: Screening
Similar to a hole in the metal

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

22. Gauss’s Law: Circuits
Can we have excess charge inside in steady state?
We already have established that in steady state for a uniform conductor, the E-field is the same magnitude and follows the wire. Thus, Gauss’s law gives no net flux and hence no net charge contained within the conductor. But current carriers are all negative. Where does the positive charge reside?

23. Gauss’s Law: Junction Between two Wires
i1=i2
n1Au1E1 = n2Au2E2
There is negative charge along the interface!
n2<n1
u2<u1
Take Gaussian surface just inside the conductor since we are not interested in surface charges. We’ve already shown that within each uniform conductor there is no charge.