1. Magnetic Forces in Moving Reference Frames
Electric force:
Two protons +e r 1 2 v
Magnetic field:
F21,m B1
Magnetic force:
F21,e E1
Coulomb force on moving charges: approximation, E is not exactly the same (address later). Recall that we have already said that two current carrying wires experience an attractive magnetic force if current direction is the same.

2. Magnetic Forces in Moving Reference Frames
Electric force: +e r 1 2 v E1
F21,e B1
F21,m
Magnetic force:
Ratio:
Coulomb force on moving charges: approximation, E is not exactly the same (address later)
=c2
(m/s)2
it is not accidental!

3. Magnetic Forces in Moving Reference Frames1 2 v E1
F21,e B1
F21,m
For v<<c the magnetic force is much
smaller than electric force
Ask students to reconcile this with the fact that two wires carrying current in the same direction are attracted to each other.
Full Lorentz force:
downward

4. Magnetic Forces in Moving Reference Frames
20 ns +e r 1 2 v E1
F21,e B1
F21,m
15 ns
Who will see protons hit
floor and ceiling first?
This is a kind of phenomenon which led Einstein to develop theory of relativity

5. Relativistic Field Transformations
Our detailed derivations are not correct for relativistic speeds,
but the ratio Fm/Fe is the same for any speed:
According to the theory of relativity:

6. Magnetic Field of a Moving Particle
Still:
Moving:
Slow case: v<<c 
Field transformation is consistent with Biot-Savart law
What about B below particle?
Electric and magnetic fields are interrelated
Magnetic fields are relativistic consequence of electric fields

7. Electric Field of a Rapidly Moving Particle

8. The Principle of Relativity
There may be different mechanisms for different observers in different reference frames, but all observers can correctly predict what will happen in their own frames, using the same relativistically correct physical laws.

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

10. 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.

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

12. Adding up the Flux

13. 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

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

15. 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

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

17. 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

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

19. 5. Gauss’s Law: Superposition

20. 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

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

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

23. 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.

24. 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:

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

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

27. 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

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

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