1. Potential Difference: Path Independence
Path independence principle:
V between two points does not depend on integration path

2. Potential Difference in Metal
In static equilibrium
What is E inside metal?
E = 0
In static equilibrium the electric field is zero at all locations along any path through a metal. i f
What is the potential difference (Vf – Vi)?
The potential difference is zero between any two locations inside the metal, and the potential at any location must be the same as the potential at any other location.
Is V zero everywhere inside a metal?
No! But it is constant

3. Potential in Metal In static equilibrium
A Capacitor with large plates and a small gap of 3 mm has a potential difference of 6 Volts from one plate to the other. E
d =3 mm +Q -Q
-3 V
+3 V
Charges are on surface
V = 6 Volt

4. Potential in Metal In static equilibrium
Insert a 1 mm thick metal slab into the center of the capacitor.
d =3 mm
+Q1
-Q1
1 mm
Metal slab polarizes and has charges +Q2 and -Q2 on its surfaces.
What are the charges Q1 and Q2? X
At X
E=0 inside metal
𝐸 1 = 𝐸 2
Q2=Q1
Now we have 2 capacitors instead of one
E inside metal slab is zero!
Ignoring the fringe fields, E = 2000V/m in each capacitor (from previous slide).
V = 4 V
V inside metal slab is zero!
Charges +Q2 and –Q2
There is no “conservation of potential”!

5. Potential in Metal
There can be a potential in metal if is NOT in static equilibrium
Metal is not in static equilibrium:
When it is in the process of being polarized
When there is an external source of mobile charges (battery)
Nonzero electric field of uniform magnitude E throughout the interior of a wire of length L.
Direction of the field follows the direction of the wire.
For each step 𝐸 ∙∆ 𝑙 , the potential difference is: V = -EL
But in a circuit a thick copper wire may have a very small electric field. B is at lower potential than A.
If a metal is not in static equilibrium, the potential isn’t constant in the metal.

6. Example: Two Different Paths in Capacitor
Need to find VAC =VC - VA
1. Straight path AC
Electric field is constant inside capacitor!

7. Example: Two Different Paths in Capacitor
Need to find VAC =VC -VA
1. Straight path AC
2. Path AB C
Does it make sense VCB=0?
No work is required to move at right angle to the electric field

8. Question 300 V/m 0 V/m 300 V/m A B 0.02m 0.03m 0.04m What is VB-VA?D
What is VB-VA?
270 V
-270 V
-18 V
6 V
-6 V

9. Question 0 V/m A B 0.02m 0.03m 0.04m x 𝑉 𝐵 − 𝑉 𝐴 =− 300 V m 0.02m−0D
𝑉 𝐵 − 𝑉 𝐴 =− 300 V m m−0
− −300 V m m−0.05m
𝑉 𝐵 − 𝑉 𝐴 =−6V− −12V
= 6V

10. A

11. Round Trip Potential Difference+
Potential difference due to a stationary point charge is independent of the path
Potential difference along a closed loop is zero

12. Predicting Possible Field Configuration
Is the following “curly” pattern of electric field possible? dl dl
is always parallel to
Cannot produce with any stationary arrangement of charges – but could be in principle, will talk in later chapters.
Move from A to A dl
This “curly” pattern of electric field is impossible to produce by arranging any number of stationary point charges!

13. Wire in a Circuit and Electric Field
In a current-carrying wire in a circuit, there can be a nonzero electric field, so there is a difference in potential between two locations in the wire.
Electric field is not confined to wire in a circuit – there is E around!
Electric field is not confined to the wire in a circuit – there must be an electric field E around the wire in the air!

14. Potential due to Point Chargerf ri +q
For final r greater than initial r, the change in potential is less than zero as expected since the path direction is the same as that of the electric field. Likewise, if we go from a larger r to a small r, the potential increases.

15. Potential of a Uniformly Charged RingQ
Method 1: Divide into point charges and add up contributions due to each charge

16. Potential of a Uniformly Charged RingQ
Method 2: Integrate electric field along a path
Note that we integrate from an initial z=infinity to a final z so that V represents the energy per unit charge required to move a point charge in from infinity to z.

17. Potential of a Uniformly Charged RingQ
What is V for z>>R ?
Is it unexpected?
The same as for a point charge!

18. Potential of a Uniformly Charged Disk
one ring:
integrate:

19. Exercise
What is the potential in the center of uniformly charged hollow sphere? R
Add contribution from each point charge on the surface.

20. Potential of a Uniformly Charged Disk
Can find :

21. Potential Difference in an Insulator1 2 3 4 5
Electric field in capacitor filled with insulator: Enet=Eplates+Edipoles
Eplates=const (in capacitor)
Edipoles complicated f(x,y,z)
Edipoles,A
Edipoles,B
∆𝑉=− 𝐴 𝐵 𝐸 ∙𝑑 𝑙 A B
Situation in insulator is more complex than in metals.
Polarized molecules contribute to net electric field
A … inside ”capacitor”
B … between two “capacitors”
Travel from B to A:
Edipoles is sometimes parallel to dl, and sometimes antiparallel to dl

22. Potential Difference in an Insulator
Instead of traveling through inside – travel outside from B to A:
Edipoles, average A B
∆ 𝑉 𝐵𝐴 =− 𝐴 𝐵 𝐸 𝑑𝑖𝑝𝑜𝑙𝑒𝑠 ∙𝑑 𝑙
Enet=Eplates+Edipoles,average
<0
Enet< Eplates
Effect of dielectric is to reduce the potential difference.

23. Dielectric Constant
Electric field in capacitor filled with insulator: Enet=Eplates-Edipoles
K – dielectric constant
Alpha is small.

24. Dielectric Constant Inside an insulator:
Dielectric constant for various insulators:
vacuum 1 (by definition)
air
typical plastic 5
NaCl 6.1
water 80
strontium titanate 310

25. Potential Difference in a Capacitor with Insulator

26. Potential Difference in Partially Filled CapacitorK -Q +Q
Talk if time permits – skip with no consequences s A B x

27. Energy Density of Electric Field
Energy can be stored in electric fields
(for small s)
Until now we have thought of energy (potential energy) as associated with interacting charges
Move one plate of a capacitor.
volume E
Field energy density:
(J/m3)
Energy expended by us was converted into energy stored in the electric field

28. Energy Density of Electric Field
In the previous slide, the “system” is the set of two plates. Work, Wexternal > 0, is done on the system by you – part of the “surroundings.”
If the force exerted by you just offsets the attractive force, Fby-plates, so that the plate moves with no gain in KE,

29. Electric Field and Potential
90V
100V x Ex
1 mm

30. Exercise
Suppose in some area of space V(x,y,z)=x2+yz. What is E(x,y,z)?

31. Potential Inside a Uniformly Charged Hollow Sphere=0
In general, integration path may be complex

32. Electron-Volt (eV) – Unit of Energy
What is the change in electric potential energy associated with moving an electron from 1Å to 2Å from a proton?
If an electron moves through a potential difference of 1 V there is a change in electric potential energy of 1 eV.
1 eV = e.(1 V) = ( C)(1 V) = 1.610-19 J

33. Example 0V
35 V E
If an electron moves from rest through a potential difference of 35 V, what would be its kinetic energy?
What if we had a proton?
0 V
35 V E
Which particle will move faster?

34. Shifting the Zero Potential
In most cases we are interested in V, not the absolute values of V