i

f

Path independence principle:V between two points does not depend on integration path

Potential Difference: Path Independence

What is E inside metal? E = 0

What is the potential difference (Vf – Vi)?

Is V zero everywhere inside a metal?No! But it is constant

In static equilibrium

Potential Difference in Metal

In static equilibrium the electric field is zero at all locations along any path through a metal.

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.

i

f

E

d =3 mm

+Q-Q

V = 6 Volt

Potential in MetalIn 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.

d =3 mm

+Q1-Q1 1 mm

Charges +Q2 and –Q2

What are the charges Q1 and Q2?

Now we have 2 capacitors instead of one

V = 4 V

There is no “conservation of potential”!

Potential in MetalIn static equilibrium Insert a 1 mm thick metal slab into the

center of the capacitor.Metal slab polarizes and has charges +Q2 and -Q2 on its surfaces.

Q2=Q1E=0 inside metal

V inside metal slab is zero!

XAt X

𝐸1=𝐸2

Ignoring the fringe fields, E = 2000V/m in each capacitor (from previous slide).

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)

If a metal is not in static equilibrium, the potential isn’t constant in the metal.

Potential in MetalThere can be a potential in metal if is NOT in static equilibrium

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 , the potential difference is: V = -EL

Question

0.02m 0.03m 0.04m

A B

300 V/m

0 V/m

300 V/m

What is VB-VA?A) 270 VB) -270 VC) -18 VD) 6 VE) -6 V

Question

0.02m 0.03m 0.04m

A B

0 V/m

0x

𝑉 𝐵−𝑉 𝐴=−(300 Vm ) (0.02 m −0 )

=

The electric field is uniform in the region below left. The point B is at <0, 0, 0> m. The point C is at <0,-2, 0>m. The potential difference along the path from B to C is +500V.

Potential difference due to a stationary point charge is independent of the path

Potential difference along a closed loop is zero

Round Trip Potential Difference

+

Is the following “curly” pattern of electric field possible?

dl

dl

dl

Predicting Possible Field Configuration

is always parallel to

This “curly” pattern of electric field is impossible to produce by arranging any number of stationary point charges!

Method 1: Divide into point charges and add up contributions due to each charge

Q

Potential of a Uniformly Charged Ring

Method 2: Integrate electric field along a path

Q

Potential of a Uniformly Charged Ring

Q

What is V for z>>R ?

The same as for a point charge!

Potential of a Uniformly Charged Ring

one ring:

integrate:

Potential of a Uniformly Charged Disk

Potential of a Uniformly Charged Disk

Can find :

Electric field in capacitor filled with insulator: Enet=Eplates+Edipoles

Eplates=const (in capacitor)

1 2 3 4 5

Edipoles,A

A B

Edipoles,B

Edipoles complicated f(x,y,z)

Travel from B to A:Edipoles is sometimes parallel to dl, and sometimes antiparallel to dl

Potential Difference in an Insulator

∆𝑉=−∫𝐴

𝐵

𝐸 ∙𝑑 �⃗�

Instead of traveling through inside – travel outside from B to A:

BAEdipoles, average

Potential Difference in an Insulator

Effect of dielectric is to reduce the potential difference.

Enet< Eplates

Enet=Eplates+Edipoles,average ∆𝑉 𝐴 𝐵=−∫𝐵

𝐴

𝐸𝑑𝑖𝑝𝑜𝑙𝑒𝑠 ∙𝑑�⃗�¿0

Electric field in capacitor filled with insulator: Enet=Eplates-Edipoles

K – dielectric constant

Dielectric Constant

Dielectric constant for various insulators:

vacuum 1 (by definition)air 1.0006typical plastic 5NaCl 6.1water 80strontium titanate 310

Inside an insulator:

Dielectric Constant

s

Potential Difference in a Capacitor with Insulator

+Q-Q

s

d

K

Potential Difference in Partially Filled Capacitor

BAx

Energy can be stored in electric fields

(for small s)

Evolume

Field energy density: (J/m3)

Energy expended by us was converted into energy stored in the electric field

Energy Density of Electric Field

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,

1 mm

90V 100Vx

Ex

Electric Field and Potential

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

Exercise

=0

In general, integration path may be complex

Potential Inside a Uniformly Charged Hollow Sphere

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?

0V 35 V

E

0 V 35 V

E

Which particle will move faster?

Example

In most cases we are interested in V, not the absolute values of V

Shifting the Zero Potential