4. Electric potential

4.1 Electric (electrostatic) potential energy

xqExFWU

Example:

2

21

r

qqkF 2

21

r

mmGF

Conservative forces

?

1

1

/104

U

Cq

cmx

CNE

JCNmCU 4426 10/101010

+++++++

a

-------

b

E

x

We can introduce potential energy:

Compare electric force and gravitational force

WUUU ab

1

4.2 Electric potential and potential difference

q

FE

electric field – force

q

UV

potential – energy

Definitions:

Example:

Units (Volt): VC

JV

?

1

100

W

Cq

VV

JVCqVUW 46 1010010

2

4.3 Electric potential and electric field

x

VEx

Example:

?

1.0

100

E

mx

VV +++++

-----

E

Units: mVCNE //

mVm

V

x

VE /1000

1.0

100

xq

U

xq

W

xq

xF

q

FE xxx

3

4.4 Potential due to a group of point charges

a) One charge

2r

QkE

0

Vr

QkVV

Usually we assume that

b) Several charges(superposition)

...21 VVV4

4.5 Equipotential surfaces

Definition: V = const

Properties: •W = 0 for any motion along any equipotential surface•The electric field, E is always perpendicular to equipotential surfaces•The electric field, E points in the direction of decreasing potential•The surface of a conductor is always equipotential•All points of a conductor have the same potential

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2D mapping of potential. Positive point charge.

30 V

20 V

10 V

E

x

y

V

E

Topographic map of Mt. Fuji:Examples:

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Example: What is the electric energy stored in a system of three charges q = 3.0 nC that form an equilateral triangle of side a = 1.0 cm?

The stored energy is: A) Positive B) Negative C) Zero

q

q

q aExplanation: This question can be reformulated:•How much energy has been put in the system to built it?•How much work was done to built it?

We have to push the charges to arrange them like this → Add energy

Unless charges are somehow fixed, they will move to a situation with less energy

Solution: External work done to bring a charge from infinity:

r

qQkVVqUUWext

For the first charge: 01 extW

akqWext /22

akqWext /2 23

JakqWtot 3.24/3 2

For the second charge:

For the therd charge:

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Example: Three charges q = 3.0 nC are initially fixed at the corners of an equilateral triangle of side a = 1.0 cm. One of them is released. Find its kinetic energy when it has doubled the distance to each of the other two charges.

q

q

q

a

q

q

q

2a

2 2 9 29

f 2

Nm (3.0 10 C)2 2(9 10 ) 8.1 J

2 2(0.01 m)C

qK k

a

i i f fK U K U

2 2

f0 2 22

q qk K ka a

8