ElectricityElectric Potential

1

Review of Work

Work Done on a Point Charge by a Point Charge

Potential Energy and Potential of Point Charges

Potential Energy and Potential of Many Point Charges

Relationship between Force, Field, Energy, Potential

Electric Power

Potential Surfaces

SI Units

Field and Potential Graphs

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2

Definition of Work

Work is defined as the amount of force acting over a distance.

The unit of work is either Joules (J) or electron-volts (eV)

Δr

F

F rW

cosW F r

φ

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3

Example of Work

The following is an example of the work done by gravity on a mass.

Δh2

Δh1

Δw

y

x

F rW

m

2121 00 hmghmghmghmgW

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4

A2

A1

Example of Work

This is an example using the the fact that integrals are areas under a curve.

JNmbhA 5252

1

2

11

JNmbhA 6432

1

2

12

JAAW 121

F (N)

0

1

-1 X (m)

0 1 2

F rW

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Definition of a Conservative Force

A force is conservative if the work it does on an object is zero when the object moves along a path and returns to its initial position.

Examples of conservative forces are gravity, elastic force, electricity and magnetism.

The following are not conservative: friction, tension, normal force.

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y

x

Example of a Conservative ForceGravity as a mass is raised and lowered

1

2

3

5Δh2

Δh1

11 hmgW

02 W

123 hmghmgW

04 W

25 hmgW

WTOTAL = 0

4

m

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11 LfW s

WTOTAL = - fs (2ΔL1 + 2ΔL2 + 2Δw )

y

x

2

1

3

45

ΔL2

ΔL1

Δw

wfW s2

123 LLfW s

wfW s 4

25 LfW s

m

Example of a Non-conservative ForceFriction on a block that is moving around a table

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8

Consider the work necessary to put together two point charges.

The force is given by Coulomb’s Law.

1 22

12

kq qF

r

The work needed to bring one charge from infinity to within a distance r12 of the other is

12

21212

2112

12

r

qkq

r

qkqdr

r

qkqW

rr

Don’t worry about the integral in the last equation. It is only there so that those who know calculus can see where the other part came from.

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The potential energy held between these two point charges is then

Now we consider the same relationship, but with electric field instead of force. This we call the potential.

F E U r V r

12

21

r

qkqWU

Since , then we see that

EF q

qVU r

kqV

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10

3

12

What if there are more point charges?

12

3

1 212

12

q qU k

r

1 313

13

q qU k

r 2 3

2323

q qU k

r

1 3 2 31 212 13 23

12 13 23

q q q qq qU U U U k k k

r r r

1

2i j

i j ij

q qU k

r

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3

12

Example 12

3

1

2i j

i j ij

q qU k

r

2nC3nC

4nC

2m

2.1m 2.1m

1 3 2 3 3 1 3 21 2 2 1

12 13 21 23 31 32

1

2

q q q q q q q qq q q qk

r r r r r r

1 3 2 3 1 3 2 31 2 1 2

12 13 23 12 13 23

12 2 2

2

q q q q q q q qq q q qU k k

r r r r r r

29

2

2nC 4nC 3nC 4nCNm 2nC 3nC8.99 10

C 2m 2.1m 2.1mU

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What about potential?

12

3

0P

12

3

110

10

qV k

r

220

20

qV k

r

330

30

qV k

r

31 210 20 30

10 20 30

qq qV V V V k k k

r r r

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3

12

Example 12

3

00

i

i

qV k

r

2nC3nC

4nC

2m

2m 2m

31 2

10 20 30

qq qkr r r

29

2

Nm 2nC 3nC 4nC8.99 10

2 2 2Cm m m

3 3 3

V

0P

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Relationship between Work, Potential Energy, Force and Potential

The table below represents the fundamental equations of electricity all of

which a derived from the electric field E.

Quantity General EquationEquation for

Point Charges

Electric Field

Force

Potential

Potential

1 212 2

12

q qF k

rF qE

2

qE k

r

FU r qV

1 2

1212

q qU k

r

EV r

q

V kr

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A surface is an equipotential surface if the electric potential at every point on the surface is the same.

As charges move on an equipotential surface the electric force does no work.

The electric field at a point is always perpendicular to the equipotential surface on which the point lies.

The electric field always points in the direction of decreasing potential.

Java AppletThis Applet can be found at http://www.slcc.edu/schools/hum_sci/physics/tutor/2220/e_fields/java/

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Common SI Units in Electricity

Electric

Potential

V J/C

Electric Field

V/m N/C

Energy or Work

eV J

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Relationship Electric Potential and Electric Field

The electric field is the slope of the electric potential.

V

s

slope of this line is -E

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Relationship Electric Potential and Electric Field

The electric potential is the area under the curve of electric field.

E

s

positive area

negativearea

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What is electric power?

Power is the change in work over time

We will use this latter when we reach the topic in circuits.

qVU qP V IV

t t t

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What is electric power?

Power is the change in work over time

We will use this latter when we reach the topic in circuits.

qVU qP V IV

t t t