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Chapter 21: Electric Charge and Electric FieldElectric Charge

Ancient Greeks ~ 600 BC Static electricity: electric charge via friction (see also fig 21.1)

(Attempted) pith ball demonstration:2 kinds of properties

2 objects with same property repel each other2 objects with different properties attract each otherboth properties are always created together

Benjamin Franklin:kinds of charges are positive and negativeby convention, negative charge associated with amber

Conservation of Charge: The algebraic sum of all the electric charges in any closed system is constant.

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Conductors and Insulators

(Objects are usually “charged” by moving electric charge around, rather than creating or destroying charge.)

Conductor:

charge passes easily through the material

=> conductors contain charges which are free to move

Insulator:

charge cannot move (easily) through material

Semiconductor:

transition between insulator and conductor, usually of interest because of exotic electrical properties

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Charging by induction (see also fig 21.6).−−−−

+ −+ − + −

−−−−

+ + +

−−−−

+ + +

++ +

Electrical Force on Neutral Insulators: Induced Polarization (see fig 21.7, n.a. )

−−−−

+ −+ −+ −+ −+ −

+ −+ −+ −+ −+ −+ −

+ −+ −+ −+ −+ −+ −

+ −

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Quantization and Conservation of ChargeMicroscopic structure of matter: Atoms (see fig 21.4)

Nucleusmost of masspositive chargecomposed of protons (each has charge = +e) and neutrons (no electric charge)

“orbiting” electrons (each has charge = −e)Atoms tend to be charge neutralcharge quantized: e = 1.6x10-19 C

Charge transfer usually in the form of addition or removal of electrons.

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Coulomb’s LawA description of the interaction between two (point) chargesThe magnitude of the Force exerted by one charge on the other

is proportional to the magnitude of each of the chargesis inversely proportional to the square of the distance between the chargesacts along a line connecting the charges

2

29

2

29

02 10910988.8

41

CmN

CmNk

rQqkF ⋅×≈⋅×===

πε

Unit of charge is the Coulomb, a new type of quantity.

How big is 1 coulomb?

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Another form of Coulomb’s Law

the force exerted by Q (“source charge”) on q (“test charge”)

q toQ from distance,

q toQ from r,unit vectoˆ

qon Force

ˆ2

=

==

=

r

rF

rrQqkF

ˆ r

F

Q

q

r

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rrQqkF ˆ2=

ˆ r

F

Q

q

r

Coulomb’s Law: the force exerted by Q on qElectric Charge and Electric Field (cont’d)

∑=

++=

++=

i ii

i rr

qQk

rr

qQkrr

qQk

FFF

ˆ

ˆˆ

2

222

212

1

1

21

r21F

Q1

q

r1

ˆ r 1

2F

Q2ˆ r 2

F

For several Sources

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•Analyze geometry/draw diagram

•calculate magnitude of each contribution

•calculate components of each contribution

•add contributions as vectors (add component by component)

Fi = kQi q

ri2

A charge q = 5.0 nC is at the origin. A charge Q1 = 2.0 nC is located 2cm to the right on the x axis and and Q2 = −3.0 nC is located 4 cm to the right on the x axis. What is the net force on q?

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.3m

.3m

.4m

Q1 = 2.0 µC

Q2 = 2.0 µC

q = 4.0 µC

Geometry

Magnitudes

Components

Net Force

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Example: Compare the electric repulsion of two electrons to theirgravitational attraction

F=k ∣Qq∣r2

versus F=G Mmr 2

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Electric Field and Electric forcesElectric field is a “disturbance” in space resulting from the presence of (source) charge, which exerts a force on a (test) charge.

ˆ r

Q

P

r

ˆ r

qPFPE

/)at ()at (

≡

Q

q

r )at () o(

qEqqnF

≡

ˆ r

F

Q

q

r

Force of interaction Source charge creates a disturbance in space

Test charge “senses” the disturbance in space

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2

2

P toQ from distance,

P toQ from r,unit vectoˆ

Pat Field

ˆ

rQkE

r

rE

rrQkE

=

=

==

=

ˆ r

Q

P

r

E

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r2

Q1r1

ˆ r 1

Q2ˆ r 2

For several Sources

∑=

++=

++=

i ii

i rrQk

rrQkr

rQk

EEE

ˆ

ˆˆ

2

222

212

1

1

21

Force Law: F = qE

2E

E

1E

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Elementary Electric Field Examples:What is the electric field 30 cm from a +4nC charge?

When a 100 volt battery is connected across two parallel conducting plates 1 cm apart, the resulting charge configuration produces a nearly uniform electric field of magnitude E = 1.00E4 N/C.

Compare the electric force on an electron in this field with its weight.

What is the acceleration of the electron?

The electron is released from rest at the top plate. What is the final speed of the electron as it hits the second plate?How long does it take the electron to travel this distance?

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Electric Field Calculations

•Analyze geometry/draw diagramElectric Field Contributions are directed away from positive charges, toward negative charges

•calculate magnitude of each contribution

•calculate components of each contribution

•add contributions as vectors (add component by component)

E i=k ∣Qi∣ri

2

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a

x

+Q

−Q

Geometry

Magnitudes

Components

Net Force

a

Electric Dipole: two equal size(Q), opposite sign charges separated by a distance (l = 2a).Determine the electric field on the x-axis

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a

x

+Q

−Q

a

( )

( )3

2322

2322

20

xpk

axpk

axaQkE

E

ax

y

x

− →

+−=

+−=

=

>>

Field of an electric dipole

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a

x

Geometry

Magnitudes

Components

a

Line of charge: uniform line of charge (charge Q, l = 2a oriented along y-axis).Determine the electric field on the x-axissee also animation http://phys23p.sl.psu.edu/phys_anim/EM/E_line_of_charge.avi

y

dQ=λdy

2222

222

22

cos

cossin

22

yxx

yxdyk

EddE

yxdyk

rdQkEd

yxrrx

ry

aQ

ady

QdQ

x

++=

=

+==

+=

==

==

λ

θ

λ

θθ

λ

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Add Components

( )

rkE

axxkE

axxQkE

axxkQ

axxak

yxdyk

dEE

r

x

x

ay

ay

xx

λ

λ

λ

λ

2 :charge of line infinite

,2

,

2

2

2222

2322

charge all

=

<<≈

>>≈

+=

+=

+=

=

∫

∫=

−=

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Next: infinite sheet of charge is composed of of a series of infinite lines of charges

Look carefully at related textbook examples of a ring of charge, and a disk of charge.

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sheet of charge composed of a series of infinite lines of chargeσ = charge per area

Electric Field due to an infinite sheet of charge

z

y

Geometry

Magnitudes

Components

2222

22

22

2

cos

22

cossin

yzy

yzdzk

EddE

yzdzk

rkEd

yzrry

rz

dz

y

++=

=

+==

+=

==

=

σ

θ

σλ

θθ

σλ

θ

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Add Components

!field! electric uniform4

12

222

2

arctan12

2

00

22

charge all

==

=

−−=

=

+=

=

∞

−∞=

∞=

−∞=∫

∫

πεεσ

σπππσ

σ

σ

k

kk

yz

yyk

yzydzk

dEE

z

z

z

yy

z

y

θ

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Interaction of an electric dipole with an electric fielddipole in a uniform electric field

x

y

z−q

−ql

F=qE

F=− qE

Ep

pElqE

FlFlRHRFr

F

z

iii

i

×=

==

+=

×==

=

∑∑

∑

τ

θθ

θθτ

ττ

sinsin

sin2

sin2

!dipole ofcenter about torque

0

p

F = F = qE

θ

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Work done rotating dipole in an electric fieldwatch directions of τ, θ an dθ

x

y

z

θl

EppEU

UpEpE

dpEW

dpEddW

⋅−=−=

∆−=−=

−=

−==

∫

θθ

θθ

θθ

θθθτθ

θ

cos)(

coscos

sin

sin

12

2

1p

dθ

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“Lines of Force”

Electric Field Lines: a means of visualizing the electric field=A line in space always tangent to the electric field at each point in spaceconcentration give indication of field strengthdirection give direction of electric field•start on positive charges, end on negative charges•electric field lines never cross

see figure 21.26, http://phys23p.sl.psu.edu/simulations/physlets/em_fieldplot.html

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+ + + + + + + + + + + + + +

− − − − − − − − − − − − − −