Physics 72 Chapter 21

Electric Charge & Electric FieldChapter 21 of Young & Freedman’s

University Physics

1

Basic Laws/ Facts

1. Charge is conserved.

2. The electric field produced by a charge distribution is given by Coulomb’s Law.

3. Magnetic monopole or charge does not exist, while electric monopole or charge does exist.

4. A changing magnetic field produces an electric field.

5. The magnetic field is related to the currents, and to changing electric fields.

6. Electric and magnetic fields produces forces on charges according to the Lorentz equation.

We will devote about Sixty Days to study the ramifications of these Six Wisdoms.

http://images.allposters.com/images/30/007_yoda.jpg

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Goals for the Day1. Discuss the dichotomy, quantization & conservation of

electric charge

2. Given the initial/final charge distribution, calculate the final/initial charge distribution using conservation principles

3. Predict charge distributions, & the resulting attraction or repulsion, in a system of charged insulators & conductors

4. Outline the process of charging

5. Calculate the net electric force on a point charge exerted by a system of point charges

3

Greeks, Amber & Wool

http://wikipedia/commons/6/68/Raffael_058.jpg; http://www.thebakken.org/electricity/images-static/amber-wool.jpg; http://en.wikipedia.org/wiki/Image:Gouttes-drops-resine-2.jpg; http://www.freewebs.com/fromagatetozoisite/Ambers.jpg; http://www.vision.smgroup.info/wp-content/uploads/2006/04/lamb.jpg

Elektron (Greek) = Amber (English) Electron

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Electron

Joseph John Thomson(1856 – 1940)

http://www.aip.org/history/electron/images/jj-equip.jpg; http://www.le.ac.uk/bs/em/images/sem_montage.jpg; http://www.engr.uky.edu/~bjhinds/facil/images/2010.jpg

Electron Microscope

Electron Microscope Images

5

Electric Charge Quantization

e = 1.602 x 10-19 Coulomb Neutron: 0 Proton: +e Electron: -e Nproton: total number of proton; integer Nelectron: total number of electron; integer Charge of an object, q = Nproton *(+e) + Nelectron*(-e)

Dichotomy positive (+) & negative (-) Positively charged object: Nproton > Nelectron

Negatively charged object: Nproton < Nelectron

Uncharged/neutral object: Nproton = Nelectron

Conservation of Electric Charge Charge is NOT created nor destroyed Charge is only transferred from one object to another Fundamental law of nature (like conservation of energy)

http://en.wikipedia.org/wiki/Image:Usdollar100front.jpg

Benjamin Franklin(1706 – 1790)

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Electric Charge: Interaction Positively charged object REPELS another

positively charged object

Negatively charged object REPELS another negatively charged object

Positively charged object ATTRACTS negatively charged object

Uncharged/neutral object ATTRACTS charged (positive or negative) object

http://www.pbs.org/kcet/wiredscience/education/wired,megavolt1.JPG; http://www.glenbrook.k12.il.us/gbssci/Phys/Class/estatics/u8l1c1.gif

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Like charges repel…Opposite charges attract…

but…

Why do protons not repel each other in the nucleus?Why do electrons not collapse into the nucleus?

Classical Electromagnetism CANNOT explain the structure of the atom…Quantum Mechanics can…

Classical Electromagnetism only explains phenomena at scales greater than 10-10 meters.

Wait a minute!!! Contradictions?...

r ≈1 Angstrom≈10-10 meters

8

Kinds of Materials: Insulator

Insulators: electrons are bound to a nearby nucleus;

electrons are not free to move about the entire material

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Kinds of Materials: Conductor

Conductors: electrons are free to move about the entire material

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Kinds of Materials

http://www.ntsb.gov/events/twa800/crossect.jpg; http://www.billfrymire.com/gallery/webJpgs/bounce-rubber-band-ball-elastic.jpg; http://en.wikipedia.org/wiki/Image:The_Earth_seen_from_Apollo_17.jpg ; http://www.friedlandindustries.com/images/new/NonferrousMetalsCopper.jpg; http://www.liquidsculpture.com/images/water/water-drop-a.jpg

copper (conductor)

rubber (insulator)

wood (insulator)

glass (insulator)water (conductor)

electric cable (conductor within an insulator)

Earth/ground (conductor)

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Kinds of Materials

Semiconductor Conductivity is dependent on the

applied voltage Conducting at certain voltages;

insulating at other voltages

Superconductors Conductivity is perfect at certain

temperature & below (very, very cold temperature)

Cannot be explainable by Classical Electromagnetism

http://www.cstl.nist.gov/div837/Division/images/semi3.gif; http://www.chemistryland.com/CHM107/AirWeBreathe/Comp/superconductor2.jpg

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ChargingBasic ideas: Charge is conserved.Charging = add/remove charge to /from an object.Mechanically add or remove (rubbing an object with another charged object).Apply electric force (repel or attract) and allow the repelled charge to conduct to somewhere else.

Ground - a very large conductor that can supply an unlimited amount and kind of charge (e.g. earth).

Earth/ground

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Charging

http://www.glenbrook.k12.il.us/gbssci/phys/Class/estatics/u8l2b.html

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http://www.glenbrook.k12.il.us/gbssci/phys/Class/estatics/u8l2b.html

Charging15

Charging16

Charging: Electroscope

Electroscope: device for detecting electric charge

17

Electric Force: Coulomb’s Law

http://en.wikipedia.org/wiki/Image:Coulomb.jpg

Charles-Augustin de Coulomb1736 – 1806

The electrostatic force, F, between two point electric charges, q1 and q2, is directly proportional to the product, *, of the magnitudes of each charge and inversely proportional to the square

of the distance, r, between the charges.

F = k q1 * q2 ř12

r212

= k (±N1 e) * (±N2 e) ř12

r212

k (proportionality constant) = 1 ≈ 9 x 109 Nm2/C2

4πεo

εo (permittivity of free space) ≈ 8.85 x 10-12 C2/Nm2

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Recall Newton’s Gravitational Law?

http://www.crystalinks.com/newton.jpg

Isaac Newton1642 – 1726

The electrostatic force, F, between two point masses, m1 and m2, is

directly proportional to the product, *, of the magnitudes of each charge and inversely proportional to the square

of the distance, r, between the masses.

F = G m1 * m2 ř12

r212

G (gravitational constant) = 6.67 x 10-11 Nm2/kg2

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Force Vector quantity Depends on charge

Attractive or Repulsive Inverse square

Central; Weaker at greater distance Can act through vacuum Stronger than gravity Obeys Newton’s 3rd Law (Action-Reaction)

F12= - F21

Independent of geometrical & mechanical properties

Electric Force: Coulomb’s Law20

Principle of Superposition

The net force is the vector sum of all the individual forces acting on a system.

…in other words…

Consider two (test charge + another charge) at a time.

Apply Coulomb’s Law on each pair (as if other charges are not there).

Sum these pair-by-pair forces by vector addition.

21

Basic Electrostatic Problem

There is charge of interest (test), Q0 …

There is a group of charges…

What is the force exerted on Q0 by the other charges?

X

22

Basic Electrostatic ProblemTo determine whether they attract/ repel,

identify the magnitude & sign of the charges.

X

q1

Q0

q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

Example:Q0 > 0 q1 > 0q2 < 0q3 < 0 q4 > 0etc…

23


Calculate the distance between the other charges & Q0.

Example:r01 = distance bet. Q0 & q1 r02 = distance bet. Q0 & q2 r03 = distance bet. Q0 & q3 r04 = distance bet. Q0 & q4

etc…

X

q1

Q0

q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

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Basic Electrostatic ProblemUse Coulomb’s Law on each pair.

X

q1q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

Q0

F01 = k Q0 * q1 ř01

r201

F02 = k Q0 * q2 ř02

r202

F03 = k Q0 * q3 ř03

r203

F04 = k Q0 * q4 ř04

r204

etc,…

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Get the net/total force, Fnet. Add the forces as vectors.

q1q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

Q0

F01 = F01xi + F01yjF02 = F02xi + F02yjF03 = F03xi + F03yjF04 = F04xi + F04yjetc,…

Fnet = F01 + F02 + F03 + F04 +…Fnet = (F01x+F02x+F03x+F04x+…)i + (F01y+F02y+F03y+F04y+…)j

Note the arrow heads (recall Q0 > 0; q1 > 0; q2 < 0; q3 < 0; q4 > 0; etc…)

Fnet on Q0

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X

q1

Q0

q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

http://www.earthsmightiest.com/images/news/animation/penguins2.jpg; http://uk.gizmodo.com/ChickenLittleSing.jpg

If you know vectors & vector addition, its chicken!

Vector: Magnitude & Direction

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Yes! Electric Fields…

Is there an alternative (better) approach to these problem?

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http://images.art.com/images/-/LeBron-James--C10126004.jpeg; http://faculty.ssfs.org/~alisonb/Rodin%20Thinker-red.JPGhttp://www.tattonpark.org.uk/NR/rdonlyres/87A00E0B-9AAA-4852-82EF-9ADCC7137969/0/medievalcannon.jpg

How about the cannon ball?

Will the rubber ball go up?

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http://www.earthsmightiest.com/images/news/animation/penguins2.jpg; http://uk.gizmodo.com/ChickenLittleSing.jpg

No! No!!! They will fall. They will fall.

…because of the earth’s “gravitational field”

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What is a Field?

http://www.dkimages.com/discover/previews/740/51217.JPG

X

Go away, Blue Charge! Red Charge is mine.

Come here, Blue Charge.

I am in love.

FIELD… a quantity that has a value at each point in space; independent of the presence of object it acts on;

does not act on the object that produced it.

Action by CONTACT

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What is a Field?

http://www.dkimages.com/discover/previews/740/51217.JPG

X

Go away, Chicken Little! Red Charge is mine.

Come here, Chicken Little.

I am in love.

FIELD… a quantity that has a value at each point in space; independent of the presence of object it acts on;

does not act on the object that produced it.

Anybody who stands here gets the attention of the

neighbors… especially of the Red Charge & Green Charge.

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“gravitational field” ↔ electric field

http://www.crystalinks.com/newton.jpg

Isaac Newton1642 – 1726

F = G m * Mearth ř12

R2earth

Charles-Augustin de Coulomb1736 – 1806

F = k Q0 * qother ř12

R20,other

g = G Mearth ř12

R2earth

F = m g

E = k qother ř12

R20,other

F = Q0 E

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There is charge of interest (test), Q’ …


What is the force exerted on Q’ by the other charges?

X

34

Basic Electrostatic ProblemTo determine whether they attract/ repel,

identify the magnitude & sign of the charges.

X

q1

Q’

q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

Example:Q’ > 0 q1 > 0q2 < 0q3 < 0 q4 > 0etc…

35


Calculate the distance between the other charges & Q0.

Example:r01 = distance bet. Q’ & q1 r02 = distance bet. Q’ & q2 r03 = distance bet. Q’ & q3 r04 = distance bet. Q’ & q4

etc…

X

q1q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

Q’

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Basic Electrostatic ProblemUse Coulomb’s Law on each pair.

X

q1q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

Q’

F01 = k Q’* q1 ř01

r201

F02 = k Q’ * q2 ř02

r202

F03 = k Q’ * q3 ř03

r203

F04 = k Q’ * q4 ř04

r204

etc,…

37


q1q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

E01 = k q1 ř01

r201

E02 = k q2 ř02

r202

E03 = k q3 ř03

r203

E04 = k q4 ř04

r204

etc,…

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Get the net/total electric field, Enet. Add them as vectors.

q1q2 q3 q6 q8

q7

q4

q5

q10

q9

q11q12 q13

E01 = E01xi + E01yjE02 = E02xi + E02yjE03 = E03xi + E03yjE04 = E04xi + E04yjetc,…

Enet = E01 + E02 + E03 + E04 +…Enet = (E01x+E02x+E03x+E04x+…)i + (E01y+E02y+E03y+E04y+…)j

Note the arrow heads (recall Q0 > 0; q1 > 0; q2 < 0; q3 < 0; q4 > 0; etc…)

Enet at this point

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Vocabulary Check: “Electrostatic”

ALL charge/ charge distribution has its corresponding electric field…

“Electrostatic” means…a situation where we assume that…

Electric field of the charge of interest (test charge) does not change the configuration of the other charges.

40


We are not interested at charge of interest (test), Q’… unless it is there


What is the Electric Field at that position?

41

Electric Field Calculations

(at this point)

(at this point)

(at this point)

42

Compute the ratio of the electric force to the gravitational force exerted by a proton on an electron in a hydrogen atom.

43

Three point charges lie on the x - axis; q1= 25nC is at the origin, q2 = -10nC is at x = 2m, and q0 = 20nC is at x = 3.5m. Find the net force on q0 due to q1 & q2.

44

Charge q1 = +25nC is at the origin, charge q2 = -15nC is on the x-axis at x = 2m,and charge q0 = +20nC is at the point x = 2m,y = 2m as shown. Find the resultant force F on q0.

45

When a test charge, q0 = 5nC, is placed at a certain point, it experiences a force Fnet = 2 × 10−4 N in the x-direction. What is the electric field E at that point?

What is the force, Fnet, on an electron placed at a point where the electric field is E = (4 × 10+4 N/C)i? (e ≈ 1.6 x × 10-19 C)

E = Fnet / q0= [(2 × 10−4 N)i]/(5 × 10−9 C) = 4 × 104 N/C)i

Fnet = q0E = (1.6 × 10−19 C)(4 × 10+4 N)i = (6.4 × 10-15 N)i

48

A positive charge q1 = +8nC is at the origin, and a second positive charge q2 = +12nC is on the x-axis at a = 4m. Find the net electric field (a) at point P1 on the x-axis at x = 7m, and (b) at point P2 on the x-axis at x = 3m.

49

A charge q is at x = a and a second charge -q is at x = a. Find the electric field on the x-axis at an arbitrary point x > a. (b) Find the limiting form of the electric field for x >> a.

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Electric Field Calculations: Discrete Charges

You will need Algebra & Trigonometry

(at this point)

(at this point)

(at this point)

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Electric Field Calculations: Continuous Charges

Now, you will need Calculus & Trigonometry.

Isaac Newton(invented Calculus)

(at this point)

(at this point)

(at this point)

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Electric Field Calculations

line

surface

volume

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Charge Densities

1 unit length

1 unit length

1 unit length

1 unit length

1 unit length

1 unit

length

linear charge density

surface charge density

volume charge density

54

E on the Axis of a Finite Line Charge

55

E off the Axis of a Finite Line Charge 56

E off the Axis of an Infinite Line Charge

inverse linear distance dependence

57

E on the Axis of a Ring Charge 58

E on the Axis of a Uniformly Charged Disc

E due to an Infinite Plane of Charge

no distance dependence

59

Electric Field Lines

Electric field lines begin on positive charges (or at infinity) & end on negative charges (or at infinity).

The lines are drawn symmetrically entering or leaving an isolated charge.

60

Electric Field Lines

Number of lines leaving a positive charge or entering a negative charge is proportional to the charge magnitude.

Density of the lines (number of lines /unit area perpendicular to the lines) at any point is proportional to the magnitude of the field at that point.

61

Electric Field LinesAt large distances from a system of charges, the field lines are equally spaced & radial, as if they came from a single point charge equal to the net charge of the system.

Field lines do not cross. (If two field lines crossed, that would indicate two directions for the electric field at the point of intersection.)

62

Electric Dipole

Two (2) charges [two “di”; charges poles] same magnitude, qopposite signclose (bound) together

- +d

+q-q

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Electric Dipole: Dipole Moment

- +d

+q-q

64

p = qdDipole Moment, P vector (arrow: from - to +)“resistance” to turning/rotation

E

Dipole in a Uniform Electric Field: Forces & Torque

- ++++++++

-------

But if it has an angle it will turn TORQUE

ZERO (0) Net Force same magnitude opposite sign

-+ +-

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Dipole in a Uniform Electric Field: Forces & Torque

- ++++++++

-------

TORQUE, τ

same magnitude opposite sign

/Fon - /= /Fon + / = FFon - - Fon + = Fnet = 0

τ = 2F(d/2)sinΘ = qEdsinΘ = qdEsinΘτ = qdEsinΘ = p x E

E

66

Dipole in a Uniform Electric Field: Potential Energy

Equilibrium (stable): Uminimum

-+ +-

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-+

Dipole Potential Energy U = -p ∙ E

Umaximum = -(-p0) ∙ E = +p0 ∙ E

U = -p ∙ E = -p0EcosΘ;

/cosΘ/>0

Uminimum = -(p0) ∙ E= -p0E

p = qd

Ed on the x-axis at an arbitrary point x > a.

Electric Field of a Dipole, Ed

- +d

+q-q

a=d/2

The limiting form of Ed for x >> a.

Ed, at x>>a

Note: Ed is different from E shown in the previous slide. Ed is the field of the dipole.E is the field acting on the dipole.

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Physics 72 Chapter 21

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