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Physics 2B for Materials and Structural EngineeringPhysics 2B for Materials and Structural Engineering

Ben Gurion University of the Negevwww.bgu.ac.il/atomchip

Lecturer: Daniel Rohrlich Teaching Assistants: Oren Rosenblatt, Irina Segel

Week 1. Charge, E and Coulomb’s law – Introduction • electrical charges, quantization and conservation • Coulomb’s law • addition of electric forces • electric fieldSources: Halliday, Resnick and Krane, 4th Edition, Chap. 27;Halliday, Resnick and Krane, 5th Edition, Chaps. 25-26;Purcell (Berkeley course 2), Chap. 1, Sects. 1.3 – 1.4 and 1.7.

Introduction

Both electricity and magnetism were known to the ancient Greeks 2600 years ago. When they rubbed a piece of amber with fur, the amber would attract straw and hair.

Introduction

Both electricity and magnetism were known to the ancient Greeks 2600 years ago. When they rubbed a piece of amber with fur, the amber would attract straw and hair.

Amber is resinthat became a fossil.

Introduction

Both electricity and magnetism were known to the ancient Greeks 2600 years ago. When they rubbed a piece of amber with fur, the amber would attract straw and hair.

ל = Bמ Dש Bענבר = חήλεκτρον = amber

Introduction

ל = Bמ Dש Bענבר = חήλεκτρον = amber

תוך יחזקאל (א,ד): ל מ] Bמ Dש Bח Bעין ה Dה כ aתוכ יב ומ] ב] aה לו ס Bג fנ Dת ו Bח Bק Bל Dת אש מ] Dדול ו aן ג aנ aעאש. aה

“ = ”amber .”ל Bמ Dש Bח“ According to the Septuagint ,

Introduction

The ancient Greeks also knew of stones that would attract iron.

Magnetite from Utah, U.S.A. © R.Weller/Cochise College.

2600 years ago, and even 300 years ago, electric and magnetic phenomena appeared to be very unusual. They appeared to have very little to do with the rest of nature, and nothing to do with each other.

2600 years ago, and even 300 years ago, electric and magnetic phenomena appeared to be very unusual. They appeared to have very little to do with the rest of nature, and nothing to do with each other.

Today, our view is about as different from this ancient view as possible: everything in the natural world, including all of biology and chemistry, depends on electricity and magnetism; also electricity and magnetism depend on each other.

2600 years ago, and even 300 years ago, electric and magnetic phenomena appeared to be very unusual. They appeared to have very little to do with the rest of nature, and nothing to do with each other.

Today, our view is about as different from this ancient view as possible: everything in the natural world, including all of biology and chemistry, depends on electricity and magnetism; also electricity and magnetism depend on each other.

Why did it take so long to see that electricity and magnetism are everywhere?

Electrical charges, quantization and conservation

Electrostatics: the study of electric charges at rest.

Electrical charges, quantization and conservation

Electrostatics: the study of electric charges at rest.

How do we know that there are two kinds of charge? (We call them “positive” and “negative”, but any other names would be as good.)

Electrical charges, quantization and conservation

Electrostatics: the study of electric charges at rest.

How do we know that there are two kinds of charge? (We call them “positive” and “negative”, but any other names would be as good.)

Since electric charges can repel as well as attract, there must be at least two kinds.

Electrical charges, quantization and conservation

Electrostatics: the study of electric charges at rest.

How do we know that there are two kinds of charge? (We call them “positive” and “negative”, but any other names would be as good.)

Since electric charges can repel as well as attract, there must be at least two kinds.

We have heard that “same charges repel, opposite charges attract”, but could it be the other way around? Could our world be a world in which “opposite charges repel, same charges attract”?

Electrical charges, quantization and conservation

Electric charge is quantized, and nobody knows why!

Not even quantum mechanics explains (so far) why charge is quantized.

• All electrons have exactly the same charge –e:

e = (1.602 176 487 ± 0.000 000 040) × 10-19 C

Our unit of charge is C and is called a Coulomb.

Electrical charges, quantization and conservation

Electric charge is quantized, and nobody knows why!

Not even quantum mechanics explains (so far) why charge is quantized.

• All electrons have exactly the same charge –e:

e = (1.602 176 487 ± 0.000 000 040) × 10-19 C

The first person to measure e was R. A. Millikan (around 1910).

Electrical charges, quantization and conservation

Electric charge is quantized, and nobody knows why!

Not even quantum mechanics explains (so far) why charge is quantized.

• All electrons have exactly the same charge –e:

e = (1.602 176 487 ± 0.000 000 040) × 10-19 C

The first person to measure e was R. A. Millikan. He sprayed droplets of oil and measured the mass m of each droplet from its fall. Then he applied an electric field of strength E to balance the droplet in mid-air, and extracted e from mg = eE.

Electrical charges, quantization and conservation

• The proton and the electron are very different, but the electron charge and the proton charge are known to be the same (except for sign) to an accuracy of one part in 1020.

So why did it take so long to see that electricity and magnetism are everywhere?

Electrical charges, quantization and conservation

Conservation of electric charge: The total electric charge in an isolated system does not change.

There are processes that change the number of charged particles in an isolated system, but no process changes the total electric charge.

Coulomb’s law

Consider two fixed point charges. One is located at r1 and has charge q1; the other is located at r2 and has charge q2. The force of the charge located at r1 on the charge at r2 has magnitude

where r12 = | r1 – r2 | and the constant k in the units of this course is

,

212

2112

r

qqkF

. /CmN 10 8.994

1 229

0

k

q1

q2

Coulomb’s law (vector formulation)

Consider two fixed point charges. One is located at r1 and has charge q1; the other is located at r2 and has charge q2. The force of the charge located at r1 on the charge at r2 is

where is a unit vector pointing from r1 to r2. By Newton’s Third Law, the force F21 of the charge located at r2 on the charge at r1 is equal in magnitude and opposite in sign.

, ˆ

4

1122

12

21

012 rF

r

12r

q1

q2F21

F12

Coulomb’s law (vector formulation)

Consider two fixed point charges. One is located at r1 and has charge q1; the other is located at r2 and has charge q2. The force of the charge located at r1 on the charge at r2 is

writing it a slightly different way. By Newton’s Third Law, the force F21 of the charge located at r2 on the charge at r1 is equal in magnitude and opposite in sign.

, 4

1123

12

21

012 rr

rrF

q1

q2F21

F12

Addition of electric forces

Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r1 and has charge q1; the second is located at r2 and has charge q2; the third is located at r3 and has charge q3. What is the total force of the charges at r1 and r2 on the point charge located at r3?

q1

q2

q3

Addition of electric forces

Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r1 and has charge q1; the second is located at r2 and has charge q2; the third is located at r3 and has charge q3. What is the total force of the charges at r1 and r2 on the point charge located at r3?

q1

q2

q3

F13

F23

Addition of electric forces

Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r1 and has charge q1; the second is located at r2 and has charge q2; the third is located at r3 and has charge q3. What is the total force of the charges at r1 and r2 on the point charge located at r3? It is

. ˆ4

1 ˆ

4

1232

23

32

0132

13

31

02313 rrFF

r

r

q1

q2

F13

F23 q3

Addition of electric forces

Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r1 and has charge q1; the second is located at r2 and has charge q2; the third is located at r3 and has charge q3. What is the total force of the charges at r1 and r2 on the point charge located at r3? It is

q1

q2

F13

F23

F13 + F23

q3

. ˆ4

1 ˆ

4

1232

23

32

0132

13

31

02313 rrFF

r

r

Addition of electric forces

Example 1 (electric dipole): Two equal and opposite point charges lie on the z-axis: charge e is at z = a and charge –e is at z = –a. What is the force Fq on a point charge q on the z-axis at arbitrary z?

–e

q

0

e

Addition of electric forces

Example 1 (electric dipole): Two equal and opposite point charges lie on the z-axis: charge e is at z = a and charge –e is at z = –a. What is the force Fq on a point charge q on the z-axis at arbitrary z?

Answer:

–e

q

0

z

z

zF

ˆ/1

1

ˆ4

4

1

ˆ 4

1

22230

2220

220

zaz

aeq

az

zaeq

az

eq

az

eqq

e

Addition of electric forces

Example 1 (electric dipole): Two equal and opposite point charges lie on the z-axis: charge e is at z = a and charge –e is at z = –a. What is the force Fq on a point charge q on the z-axis at arbitrary z?

Answer:

–e

q

0

z

z

zF

ˆ.../11

ˆ4

4

1

ˆ 4

1

2223

0

2220

220

zaz

aeq

az

zaeq

az

eq

az

eqq

e

Addition of electric forces

Answer:

ˆ

1

ˆ4

4

1

ˆ 4

1

30

2220

220

z

z

zF

z

aeq

az

zaeq

az

eq

az

eqq

–e

q

0 for z >> a.

e

Addition of electric forces

Let’s continue this example with the charge q on the x-axis at arbitrary x. What is the force Fq on the charge q?

–e

q0

e

Addition of electric forces

Let’s continue this example with the charge q on the x-axis at arbitrary x. What is the force Fq on the charge q?

e

–e

0 . ˆ ][2

2)ˆ(

,sin2)ˆ(

2/3220

22

z

z

zF

ax

eqaax

aF

Fq

θθ

q

Answer: Each charge on the z-axis produces a force of magnitude F = eq/4πε0(x2 + a2) but the x-components of these forces cancel. The net force is down:

Addition of electric forces

Example 2: An infinite, stationary straight line carries uniform charge per unit length λ. Off the wire, a distance L from it (closest approach) is a fixed point charge of magnitude q. What is the force of the line on the point charge?

q

L

Addition of electric forces

Example 2: An infinite, stationary straight line carries uniform charge per unit length λ. Off the wire, a distance L from it (closest approach) is a fixed point charge of magnitude q. What is the force of the line on the point charge?

q0

Answer: First we choose convenient coordinates.z

xL

Addition of electric forces

Example 2: An infinite, stationary straight line carries uniform charge per unit length λ. Off the wire, a distance L from it (closest approach) is a fixed point charge of magnitude q. What is the force of the line on the point charge?

q0

Answer: First we choose convenient coordinates.Now consider two infinitesimal line elements of length dz and charge λdz symmetrically spaced above and below the origin on the z-axis.

z

x

λdz

λdz

L

Addition of electric forces

q0

Each line element produces a force of magnitude F = qλdz/4πε0 (z2 + L2) but the z-components of these forces cancel. The net force is horizontal:

z

xθθ

. ˆ][2

1

ˆ4

12

ˆcos2

2/3220

22220

x

x

xF

Lz

dzqLLz

L

Lz

dzq

Fnet

L

Addition of electric forces

q0

For the total force on the point charge q, we integrate this expression:

z

θθ

L

. ][2

ˆ

02/322

0dz

Lz

qLq

x

F

Addition of electric forces

Substitute z = L tan θ, dz = L dθ /cos2θ:

. ][2

ˆ

02/322

0dz

Lz

qLq

x

F

cos]tan[2

ˆ

][2

ˆ

2

2/

02/3222

0

02/322

0

Ld

LL

qL

dzLz

qLq

x

xF

Addition of electric forces

Substitute z = L tan θ, dz = L dθ /cos2θ:

Addition of electric forces

x

. ˆ2

cos2

ˆ

cos]tan[2

ˆ

][2

ˆ

0

2/

00

2

2/

02/3222

0

02/322

0

xx

x

xF

L

qdq

L

Ld

LL

qL

dzLz

qLq

Electric field

In general, a set of fixed point charges q1, q2, q3,… located at r1, r2, r3,… produces an electric force Fq on a fixed point charge q located at r. If we divide Fq by q, we get the electric field E(r) at the point r arising from the charges q1, q2, q3,… at r1, r2, r3,…:

q1

q2

q3

q

Fq

q4q5

q6

q7

Electric field

In general, a set of fixed point charges q1, q2, q3,… located at r1, r2, r3,… produces an electric force Fq on a fixed point charge q located at r. If we divide Fq by q, we get the electric field E(r) at the point r arising from the charges q1, q2, q3,… at r1, r2, r3,…:

q1

q2

q3

E(r)

q4q5

q6

q7

Electric field

Example 1: What is the electric field of an electron?

Electric field

Example 1: What is the electric field of an electron?

Answer: The force of an electron at the origin on a point charge at r is

therefore E(r) is

, ˆ4

12

0rF

r

eqq

. ˆ4

1)(

20

rrEr

e

Electric field

Example 2: What is the electric field of a line of charge with uniform linear charge density λ?

Answer: We found that the force due to the line of charge on a point charge q at a distance L from the line is

where ρ is the radial coordinate with respect to the line, therefore E(ρ) is

, ˆ2 0

ρF

. ˆ2

)(0

ρE

Electric field

What difference does it make whether we talk about point charges that produce an electric force, or about an electric field?

If the charges don’t move, it doesn’t make a difference. But if the charges move, it does make a difference. We will see that the electric field E(r) takes on a life of its own; it is not simply a function of where the charges are.

Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 4(a):

Two small balls of mass m and charge q hang from the same point on threads of length L. At equilibrium, their separation is x and their angular separation is 2θ. Assuming θ small so that sin θ ≈ θ ≈ tan θ, what is x?

θθ

x

L

Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 4(a):

Two small balls of mass m and charge q hang from the same point on threads of length L. At equilibrium, their separation is x and their angular separation is 2θ. Assuming θ small so that sin θ ≈ θ ≈ tan θ, what is x?

θθ

x

Answer: Let T denote the tension in the threads. The total force must vanish in the vertical and horizontal directions, hence

L

3/1

0

2

20

2

20

2

2

2/tan

4

4sin and cos

mg

Lqx

L

x

mgx

q

x

qTmgT

T

Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:

Two point charges q are held on the z-axis at points z = ±a. Where on the xy-plane – at what distance R from the origin – is the electric field strongest?

Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:

Two point charges q are held on the z-axis at points z = ±a. Where on the xy-plane – at what distance R from the origin – is the electric field strongest?

0

z

θθ

R

Answer: We use our previous calculation, replacing the two line elements with two point charges. The electric field at a distance R from the z-axis isq

q

, ][2

1)(

2/3220 aR

qRRE

Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:

Two point charges q are held on the z-axis at points z = ±a. Where on the xy-plane – at what distance R from the origin – is the electric field strongest?

0

z

θθ

R

and we maximize it by requiring dE/dR = 0:

q

q

, ][2

1)(

2/3220 aR

qRRE

, ][

2

2

3

][2

1

)(0

2/522

2

2/3220

aR

qR

aR

q

REdR

d

Halliday, Resnick and Krane, 5th Edition, Chap. 25, Prob. 11:

0

z

θθ

R

q

q

, ][

2

2

3

][2

10

2/522

2

2/3220

aR

qR

aR

q

so

. 2/

3 222

aR

RaR