Physics 2B for Materials and Structural Engineering Ben Gurion University of the Negev 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 field Sources: Halliday, Resnick and Krane, 4 th Edition, Chap. 27; Halliday, Resnick and Krane, 5 th Edition, Chaps ; 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 resin that 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. ענבר = חַשְׁמַל = ήλεκτρον = amber

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. ענבר = חַשְׁמַל = ήλεκτρον = amber יחזקאל (א,ד): עָנָן גָּדוֹל וְאֵשׁ מִתְלַקַּחַת וְנֹגַהּ לוֹ סָבִיב וּמִתּוֹכָהּ כְּעֵין הַחַשְׁמַל מִתּוֹךְ הָאֵשׁ. ” = “amber”. חַשְׁמַל“ 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.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.

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 = ( ± ) × 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 = ( ± ) × 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 = ( ± ) × 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 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 r 1 and has charge q 1 ; the other is located at r 2 and has charge q 2. The force of the charge located at r 1 on the charge at r 2 has magnitude where r 12 = | r 1 – r 2 | and the constant k in the units of this course is q1q1 q2q2

Coulomb’s law (vector formulation) Consider two fixed point charges. One is located at r 1 and has charge q 1 ; the other is located at r 2 and has charge q 2. The force of the charge located at r 1 on the charge at r 2 is where is a unit vector pointing from r 1 to r 2. By Newton ’ s Third Law, the force F 21 of the charge located at r 2 on the charge at r 1 is equal in magnitude and opposite in sign. q1q1 q2q2 F 21 F 12

Coulomb’s law (vector formulation) Consider two fixed point charges. One is located at r 1 and has charge q 1 ; the other is located at r 2 and has charge q 2. The force of the charge located at r 1 on the charge at r 2 is writing it a slightly different way. By Newton ’ s Third Law, the force F 21 of the charge located at r 2 on the charge at r 1 is equal in magnitude and opposite in sign. q1q1 q2q2 F 21 F 12

Addition of electric forces Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r 1 and has charge q 1 ; the second is located at r 2 and has charge q 2 ; the third is located at r 3 and has charge q 3. What is the total force of the charges at r 1 and r 2 on the point charge located at r 3 ? q1q1 q2q2 q3q3

Addition of electric forces Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r 1 and has charge q 1 ; the second is located at r 2 and has charge q 2 ; the third is located at r 3 and has charge q 3. What is the total force of the charges at r 1 and r 2 on the point charge located at r 3 ? q1q1 q2q2 q3q3 F 13 F 23

Addition of electric forces Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r 1 and has charge q 1 ; the second is located at r 2 and has charge q 2 ; the third is located at r 3 and has charge q 3. What is the total force of the charges at r 1 and r 2 on the point charge located at r 3 ? It is q1q1 q2q2 F 13 F 23 q3q3

Addition of electric forces Electric forces add like vectors! Suppose we have three fixed point charges. One is located at r 1 and has charge q 1 ; the second is located at r 2 and has charge q 2 ; the third is located at r 3 and has charge q 3. What is the total force of the charges at r 1 and r 2 on the point charge located at r 3 ? It is q1q1 q2q2 F 13 F 23 F 13 + F 23 q3q3

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 F q on a point charge q on the z-axis at arbitrary z? –e–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 F q on a point charge q on the z-axis at arbitrary z? Answer: –e–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 F q on a point charge q on the z-axis at arbitrary z? Answer: –e–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 F q on a point charge q on the z-axis at arbitrary z? Answer: –e–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 F q on the charge q? –e–e q 0 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 F q on the charge q? e –e–e 0 θ θ q Answer: Each charge on the z-axis produces a force of magnitude F = eq/4πε 0 (x 2 + a 2 ) 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? q 0 Answer: First we choose convenient coordinates. z x 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? q 0 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 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? q 0 Each line element produces a force of magnitude F = qλdz/4πε 0 (z 2 + L 2 ) but the z-components of these forces cancel. The net force is horizontal: z x θ θ 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? q 0 For the total force on the point charge q, we integrate this expression: z θ θ 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? Substitute z = L tan θ, dz = L dθ /cos 2 θ:

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? Substitute z = L tan θ, dz = L dθ /cos 2 θ:

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? x

Electric field In general, a set of fixed point charges q 1, q 2, q 3,… located at r 1, r 2, r 3,… produces an electric force F q on a fixed point charge q located at r. If we divide F q by q, we get the electric field E(r) at the point r arising from the charges q 1, q 2, q 3,… at r 1, r 2, r 3,…: q1q1 q2q2 q3q3 q FqFq q4q4 q5q5 q6q6 q7q7

Electric field In general, a set of fixed point charges q 1, q 2, q 3,… located at r 1, r 2, r 3,… produces an electric force F q on a fixed point charge q located at r. If we divide F q by q, we get the electric field E(r) at the point r arising from the charges q 1, q 2, q 3,… at r 1, r 2, r 3,…: q1q1 q2q2 q3q3 E(r)E(r) q4q4 q5q5 q6q6 q7q7

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

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

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, 5 th 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? θ θ x L

Halliday, Resnick and Krane, 5 th 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? θ θ x Answer: Let T denote the tension in the threads. The total force must vanish in the vertical and horizontal directions, hence L T

Halliday, Resnick and Krane, 5 th 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, 5 th 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 θ θ 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 is q q

Halliday, Resnick and Krane, 5 th 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 θ θ R and we maximize it by requiring dE/dR = 0: q q

Halliday, Resnick and Krane, 5 th 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 θ θ R q q so