Welcome…

1. Welcome… …to Physics 104.

2. Prologue Some things you will recall from “last”* semester…  Newton’s Laws  energy and its conservation *or whenever you took your previous physics class

3.  momentum and its conservation (linear and angular) These “things” aren’t going to go away!

4. Electric Charge Static Electricity There are two kinds of charge. - + Properties of charges  like charges repel  unlike charges attract  charges can move but charge is conserved “Law” of conservation of charge: the net amount of electric charge produced in any process is zero.

5. Coulomb’s Law Coulomb’s “Law” quantifies the magnitude of the electrostatic force. Coulomb’s “Law” gives the force (in Newtons) between charges q1 and q2, where r12 is the distance in meters between the charges, and k=9x109 N·m2/C2.

6. Force is a vector quantity. The equation on the previous slide gives the magnitude of the force. If the charges are opposite in sign, the force is attractive; if the charges are the same in sign, the force is repulsive. Also, the constant k is equal to 1/40, where 0=8.85x10-12 C2/N·m2. I could write Coulomb’s “Law” like this… attractive for unlike repulsive for like Remember, a vector has a magnitude and a direction.

7. + - + - + - The equation is valid for point charges. If the charged objects are spherical and the charge is uniformly distributed, r12 is the distance between the centers of the spheres. r12 - + If more than one charge is involved, the net force is the vector sum of all forces (superposition). For objects with complex shapes, you must add up all the forces acting on each separate charge (turns into calculus!).

8. We could have agreed that in the formula for F, the symbols q1 and q2 stand for the magnitudes of the charges. In that case, the absolute value signs would be unnecessary. However, in later equations the sign of the charge will be important, so we really need to keep the magnitude part. On your diagrams, show both the magnitudes and signs of q1 and q2. Your starting equation is this version of the equation: which gives you the magnitude F12 and tells you that you need to figure out the direction separately.

9. Solving Problems Involving Coulomb’s “Law” and Vectors You may wish to review vectors (on your own). Example: Calculate the net electrostatic force on charge Q3 due to the charges Q1 and Q2. y Q3=+65C 30 cm 60 cm =30º x Q1=-86C Q2=+50C 52 cm

10. Step 0: Think! This is a Coulomb’s “Law” problem (all we have to work with, so far). We only want the forces on Q3. Don’t worry about other forces. Forces are additive, so we can calculate F32 and F31 and add the two. If we do our vector addition using components, we must resolve our forces into their x- and y-components.

11. F32 F31 Step 1: Diagram y Draw a representative sketch—done. Draw and label relevant quantities—done. Draw axes, showing origin and directions—done. Q3=+65C 30 cm 60 cm =30º x Q1=-86C Q2=+50C 52 cm Draw and label forces (only those on Q3). Draw components of forces which are not along axes.

12. F32 F31 Step 2: Starting Equation y Q3=+65C 30 cm 60 cm =30º x Q1=-86C Q2=+50C 52 cm <complaining> “Do I have to put in the absolute value signs?” Yes. Unless you like losing points.

13. F32 F31 Step 3: Replace Generic Quantities by Specifics y Q3=+65C r31=60 cm r32=30 cm =30º x Q1=-86C Q2=+50C 52 cm (from diagram) Can you put numbers in at this point? OK for this problem. You would get F32,y = 330 N and F32,x = 0 N.

14. F32 F31 Step 3 (continued) y Q3=+65C r31=60 cm r32=30 cm =30º x Q1=-86C Q2=+50C (+ sign comes from diagram) 52 cm (- sign comes from diagram) Can you put numbers in at this point? OK for this problem. You would get F31,x = +120 N and F31,y = -70 N.

15. F32 F3 F31 You know how to calculate the magnitude F3 and the angle between F3 and the x-axis. (If not, holler!) Step 3: Complete the Math y Q3=+65C The net force is the vector sum of all the forces on Q3. 30 cm 60 cm =30º x Q1=-86C Q2=+50C 52 cm F3x = F31,x + F32,x = 120 N + 0 N = 120 N F3y = F31,y + F32,y = -70 N + 330 N = 260 N

16. I did a sample Coulomb’s law calculation using three point charges. How do you apply Coulomb’s law to objects that contain distributions of charges? We’ll use another tool to do that…

17. Today’s agendum: • The electric field. • You must be able to calculate the force on a charged particle in an electric field. • Electric field of due to a point charge. • You must be able to calculate electric field of a point charge. • Motion of a charged particle in a uniform electric field. • You must be able to solve for the trajectory of a charged particle in a uniform electric field. • The electric field due to a collection of point charges. • You must be able to calculate electric field of a collection of point charges. • The electric field due to a continuous line of charge. • You must be able to calculate electric field of a continuous line of charge.

18. Coulomb’s “Law”: The Big Picture Coulomb's “Law” quantifies the interaction between charged particles. r12 - + Q1 Q2 Coulomb’s “Law” was discovered through decades of experiment. By itself, it is just “useful." Is it part of something bigger?

19. is the local gravitational field. On earth, it is 9.8 N/kg, directed towards the center of the earth. What we calledg = 9.8 m/sec2 is the magnitude of the gravitational field. Gravitational Fields You experienced gravitational fields in Physics 103. Units of g are actually N/kg!

20. The Electric Field Coulomb's “Law” (demonstrated in 1785) shows that charged particles exert forces on each other over great distances. How does a charged particle "know" another one is “there?” Action At A Distance Viewpoint Electric, magnetic, and gravitational forces result of direct and instantaneous interaction between particles. Relativity theory shows why this viewpoint is wrong. Faraday developed the correct explanation.

21. Faraday, beginning in 1830's, was the leader in developing the idea of the electric field. Here's the idea: F12  A charged particle emanates a "field" into all space. + F13  Another charged particle senses the field, and “knows” that the first one is there. F31 + - F21 unlike charges attract like charges repel

22. The idea of an electric field is good for a number of reasons: •  It makes us feel good, like we’ve actually explained something. F12 + • OK, that was a flippant remark. There are serious reasons why the idea is “good.” F13 •  We can develop a theory based on this idea. From this theory may spring unimagined inventions. F31 + - F21 • If the theory explains past observations and leads to new predictions, the idea was “good.” unlike charges attract like charges repel

23. We define the electric field by the force it exerts on a test charge q0: The subscript “0” reminds you the force is on the “test charge.” I won’t require the subscripts when you use this equation for boardwork or on exams. This is your second starting equation. By convention the direction of the electric field is the direction of the force exerted on a POSITIVE test charge. The absence of absolute value signs around q0 means you must include the sign of q0 in your work. If the test charge is "too big" it perturbs the electric field, so the “correct” definition is You won’t be required to use this version of the equation. Any time you know the electric field, you can use this equation to calculate the force on a charged particle in that electric field.

24. The units of electric field are newtons/coulomb. Subsequently, you will learn that the units of electric field can also be expressed as volts/meter: The electric field exists independent of whether there is a charged particle around to “feel” it.

25. + Remember: the electric field direction is the direction a + charge would feel a force. A + charge would be repelled by another + charge. Therefore the direction of the electric field is away from positive (and towards negative). http://regentsprep.org/Regents/physics/phys03/afieldint/default.htm

26. Today’s agendum: • The electric field. • You must be able to calculate the force on a charged particle in an electric field. • Electric field of due to a point charge. • You must be able to calculate electric field of a point charge. • Motion of a charged particle in a uniform electric field. • You must be able to solve for the trajectory of a charged particle in a uniform electric field. • The electric field due to a collection of point charges. • You must be able to calculate electric field of a collection of point charges. • The electric field due to a continuous line of charge. • You must be able to calculate electric field of a continuous line of charge.

27. The Electric Field Due to a Point Charge Coulomb's law says ... which tells us the electric field due to a point charge q is …or just… This is your third starting equation.

28. We define as a unit vector from the source point to the field point: source point + field point The equation for the electric field of a point charge then becomes: You may start with either equation for the electric field (this one or the one on the previous slide).

29. Example: calculate the electric field at the electron’s distance away from the proton in a hydrogen atom (5.3x10-11 m). To be worked at the blackboard. For comparison, air begins to break down and conduct electricity at about 30 kV/cm, or 3x106 V/m.

30. Today’s agendum: • The electric field. • You must be able to calculate the force on a charged particle in an electric field. • Electric field of due to a point charge. • You must be able to calculate electric field of a point charge. • Motion of a charged particle in a uniform electric field. • You must be able to solve for the trajectory of a charged particle in a uniform electric field. • The electric field due to a collection of point charges. • You must be able to calculate electric field of a collection of point charges. • The electric field due to a continuous line of charge. • You must be able to calculate electric field of a continuous line of charge.

31. If E is constant, then a is constant, and you can use the equations of kinematics* (remember way back to the beginning of Physics 103?). Motion of a Charged Particle in a Uniform Electric Field A charged particle in an electric field experiences a force, and if it is free to move, an acceleration. - - - - - - - - - - - - - If the only force is due to the electric field, then - E F + + + + + + + + + + + + +

32. Example: a proton and an electron enter a region of uniform electric field. Describe their motion. Direction of forces? Magnitudes of accelerations? Shape of trajectories?

33. Example: an electron moving with velocity v0 in the positive x direction enters a region of uniform electric field that makes a right angle with the electron’s initial velocity. Express the position and velocity of the electron as a function of time. y - - - - - - - - - - - - - x - E v0 + + + + + + + + + + + + + To be worked at the blackboard.

34. Today’s agendum: • The electric field. • You must be able to calculate the force on a charged particle in an electric field. • Electric field of due to a point charge. • You must be able to calculate electric field of a point charge. • Motion of a charged particle in a uniform electric field. • You must be able to solve for the trajectory of a charged particle in a uniform electric field. • The electric field due to a collection of point charges. • You must be able to calculate electric field of a collection of point charges. • The electric field due to a continuous line of charge. • You must be able to calculate electric field of a continuous line of charge.

35. Example: calculate the electric field at position P due to the two protons shown. P + + D D To be worked at the blackboard. Example: field of an electric dipole. No time to work today. Will work in Lecture 3. Study example 21.9 in your text.

36. Today’s agendum: • The electric field. • You must be able to calculate the force on a charged particle in an electric field. • Electric field of due to a point charge. • You must be able to calculate electric field of a point charge. • Motion of a charged particle in a uniform electric field. • You must be able to solve for the trajectory of a charged particle in a uniform electric field. • The electric field due to a collection of point charges. • You must be able to calculate electric field of a collection of point charges. • The electric field due to a continuous line of charge. • You must be able to calculate electric field of a continuous line of charge.

37. + + + + + + + + + + + + + + + + Electric Field Due To A Line of Charge Think of a line of charge as a collection of very tiny point charges all lined up. The “official” term for “as tiny as you can imagine” is “infinitesimal.” We get the electric field for the line of charge by adding the electric fields for all the infinitesimal point charges. These words are meant to remind you of things you have learned in calculus.

38. is the linear density of charge (amount of charge per unit length).  may be a function of position. Think   length.  times the length of line segment is the total charge on the line segment. Consider charge uniformly distributed along a line (e.g., electrons on a thread).  dx x dx If charge is distributed along a straight line segment parallel to the x-axis, the amount of charge dq on a segment of length dx is dx.

39. I’m assuming positively charged objects in these “distribution of charges” slides. dE P r’ x dq The electric field at point P due to the charge dq is

40. E P r’ x dq The electric field at P due to the entire line of charge is The integration is carried out over the entire length of the line, which need not be straight. Also,  could be a function of position, and can be taken outside the integral only if the charge distribution is uniform.

41. Example: calculate the electric field due to an infinite line of positive charge. There are two approaches to the mathematics of this problem. One approach is that of example 21.11, where an equation for the electric field for an infinite line of charge is derived. Thus, if this were given for homework, you would need to repeat this derivation! If you use the text’s approach, you must evaluate this indefinite integral, which is in appendix B (page A4) of your text: If you need it, you can look this integral up; your instructor will give it to or let you look it up in the text.

42. Example: calculate the electric field due to an infinite line of positive charge.

43. Today’s agendum: • Review. • The electric field of a dipole. • You must be able to calculate the electric field of a dipole. • The electric field due to a collection of point charges (continued). • You must be able to calculate the electric field of a collection of point charges. • Electric field lines. • You must be able to draw electric field lines, and interpret diagrams that show electric field lines. • A dipole in an external electric field. • You must be able to calculate the moment of an electric dipole, the torque on a dipole in an external electric field, and the energy of a dipole in an external electric field.

44. The Big Picture, Part I In Lecture 1 you learned Coulomb's Law: r12 - + Q1 Q2 Coulomb’s Law quantifies the force between charged particles.

45. The Big Picture, Part II In Lecture 2 you learned about the electric field. • There were two kinds of problems you had to solve: • 1. Given an electric field, calculate the force on a charged particle. • 2. Given one or more charged particles, calculate the electric field they produce.

46. The Big Picture, Part II 1. Given an electric field, calculate the force on a charged particle. - E F You may not be given any information about where this electric field “comes from.”

47. The Big Picture, Part II 2. Given one or more charged particles, calculate the electric field they produce. 2 slides from now we’ll do this for a dipole. source point + field point Example: electric field of a point charge.

48. Today’s agendum: • Review. • The electric field of a dipole. • You must be able to calculate the electric field of a dipole. • The electric field due to a collection of point charges (continued). • You must be able to calculate the electric field of a collection of point charges. • Electric field lines. • You must be able to draw electric field lines, and interpret diagrams that show electric field lines. • A dipole in an external electric field. • You must be able to calculate the moment of an electric dipole, the torque on a dipole in an external electric field, and the energy of a dipole in an external electric field.

49. - +q + -q d A Dipole A combination of two electric charges with equal magnitude and opposite sign is called a dipole. The charge on this dipole is q (not zero, not +q, not –q, not 2q). The distance between the charges is d. Dipoles are “everywhere” in nature. This is an electric dipole. Later in the course we’ll study magnetic dipoles, which, as you might guess, have a north and a south magnetic pole.

50. The Electric Field of a Dipole Example: calculate the electric field at point P, which lies on the perpendicular bisector a distance L from a dipole of charge q. P L to be worked at the blackboard - +q + -q d