Physics 122B Electricity and Magnetism

1. Physics 122B Electricity and Magnetism Lecture 3 (Knight: 26.2-3)Electric Field, Field Lines andCharge Distributions March 30, 2007 Martin Savage

2. Electric Charge Distributions Physics 122B - Lecture 3

3. Electric Field of a Point Charge Positive Negative Physics 122B - Lecture 3

4. Electric Field Superposition q Therefore, the electric field of any charge distribution can be calculated by dividing the distribution into point charges and superimposing these. Physics 122B - Lecture 3

5. The Electric Field ofMultiple Point Charges Component electric fields may be superimposed to obtain the net field. This is very useful in calculating the field produced by a number of point charges. This is really three equations: Physics 122B - Lecture 3

6. The E-Field of 3 Point Charges in the plane of reflection symmetry q1=q2=q3=q Physics 122B - Lecture 3

7. The E-Field of 3 Point Charges in the plane of reflection symmetry Near-field limit: Far-field limit: Physics 122B - Lecture 3

8. The Electric Dipole Permanent Dipole Induced Dipole Physics 122B - Lecture 3

9. Electric Field of a Dipole Y >> s Physics 122B - Lecture 3

10. Q -Q 2Q Field Map vs. Field Lines Field lines start on positive charges. weak Field lines stop on negative charges. More charge Þ more field lines. strong Field lines never cross. Physics 122B - Lecture 3

11. A Prelude toMaxwell’s Equations Suppose you come across a vector field that looks something like this. What are the identifiable structures in this field? 1. An “outflow” structure: 2. An “inflow” structure: 3. An “clockwise circulation” structure: 4. An “counterclockwise circulation” structure: Maxwell’s Equations will tell us that the “flow” structures are charges (+ and -) and the “circulation” structures …. we will come back to this later…do no occur in static situations Physics 122B - Lecture 3

12. The Dipole Field Field Map Field Lines Physics 122B - Lecture 3

13. Two Positive Charges Field Map Field Lines Physics 122B - Lecture 3

14. Charge Distributions Symmetry Principle: An electric field must have the samesymmetries as the charge distribution that produced it. Physics 122B - Lecture 3

15. A(yes)A E (no) 720 E Types of Symmetry Translational symmetry: translation along a line does not change object. Full rotational symmetry:any rotation about any axis does not change object: Sphere Cylindrical symmetry:any rotation about one axis does not change object: Cylinder Reflectional symmetry: mirror image reflection is same as object. Partial rotational symmetry: 720 rotation about one axis does not change object: Star Physics 122B - Lecture 3

16. Computing E-Fields of Charged Objects Using Coulomb’s Law • Choose a coordinate system that will facilitate integration. • Use any applicable symmetries to set E-field components to zero or equal to each other. • Break up the object into point-like elements. • Write the Coulomb’s Law contribution to the E-field from a representative point-like element. • Integrate over the entire object to get the E-field. Physics 122B - Lecture 3

17. Example:E-Field of a Charged Line A thin uniformly charged rod of length L has a total charge Q. Find the electric field at a distance r from the axis of the rod in the plane that bisects the rod. Strategy: Break up the rod into small charge elements, use Coulomb’s Law to calculate the E field of each, and add them up. Physics 122B - Lecture 3

18. Example:E-Field of a Charged Line (2) Consider the x component of the electric field contribution dEx produced by a small segment i of the rod located at yi with length dy. Its charge DQi will be (Q/L)dy. Physics 122B - Lecture 3

19. An Infinite Line of Charge…the limit of a finite line-charge Top view; field lines spread in a plane + Side view; field lines are parallel. Field of an infinite line of charge falls off as 1/r (not 1/r2). Physics 122B - Lecture 3

20. Question Which of the following actions will increase the electric field strength at the position of the dot? (a) Rod longer, net charge the same;(b) Rod shorter, net charge the same;(c) Rod wider, net charge the same;(d) Rod narrower, net charge the same;(e) Move dot farther from rod. Physics 122B - Lecture 3

21. Computing Charged ObjectE-Fields Using Coulomb’s Law • Choose a coordinate system that will facilitate integration. • Use any applicable symmetries to set E-field components to zero or equal to each other. • Break up the object into point-like elements. • Write the Coulomb’s Law contribution to the E-field from a representative point-like element. • Integrate over the entire object to get the E-field. Physics 122B - Lecture 3

22. The E-Field of a Charged Ring A thin uniformly charged ring of radiusRhas a total chargeQ. Find the electric field on the axis of the ring .. the z-axis …(perpendicular to the page). The linear charge density of the ring isl = Q/(2pR).The system has cylindrical symmetry for rotations about the axis, soalong the z-axisEx=Ey=0and we need only to findEz. Consider a small segment of the circumference of the ring of widthdl = R df. The contribution to the electric field on the z-axis is : Physics 122B - Lecture 3

23. The on-axis E-Field of a Charged Ring (2) Near-field limit: linear Far-field limit: 1/r2 (same as point charge) Physics 122B - Lecture 3

24. End of Lecture 3 • Before the next lecture, read Knight,Chapters 26.4 through 26.5 • Lecture Homework #1 is on the Tychosystem and is due at 10pm on Wednesday