1. 1 W04D1 Electric Potential and Gauss’ Law Equipotential Lines Today’s Reading Assignment Course Notes: Sections 3.3-3.4, 4.4-4.6. 4.10.5

2. Announcements Exam One Thursday Feb 28 7:30-9:30 pm Room Assignments (See Stellar webpage announcements) Review Tuesday Feb 26 from 9-11 pm in 26-152 PS 3 due Tuesday Tues Feb 26 at 9 pm in boxes outside 32-082 or 26-152 2

3. 3 Outline Continuous Charge Distributions Review E and V Deriving E from V Using Gauss’s Law to find V from E Equipotential Surfaces

4. 4 Continuous Charge Distributions

5. 5 Break distribution into infinitesimal charged elements of charge dq. Electric Potential difference between infinity and P due to d q. Superposition Principle: Reference Point:

6. 6 Group Problem Consider a uniformly charged ring with total charge Q. Find the electric potential difference between infinity and a point P along the symmetric axis a distance z from the center of the ring.

7. 7 Group Problem: Charged Ring Choose

8. 8 Electric Potential and Electric Field Set of discrete charges: Continuous charges: If you already know electric field (Gauss’ Law) compute electric potential difference using

9. 9 Using Gauss’s Law to find Electric Potential from Electric Field If the charge distribution has a lot of symmetry, we use Gauss’s Law to calculate the electric field and then calculate the electric potential V using

10. 10 Group Problem: Coaxial Cylinders A very long thin uniformly charged cylindrical shell (length h and radius a) carrying a positive charge +Q is surrounded by a thin uniformly charged cylindrical shell (length h and radius a ) with negative charge -Q, as shown in the figure. You may ignore edge effects. Find V(b) – V(a).

11. 11 Worked Example: Spherical Shells These two spherical shells have equal but opposite charge. Find for the regions (i) b < r (ii) a < r < b (iii) 0 < r < a Choose

12. 12 Electric Potential for Nested Shells From Gauss’s Law Use Region 1: r > b r No field  No change in V!

13. 13 Electric Potential for Nested Shells Region 2: a < r < b r Electric field is just a point charge. Electric potential is DIFFERENT – surroundings matter

14. 14 Electric Potential for Nested Shells Region 3: r < a r Again, potential is CONSTANT since E = 0, but the potential is NOT ZERO for r < a.

15. 15 Group Problem: Charge Slab Infinite slab of thickness 2d, centered at x = 0 with uniform charge density. Find

16. 16 Deriving E from V A = (x,y,z), B=(x+Δx,y,z) E x = Rate of change in V with y and z held constant

17. 17 Gradient (del) operator: If we do all coordinates: Deriving E from V

18. 18 Concept Question: E from V Consider the point-like charged objects arranged in the figure below. The electric potential difference between the point P and infinity and is From that can you derive E(P)? 1.Yes, its kQ/a 2 (up) 2.Yes, its kQ/a 2 (down) 3.Yes in theory, but I don’t know how to take a gradient 4.No, you can’t get E(P) from V(P)

19. 19 Concept Question Answer: E from V The electric field is the gradient (spatial derivative) of the potential. Knowing the potential at a single point tells you nothing about its derivative. People commonly make the mistake of trying to do this. Don’t! 4. No, you can’t get E(P) from V(P)

20. 20 Group Problem: E from V Consider two point like charged objects with charge –Q located at the origin and +Q located at the point (0,a). (a)Find the electric potential V(x,y)at the point P located at (x,y). (b)Find the x-and y-components of the electric field at the point P using

21. 21 Concept Question: E from V The graph above shows a potential V as a function of x. The magnitude of the electric field for x > 0 is 1.larger than that for x < 0 2.smaller than that for x < 0 3.equal to that for x < 0

22. 22 Concept Question Answer: E from V The slope is smaller for x > 0 than x < 0 Translation: The hill is steeper on the left than on the right. Answer: 2. The magnitude of the electric field for x > 0 is smaller than that for x < 0

23. 23 Concept Question: E from V The above shows potential V(x). Which is true? 1.E x > 0 is positive and E x < 0 is positive 2.E x > 0 is positive and E x < 0 is negative 3.E x > 0 is negative and E x < 0 is negative 4.E x > 0 is negative and E x < 0 is positive

24. 24 Concept Question Answer: E from V E is the negative slope of the potential, positive on the right, negative on the left, Translation: “Downhill” is to the left on the left and to the right on the right. Answer: 2. E x > 0 is positive and E x < 0 is negative

25. 25 Group Problem: E from V A potential V(x,y,z) is plotted above. It does not depend on x or y. What is the electric field everywhere? Are there charges anywhere? What sign?

26. 26 Equipotentials

27. 27 Topographic Maps

28. 28 Equipotential Curves: Two Dimensions All points on equipotential curve are at same potential. Each curve represented by V(x,y) = constant

29. 29 Direction of Electric Field E E is perpendicular to all equipotentials Constant E fieldPoint ChargeElectric dipole

30. 30 Direction of Electric Field E http://web.mit.edu/viz/EM/visualizations/electrostatics/InteractingCharges/zoo/zoo.htm E is perpendicular to all equipotentials Field of 4 charges Equipotentials of 4 charges

31. 31 Properties of Equipotentials E field lines point from high to low potential E field lines perpendicular to equipotentials E field has no component along equipotential The electrostatic force does zero work to move a charged particle along equipotential