1 W04D1 Electric Potential and Gauss’ Law Equipotential Lines Today’s Reading Assignment Course Notes: Sections ,

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 PS 3 due Tuesday Tues Feb 26 at 9 pm in boxes outside or

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 Continuous Charge Distributions

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 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 Group Problem: Charged Ring Choose

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

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

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

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 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 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 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 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 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 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 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 Equipotentials

27 Topographic Maps

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

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

30 Direction of Electric Field E E is perpendicular to all equipotentials Field of 4 charges Equipotentials of 4 charges

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