General Procedure for Calculating Electric Field of Distributed Charges

1. General Procedure for Calculating Electric Field of Distributed Charges • Cut the charge distribution into pieces for which the field is known • Write an expression for the electric field due to one piece • (i) Choose origin • (ii) Write an expression for E and its components • Add up the contributions of all the pieces • (i) Try to integrate symbolically • (ii) If impossible – integrate numerically • Check the results: • (i) Direction • (ii) Units • (iii) Special cases

2. A Uniformly Charged Thin Ring Distance dependence: Far from the ring (z>>R): Ez~1/z2 Close to the ring (z<<R): Ez~z

3. A total charge Q is uniformly distributed over a half ring with radius R. The total charge inside a small element dθ is given by: dθ θ R A. B. C. Q E. D. Clicker Question • Choice One • Choice Two • Choice Three • Choice Four • Choice Five • Choice Six

4. A total charge Q is uniformly distributed over a half ring with radius R. The ycomponent of electric field at the center created by a short element dθ is given by: dθ θ R B. A. D. C. Q +y Clicker Question • Choice One • Choice Two • Choice Three • Choice Four

5. A Uniformly Charged Disk Along z axis Close to the disk (0 < z < R) Very close to disk (0 < z << R) If z/R is extremely small Approximations:

6. Field Far From the Disk Exact For z>>R Point Charge

7. Uniformly Charged Disk Edge On

8. Capacitor -Q +Q s Two uniformly charged metal disks of radius R placed very near each other A single metal disk cannot be uniformly charged: charges repel and concentrate at the edges Two disks of opposite charges, s<<R: charges distribute uniformly: Almost all the charge is nearly uniformly distributed on the inner surfaces of the disks; very little charge on the outer surfaces. We will calculate E both inside and outside of the disk close to the center

9. Step 1: Cut Charge Distribution into Pieces -Q +Q E+ Enet s E- We know the field for a single disk There are only 2 “pieces”

10. Step 2: Contribution of one Piece E+ Enet s E- Origin: left disk, center Location of disks:z=0, z=s Distance from disk to “2” z, (s-z) Left: Right: z 0

11. Step 3: Add up Contributions Location: “2”(inside a capacitor)  Does not depend on z

12. Step 3: Add up Contributions E+ Enet s E- z 0 Location: “3”(fringe field) For s<<R: E1=E30 Fringe field is very small compared to the field inside the capacitor. Far from the capacitor (z>>R>>s): E1=E3~1/z3(like dipole)

13. Electric Field of a Capacitor E+ Enet s E- z 0 Inside: Fringe: Step 4:check the results:  Units:

14. Clicker Question Which arrow best represents the field at the “X”? A) B) C) E=0 D) E)

15. Electric Field of a Spherical Shell of Charge Field inside: Field outside: (like point charge)

16. E of a Sphere Outside E2 E3 E6 E5 E1+E4  Direction: radial - due to the symmetry Divide into 6 areas:

17. E of a Sphere Inside  Magnitude: E=0 Note:E is not always 0 inside – other charges in the Universe may make a nonzero electric field inside.

18. E of a Sphere Inside E=0: Implications Fill charged sphere with plastic. Will plastic be polarized? No! Solid metal sphere: since it is a conductor, any excess charges on the sphere arranges itself uniformly on the outer surface. There will be no field nor excess charges inside! In general: there is no electric field inside metals

19. Integrating Spherical Shell Divide shell into rings of charge, each delimited by the angle  and the angle + From ring to point: d=(r-Rcos) Surface area of ring: R  R Rsin  Rcos d r Q A mess of math

20. Exercise A solid metal ball bearing a charge –17 nC is located near a solid plastic ball bearing a uniformly distributed charge +7 nC (on surface). Show approximate charge distribution in each ball. Metal -17 nC Plastic +7 nC What is electric field field inside the metal ball?

21. Exercise Two uniformly charged thin plastic shells. Find electric field at 3, 7 and 10 cm from the center 3 cm: E=0 7 cm: 10 cm:

22. A Solid Sphere of Charge What if charges are distributed throughout an object? Step 1: Cut up the charge into shells R For each spherical shell: outside: r E inside:dE = 0 Outside a solid sphere of charge: for r>R

23. A Solid Sphere of Charge Inside a solid sphere of charge: R E r for r<R Why is E~r?  On surface:

24. Patterns of Fields in Space What is in the box? vertical charged plate? no charges?

25. Patterns of Fields in Space Box versus open surface …no clue… Seem to be able to tell if there are charges inside Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.