1. Question -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal block as shown below. Select the diagram that represents the final state of the metal.

2. Chapter 16 Electric Field of Distributed Charges

3. Distributed Charges

4. Length: L Charge: Q What is the pattern of electric field around the rod? Cylindrical symmetry Uniformly Charged Thin Rod Could the rod be a conductor and be uniformly charged?

5. General Procedure for Calculating Electric Field of Distributed Charges 1.Cut the charge distribution into pieces for which the field is known 2.Write an expression for the electric field due to one piece (i) Choose origin (ii) Write an expression for  E and its components 3.Add up the contributions of all the pieces (i) Try to integrate symbolically (ii) If impossible – integrate numerically 4.Check the results: (i) Direction (ii) Units (iii) Special cases

6. Apply superposition principle: Divide rod into small sections  y with charge  Q Assumptions: Rod is so thin that we can ignore its thickness. If  y is very small –  Q can be considered as a point charge Step 1: Divide Distribution into Pieces

7. What variables should remain in our answer? ⇒ origin location, Q, x, y 0 What variables should not remain in our answer? ⇒ rod segment location y,  Q y – integration variable Vector r from the source to the observation location: Step 2: E due to one Piece

8. Magnitude of r: Unit vector r: Magnitude of  E: Step 2: E due to one Piece

9. Vector ΔE:

10. Step 2: E due to one Piece  Q in terms of integration  y:

11. Step 2: E due to one Piece

12. Simplified problem: find electric field at the location Step 3: Add up Contribution of all Pieces

13. Numerical summation: Assume: L=1 m,  y=0.1 m, x=0.05m if Q=1 nC  E x =286 N/C Increase precision: 10 slices 31.75 [Q/(   )] 20 slices 39.31 [Q/(   )] 50 slices 39.80 [Q/(4π  0 )] 100 slices 39.80 [Q/(4π  0 )] Step 3: Add up Contribution of all Pieces

14. Integration: taking an infinite number of slices  definite integral Step 3: Add up Contribution of all Pieces

15. Evaluating integral: Cylindrical symmetry: replace x  r Step 3: Add up Contribution of all Pieces

16. In vector form: Step 4: Check the results: Direction:  Units:  Special case r>>L:  E of Uniformly Charged Thin Rod At center plane

17. Very long rod: L>>r Q/L – linear charge density 1/r dependence! Special Case: A Very Long Rod

18. At distance r from midpoint along a line perpendicular to the rod: For very long rod: Field at the ends: Numerical calculation E of Uniformly Charged Rod

19. General Procedure for Calculating Electric Field of Distributed Charges 1.Cut the charge distribution into pieces for which the field is known 2.Write an expression for the electric field due to one piece (i) Choose origin (ii) Write an expression for  E and its components 3.Add up the contributions of all the pieces (i) Try to integrate symbolically (ii) If impossible – integrate numerically 4.Check the results: (i) Direction (ii) Units (iii) Special cases

20. Origin: center of the ring Location of piece: described by , where  = 0 is along the x axis. Step 1: Cut up the charge distribution into small pieces Step 2: Write E due to one piece A Uniformly Charged Thin Ring

21. Step 2: Write  E due to one piece A Uniformly Charged Thin Ring

22. Step 2: Write  E due to one piece Components x and y: A Uniformly Charged Thin Ring

23. Step 2: Write  E due to one piece Component z: A Uniformly Charged Thin Ring

24. Step 3: Add up the contributions of all the pieces A Uniformly Charged Thin Ring

25. Step 4: Check the results Direction  Units  Special cases: Center of the ring (z=0): E z =0  Far from the ring (z>>R):  A Uniformly Charged Thin Ring

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

27. Electric field at other locations:needs numerical calculation A Uniformly Charged Thin Ring

28. Section 16.5 – Study this! A Uniformly Charged Disk