1. Electric Fields Review Coulomb’s Law Electric Fields
Multiple-charge Electric Fields
Continuous-charge Electric Fields
Parallel Plate Fields
Parallel Plate Examples
Gauss’s Law

2. Coulomb’s Law Fundamental property charge Positive and negative charge
Units Coulombs (C)
Force Law

3. Compare with gravitation
Fundamental property mass
Positive mass only
Units kilograms (kg)
Force Law

4. “Field” concept (we did this in gravity)
Newton’s Universal Law of Gravitation
𝐹= 𝐺 𝑚 1 𝑚 2 𝑟 2
One-step process (tedious)
𝐹 𝑥 = 𝐺𝑀 𝑚 𝑥 𝑟 2
𝐺𝑀 𝑚 𝑐𝑎𝑟 𝑟 𝐺𝑀 𝑚 𝑡𝑟𝑢𝑐𝑘 𝑟 𝐺𝑀 𝑚 𝑝𝑒𝑑 𝑟 2
Two-step process (simple)
𝑔= 𝐺𝑀 𝑟 2 =9.8 𝑁/𝑘𝑔
𝐹 𝑥 = 𝑚 𝑥 𝑔
𝑚 𝑐𝑎𝑟 𝑔 𝑚 𝑡𝑟𝑢𝑐𝑘 𝑔 𝑚 𝑝𝑒𝑑 𝑔

5. “Field” concept - Electrical
Cellphone signal
Multiple users sharing same tower.
Why calculate each phone separately?
2 step process
1. Calculate common “field”.
2. Calculate each phone’s interaction with that “field”.
“Field” equals # of “bars” you have!

6. Electrostatic vs. Gravitational Field
In gravitation we calculate
𝐹= 𝐺𝑀 𝑟 2 𝑚
Bracket part becomes “g”
In electrostatic we calculate
𝐹= 𝑘𝑄 𝑟 2 𝑞
Bracket part becomes “E”
Field line point away from (+), toward (-)
F=qE (+) moves with field, (-) moves against field

7. Electric Field Animation

8. Electrostatic Field and Force Direction
Define: Field of +q1 points outward, -q1 points inward
Force on q2 (magnitude & direction) given by 𝐹= 𝑞 2 𝐸
Result: +q2 feels force with field, -q2 feels force against field
So like charges repel, opposite charges attract + -

9. What if gravity was attractive/repulsive???
Positive and negative mass (!!)
(don’t worry, there isn’t such a thing)
“Negative” mass falls to ceiling
“Positive” mass falls to floor g

10. Electrostatic vs. Gravitational Field
Electric Field
Gravitational Field
Field Concept
Can be anything
Usually 9.8 m/s2
Force & field
F=q2E
Force in field/opposite direction
F= m2g
Force in field direction
Field Definition
Ratio of Force/charge
Ratio of Force/mass,
(simplifies to acceleration)
Strength
Electrostatic so strong appears on circuit-board scale
Gravity so weak only appears on planetary scale
Comment
“E” quite interesting
Varies all over the place
“g” usually boring
usually constant 9.8 m/s2
Superposition
Can superimpose continuously
Wires, electrodes, circuit boards
Can superimpose discretely
Planets, etc

11. Electric Field - Example 16-8 (1)
Field from Q1
𝐸 1 =𝑘 𝑞 1 𝑟 =9∙ 𝑁 𝑚 2 𝐶 ∙ 10 −6 𝐶 𝑚 2 =5.625 ∙ 𝑁/𝐶 𝑡𝑜 𝑙𝑒𝑓𝑡
Field from Q2
𝐸 2 =𝑘 𝑞 2 𝑟 =9∙ 𝑁 𝑚 2 𝐶 ∙ 10 −6 𝐶 𝑚 2 = ∙ 𝑁 𝐶 𝑡𝑜 𝑙𝑒𝑓𝑡
𝐸 𝑡𝑜𝑡𝑎𝑙 =6.3 ∙ 𝑁 𝐶 𝑡𝑜 𝑙𝑒𝑓𝑡

12. Electric Field - Example 16-8 (2)
Force on proton at P
𝐹=𝑞𝐸= +1.6∙ 10 −19 𝐶 6.3 ∙ 𝑁 𝐶 =1∙ 10 −10 𝑁 𝑡𝑜 𝑙𝑒𝑓𝑡
Acceleration of proton at P
𝑎= 𝐹 𝑚 = 1∙ 10 −10 𝑁 1.67∙ 10 −27 𝑘𝑔 =5.99 ∙ 𝑚 𝑠 𝑡𝑜 𝑙𝑒𝑓𝑡
Force on electron at P
𝐹=𝑞𝐸= −1.6∙ 10 −19 𝐶 6.3 ∙ 𝑁 𝐶 =1∙ 10 −10 𝑁 𝑡𝑜 𝑟𝑖𝑔ℎ𝑡
Acceleration of electron at P
𝑎= 1∙ 10 −10 𝑁 9.11∙ 10 −31 𝑘𝑔 =1.09 ∙ 𝑚 𝑠 𝑡𝑜 𝑟𝑖𝑔ℎ𝑡

13. Electric Field - Example 16-9 (1)
Get magnitudes
𝐸 2 =𝑘 𝑄 2 𝑟 = 9∙ 𝑁 𝑚 2 𝐶 ∙ 10 −6 𝐶 𝑚 2 =5∙ 𝑁/𝐶
𝐸 1 =𝑘 𝑄 1 𝑟 = 9∙ 𝑁 𝑚 2 𝐶 ∙ 10 −6 𝐶 𝑚 2 =1.25 ∙ 𝑁/𝐶

14. Electric Field - Example 16-9 (2)
XY table
Field
X-component
Y-component E2
0 N/C
+5 x 106 N/C E1
+1.1 x 106 N/C
x 106 N/C
Total
x 106 N/C

15. Continuous charge distributions
Coulomb’s Law plus a lot of integral calculus!
(don’t worry, I’ll just show results)
Field of line of charge
Field of ring of charge
Field of sheet of charge
Field between 2 sheets of charge

16. 1- Field of Line of Charge

17. 2 – Field of Ring of Charge

18. 3 – Field of Sheet of Charge

19. 4- Continuous charge distributions - Summary

20. 5 – Field between 2 Sheets of Charge
In region between parallel plates
𝐸= 𝑄 𝜀 𝑜 𝐴 = 𝜎 𝜀 𝑜
𝑄=𝑐ℎ𝑎𝑟𝑔𝑒 𝑜𝑛 𝑒𝑎𝑐ℎ 𝑝𝑙𝑎𝑡𝑒
𝜎=𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑛 𝑒𝑎𝑐ℎ 𝑝𝑙𝑎𝑡𝑒
𝐴=𝐴𝑟𝑒𝑎 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑙𝑎𝑡𝑒
𝜀 𝑜 = 1 4𝜋𝑘 = 8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2
Note: now the field is constant
Coulomb’s Law Lots of integral calculus = Constant fields
𝐸= 𝑘𝑄 𝑟 𝑑𝑥 𝜎 𝜀 𝑜
(and we just show results)

21. Field between 2 Sheets of Charge
In region between parallel plates
𝐸= 𝑄 𝜀 𝑜 𝐴 = 𝜎 𝜀 𝑜
Note field is constant between plates
Uses for parallel plates
Capacitors (charge storage)
Beam accelerators / deflectors (old TVs, high-energy labs)
Semiconductor junctions
Photocopy machines

22. Example - Photocopy Field
Electrostatic for twice weight
𝑞𝐸−2𝑚𝑔=0
𝐸= 2𝑚𝑔 𝑞
= 2 9∙ 10 −16 𝑘𝑔 𝑚 𝑠 ∙ 10 −19 𝐶 =5500 𝑁/𝐶

23. Examples Problem 23 Problem 24
𝐹=𝑞𝐸= −1.6∙ 10 −19 𝐶 𝑁 𝐶 =−4.16∙ 10 −16 𝑁 𝑤𝑒𝑠𝑡
Problem 24
𝐸= 𝐹 𝑞 = 3.75∙ 10 −14 𝑁 1.6∙ 10 −19 𝐶 =234,000 𝑁 𝐶 𝑠𝑜𝑢𝑡ℎ

24. Examples 𝐹=𝑚𝑎 Problem 27 Problem 31
𝐹=𝑞𝐸= −1.6∙ 10 −19 𝐶 𝑁 𝐶 =−1.2∙ 10 −16 𝑁
𝑎= 𝐹 𝑚 = −1.2∙ 10 −16 𝑁 9.1∙ 10 −31 𝑘𝑔 =1.32∙ 𝑚 𝑠 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
Problem 31
𝐹=𝑚𝑎
𝐸= 𝐹 𝑞

25. Field Lines
Field lines start at (+) charge, end at (-) charge. Without charge, field lines don’t exist.
A (+) test charge moves in the direction of the field lines, a (-) test charge moves opposite the direction of field lines.
The cross-sectional density of field lines is proportional to the electric field strength. 𝐸∝ 𝐿𝑖𝑛𝑒𝑠 𝐴𝑟𝑒𝑎

26. Gauss’s Law
Number of field lines entering or leaving any volume proportional to charge enclosed.
𝐸∙𝑑𝐴 𝑐𝑜𝑠𝜃= 𝑞 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 𝜀 𝑜
Evaluate over entire closed surface
Field lines perpendicular to surface
Need to know symmetry

27. Gauss’s Law Examples
Point charge
Line of charge
Sheet of charge

28. Gauss’s Law 1 – Point charge
𝐸∙𝑑𝐴 𝑐𝑜𝑠𝜃= 𝑞 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 𝜀 𝑜
For spherical surface
𝐸 4𝜋 𝑟 2 = 𝑞 𝜀 𝑜
𝐸 = 𝑞 4𝜋 𝑟 2 𝜀 𝑜

29. Gauss’s Law 2 – Line charge
𝐸∙𝑑𝐴 𝑐𝑜𝑠𝜃= 𝑞 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 𝜀 𝑜
For cylindrical surface
𝐸 2𝜋𝑟𝑙= 𝜆𝑙 𝜀 𝑜
𝐸 = 𝜆 2𝜋𝑟𝜀 𝑜

30. Gauss’s Law 3 – Sheet charge
𝐸∙𝑑𝐴 𝑐𝑜𝑠𝜃= 𝑞 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 𝜀 𝑜
For pillbox surface
𝐸 2𝐴= 𝜎𝐴 𝜀 𝑜
𝐸 = 𝜎 2𝜀 𝑜