1. Chapter 23
Electric Potential

2. Goals for Chapter 23
To calculate electric potential energy of a group of charges
To understand electric potential
To calculate electric potential due to a collection of charges
To use equipotential surfaces to understand electric potential
To calculate E field using electric potential

3. Introduction How is electric potential related to welding?
Electric potential energy is an integral part of our technological society.
What is the difference between electric potential and electric potential energy?
How is electric potential energy related to charge and the electric field?

4. Electric potential energy in a uniform field
Behavior of point charge in uniform electric field is analogous to the motion of a baseball in a uniform gravitational field.

5. Electric potential energy in a uniform field
Behavior of point charge in uniform electric field is analogous to the motion of a baseball in a uniform gravitational field.
Ball speeds up g

6. Electric potential energy in a uniform field
Behavior of point charge in uniform electric field is analogous to the motion of a baseball in a uniform gravitational field.
+ charge speeds up E

7. A positive charge moving in a uniform field
If positive charge moves in direction of field, potential energy decreases (KE increases!)

8. A positive charge moving in a uniform field
If positive charge moves in direction of field, potential energy decreases (KE increases!)
If + charge moves opposite field, potential energy increases.

9. When a positive charge moves in the direction of the electric field,
Q23.1
When a positive charge moves in the direction of the electric field,
Motion
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases. +q
Answer: B

10. A23.1
When a positive charge moves in the direction of the electric field,
Motion
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases. +q

11. A negative charge moving in a uniform field
Positive charge movement in an E field is like normal mass moving in a uniform gravitational field.
Move with the field direction, KE increases
Move against the field direction, U increases
Overall, total energy U + KE is constant.
With negative charges, just reverse the sign! 

12. A negative charge moving in a uniform field
If negative charge moves in direction of E field, potential energy increases…

13. A negative charge moving in a uniform field
If negative charge moves in direction of field, potential energy increases, but if - charge moves opposite field, potential energy decreases.

14. When a negative charge moves in the direction of the electric field,
Q23.3
When a negative charge moves in the direction of the electric field,
Motion –q
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases.
Answer: C

15. A23.3
When a negative charge moves in the direction of the electric field,
Motion –q
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases.

16. A negative charge moving in a uniform field
Positive charge movement in an E field:
Move with the field direction, KE increases
Move against the field direction, U increases
Overall, total energy U + KE is constant.
Negative charges movement in an E field:
Move with the field direction, KE decreases
Move against the field direction, U decreases

17. Q23.2
When a positive charge moves opposite to the direction of the electric field,
Motion +q
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases.
Answer: C

18. A23.2
When a positive charge moves opposite to the direction of the electric field,
Motion +q
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases.

19. Q23.4
When a negative charge moves opposite to the direction of the electric field,
Motion –q
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases.
Answer: B

20. A23.4
When a negative charge moves opposite to the direction of the electric field,
Motion –q
A. the field does positive work on it and the potential energy increases.
B. the field does positive work on it and the potential energy decreases.
C. the field does negative work on it and the potential energy increases.
D. the field does negative work on it and the potential energy decreases.

21. Electric potential energy of two point charges
What is the change in electric potential energy of charge q0 moving along a radial line? b
Test charge q0 moves from a to b in the field created by q a
+ q (fixed)

22. Electric potential energy of two point charges
The electric potential energy of charge q0 moving along a radial line
NOTE!! This force is NOT constant! It decreases with distance away from +q ! You HAVE to integrate!

23. Electric potential energy of two point charges
The electric potential energy of charge q0 moving along a radial line
NOTE!! This force is NOT constant! It decreases with distance away from +q ! You HAVE to integrate!

24. Electric potential energy of two point charges
The electric potential energy of charge q0 moving along a radial line

25. Graphs of the potential energy
Electrical Potential Energy
Define U = r = ∞
Sign depends on charges

26. Electric potential energy of two point charges
Electric potential is same whether q0 moves along an arbitrary path!

27. Electric potential energy of two point charges
Electric potential is same whether q0 moves along an arbitrary path!

28. Example 23.1 – Conservation of Energy
Positron (+ electron) moves near alpha particle (+2e with mass m=6.64 x kg)
Interacts electrically; at r = 1.00 x m moving away at v = 3.00 x 106 m/s.
What is positron speed when twice as far? At ∞ ?
Assume a particle doesn’t move.

29. Example 23.1 – Conservation of Energy
Positron moves near alpha particle (m=6.64 x kg)
Interacts electrically; at r = 1.00 x m moving away at v = 3.00 x 106 m/s.
What is positron speed when twice as far? At ∞ ?
Assume a particle doesn’t move.

30. Example 23.1 – Conservation of Energy
Positron moves near alpha particle (m=6.64 x kg)
Interacts electrically; at r = 1.00 x m moving away at v = 3.00 x 106 m/s.
What is positron speed when twice as far? At ∞ ?
Assume a particle doesn’t move.

31. Example 23.1 – Conservation of Energy
Positron moves near alpha particle (m=6.64 x kg)
Interacts electrically; at r = 1.00 x m moving away at v = 3.00 x 106 m/s.
What is positron speed when twice as far? At ∞ ?
Assume a particle doesn’t move.

32. Problem 23. 1
A point charge with a charge q1 = 3.90μC is held stationary at the origin. A second point charge with a charge q2 = -4.80μC moves from the point x= 0.170m , y=0 to the point x= 0.230m , y= 0.270m .
How much work is done by the electric force on q2?

33. Electrical potential with several point charges
Electrical PE associated with q0 depends on other charges and distances from q0
Key example 23.2:
Two point charges on x axis
q1 = -e at x = 0
q2 = +e at x = a
Work done to bring q3 = +e from ∞ to x = 2a?

34. Potential from Superposition of Charges

35. D. not enough information given to decide
Q23.5
The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is
Charge #2 +q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide
Answer: B

36. A23.5
The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is
Charge #2 +q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide

37. D. not enough information given to decide
Q23.6
The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is
Charge #2 –q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide
Answer: B

38. A23.6
The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is
Charge #2 –q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide

39. D. not enough information given to decide
Q23.7
The electric potential due to a point charge approaches zero as you move farther away from the charge.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is
Charge #2 +q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide
Answer: A

40. A23.7
The electric potential due to a point charge approaches zero as you move farther away from the charge.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is
Charge #2 +q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide

41. D. not enough information given to decide
Q23.8
The electric potential due to a point charge approaches zero as you move farther away from the charge.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is
Charge #2 –q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide
Answer: B

42. A23.8
The electric potential due to a point charge approaches zero as you move farther away from the charge.
If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is
Charge #2 –q
Charge #1 +q y –q x
Charge #3
positive.
negative.
C. zero.
D. not enough information given to decide

43. Problem 23. 8
Three equal 1.70-μC point charges are placed at the corners of an equilateral triangle whose sides are 0.600m long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

44. Electric potential Potential is potential energy per unit charge.
Think of potential difference between points a and b in either of two ways.

45. Electric potential Potential is potential energy per unit charge.
One Way: Potential of a with respect to b (Vab = Va – Vb)
work done by the electric force on a unit charge that moves from a to b in the field.
if b is at a lower potential than a, work done moving + charge is positive (it speeds up!)

46. Electric potential Potential is potential energy per unit charge.
Potential of a with respect to b (Vab = Va – Vb) equals:
work done by the electric force on a unit charge that moves from a to b in the field.
Think of gravity doing positive work on a falling mass
Note limits of integration, and their order!

47. Electric potential analogy to gravity…
Fg and dl vectors are in the same direction
Work is positive
Potential decreases
Va > Vb dl g b

48. Electric potential analogy to gravity…
Fe and dl vectors are in the same direction
Work is positive
Potential decreases
Va > Vb + dl E b

49. Electric potential Potential is potential energy per unit charge.
Another Way: Potential of a with respect to b (Vab = Va – Vb)
work done by you to move a unit charge slowly the other way, from b to a against the electric force.
Think of lifting a mass against gravitational force!

50. Electric potential analogy to gravity…
Fg and dl vectors are in opposite directions
Work is negative
Potential increases
Va > Vb dl g b

51. Electric potential Potential is potential energy per unit charge.
Think of potential difference between points a and b in either of two ways. Potential of a with respect to b (Vab = Va – Vb) equals:
work done by the electric force when a unit charge moves from a to b.
work done by you to move a unit charge slowly from b to a against the electric force.

52. Potential due to charges
Potential is a scalar
Three “standard” forms

53. Finding electric potential from the electric field
Move in direction of E field electric potential decreases, but if you move opposite the field, the potential increases.
For EITHER + or – charge!

54. Potential due to two point charges
Example 23.3:
Proton moves .50 m in a straight line between a and b.
E = 1.5x107 V/m from a to b.
Force on proton?
Work done by field?
Va – Vb?

55. Problem
A small particle has charge -3.10μC and mass 2.40×10−4kg . It moves from point A, where the electric potential is VA = 300V , to point B, where the electric potential VB = 990V is greater than the potential at point A. The electric force is the only force acting on the particle. The particle has a speed of 4.90m/s at point A.
What is its speed at point B?

56. Potential due to two point charges
Example 23.4:
Dipole with q1 = +12 nC & q2 = -12 nC
d = 10 cm
Va ? Vb ? Vc ?

57. Finding potential by integration
Example 23.6: Potential at a distance r from a point charge?
Here a = r, b = 

58. Finding potential by integration
Example 23.6: Potential at a distance r from a point charge?
Or equally
Here a = r, b = 

59. Moving through a potential difference
Example 23.7 combines electric potential with energy conservation.
Dust particle m = 5.0 x 10-9 kg, charge +2.0 nC
Starts at rest. Moves from a to b. Speed at b?

60. Moving through a potential difference
Field is NOT constant! Forces are NOT constant!!
Use Energy Conservation instead…
Ka+Ua= Kb+Ub where U = q0V & V = Skq/r

61. Calculating electric potential
Example (a charged conducting sphere).
Total charge q in sphere of radius R
V = ? everywhere

62. Calculating electric potential
Example (a charged conducting sphere).
Total charge q in sphere of radius R
Start with E field!
E outside R looks like pt charge!
E inside = 0 (conductor!

63. Calculating electric potential
Example (a charged conducting sphere).
Total charge q in sphere of radius R
V = ? Everywhere
V inside must be?
NOT zero!
Just CONSTANT!!
V outside = point charge!

64. Oppositely charged parallel plates
Example 23.9: Find potential at any height y between 2 oppositely charged plates

65. Example 23.9: Oppositely charged parallel plates

66. Example 23.9: Oppositely charged parallel plates

67. An infinite line charge or conducting cylinder
Example – Potential at r away from line?

68. An infinite line charge or conducting cylinder
Example – Potential at r away from line?

69. An infinite line charge or conducting cylinder
Example – Potential at r away from line?

70. An infinite line charge or conducting cylinder
Example – Potential at r away from line?

71. An infinite line charge or conducting cylinder
Example – Potential at r away from line?
If Vb = 0 at infinite distance, won’t Va go to infinity at r = 0??
Yes!
So define V= 0 elsewhere

72. An infinite line charge or conducting cylinder
Consider cylinder…
Same result for r > R
What about inside conducting cylinder??

73. A ring of charge
Charge Q uniformly around thin ring of radius a. VP=?

74. A finite line of charge
Example – Potential at P a distance x from ring?

75. Equipotential surfaces and field lines
Equipotential surface is surface on which electric potential is same at every point.
Field lines & equipotential surfaces are always mutually perpendicular.

76. Equipotential surfaces and field lines
Equipotential surface is surface on which electric potential is same at every point.
Field lines & equipotential surfaces are always mutually perpendicular.

77. Equipotential surfaces and field lines
Equipotential surface is surface on which electric potential is same at every point.
Field lines & equipotential surfaces are always mutually perpendicular.

78. Equipotentials and conductors
When all charges are at rest:
Surface of conductor is always an equipotential surface.
E field just outside conductor is always perpendicular to surface
Entire solid volume of conductor is at same potential.

79. Potential gradient Work/unit charge Consider a uniform vector E field
Consider moving a unit charge along x-axis a very small distance from x to x+Δx (at constant y and z)
Work done against field from x to x+Δx:
Work/unit charge
Work against field by you

80. Potential gradient Consider a uniform vector E field
Consider moving a unit charge along x-axis a very small distance from x to x+Δx (at constant y and z) in this field
YOU have to do work against field from x to x+Δx
Remember Work by you against field = DPE
You lift mass up against gravity = gain in grav. PE!
You push + charge against E field = gain in elec. PE!
Your work against the field increases the potential energy.

81. Potential gradient Consider a uniform vector E field
Consider moving a unit charge along x-axis a small distance from x to x+Δx (at constant y and z)
But Work (per unit charge) done against field = D V
Initial position
Final position

82. Potential gradient Consider a uniform vector E field
Consider moving a unit charge along x-axis a small distance from x to x+Δx (at constant y and z)
But Work (per unit charge) done against field = D V!
In the limit as Δx goes to 0…

83. Potential gradient
So uniform vector E field is the gradient of a scalar Potential

84. Potential gradient
Create E as the gradient of the Potential

85. Potential gradient
Create E as the gradient of the Potential

86. Potential gradient
Analogy to Gravitational Potential Gradient = direction of force along steepest slope
Lines of constant height = constant “gh” = (Potential Energy/unit mass)
Top down view

87. Potential gradient
Analogy to Gravitational Potential Gradient = direction of force along steepest slope
Lines of equal gravitational potential (regardless of mass)
Top down view
Side view

88. Potential gradient
Analogy to Gravitational Potential Gradient = direction of force along steepest slope
Biggest change in (m)gh for small distance sideways
Top down view
(m)gh decreases in y direction quickly with small movement in x

89. Potential gradient
Analogy to Gravitational Potential Gradient = direction of force along steepest slope
Steepest slope means largest net force in direction of gravitational field
Top down view
Largest negative change in (m)gh with small change in horizontal direction

90. Largest negative change in (m)gh with small change in xy direction
Potential gradient
Analogy to Gravitational Potential Gradient = direction of force along steepest slope
Steepest slope means largest net force in direction of gravitational field
Top down view
Largest negative change in (m)gh with small change in xy direction

91. Potential gradient Create E as the gradient of the Potential 3V
Lines of constant electric potential (Volts) (Potential Energy/unit charge) 4V 5V 6V

92. Potential gradient Create E as the gradient of the Potential 3V
Lines of constant electric potential (Volts) (Potential Energy/unit charge) 4V 5V 6V
Largest negative change in Volts with small change in horizontal direction = direction of E field!

93. Potential gradient Create E as the gradient of the Potential
Potential decreases in direction of + E field
Va – Vb is positive if E field points from a to b
“Grad V” is from final position – initial = Vb – Va <0
So is positive!