1. Chapter 21
Electric Potential

2. Conservation of energy
Conservative force
Conservation of energy +
charged
Potential (energy)
air → conductor
Potential difference
(voltage)
discharge 2

3. Coulomb force is conservative
Comparing :
Electrostatic / Coulomb force is conservative
Work doesn’t depend on the path:
The circulation theorem of electric field 3

4. Define electric potential:
Conservative force
→ potential energy U
U depends on the charge!
Define electric potential: + -
Potential difference / voltage: 4

5. Electric potential & electric field
Difference in potential energy between a and b: a b
If position b is the zero point:
Relationship between electric field and potential
Unit of electric potential: Volt (V) 5

6. It depends on the chosen of zero point
Electric potential:
It depends on the chosen of zero point
Usually let V = 0 at infinity:
For the field created by a point charge : Q . a 6

7. Properties of electric potential
1) Va → potential energy per unit (+) charge
2) Va depends on the zero point, Vba does not
3) Va and Vba are scalars, differ from E vector 7

8. Calculation of electric potential
If the electric field is known:
Potential of point charge:
(V = 0 at infinity)
System of several point charges: r . A 8

9. Potential of electric dipole
Example1: (a) Determine the potential at o, a, b. (b) To move charge q from a to b, how much work must be done?
Solution: (a) . h l
(b) 9

10. . Charged conductor sphere
Example2: Determine the potential at a distance r from the center of a uniformly charged conductor sphere (Q, R) for (a) r > R; (b) r = R; (c) r < R. . a +
Solution: All charges on surface: Q
(a) V at r > R:
Same as point charge 10

11. Same potential at any point!. b
(b) V at r = R: + Q . c
(c) V at r < R:
Same potential at any point!
Discussion: + - ?
(1) Conductor: equipotential body
(2) Charges on holey conductor 11

12. Uniformly charged rod
Example3: A thin rod is uniformly charged (λ, L). Determine the potential for points along the line outside of the rod.
Solution: Choose infinitesimal charge dQ
Potential at point b? . b dx dQ x a d L 12

13. Charged ring
Example4: A thin ring is uniformly charged (Q, R). Determine the potential on the axis.
Solution: R x Q
. a r dQ
or:
Discussion:
(1) Not uniform?
(2) x >> R
(3) other shapes 13

14. Conical surface
Question: Charges are uniformly distributed on a conical surface (σ, h, R). Determine the electric potential at top point a. (V = 0 at infinity) h R a 14

15. Electrostatic induction
Question: Put a charge q nearby a conductor sphere initially carrying no charge. Determine the electric potential on the sphere. - + Q R b q o
Connect to the ground? 15

16. Air can become ionized due to high electric field
*Breakdown voltage
Air can become ionized due to high electric field
Air → conducting → charge flows
Breakdown of air:
Breakdown voltage for a spherical conductor? Q R 16

17. Equipotential surfaces
Visualize electric potential: equipotential surfaces
Points on surface → same potential
(1) Surfaces ⊥ field at any point.
(2) Move on surface, no work done
(3) Move along any field line, V ↘
(4) Surfaces dense → strong field 17

18. (1) Equipotential surfaces for dipole system:
Some examples
(1) Equipotential surfaces for dipole system:
(2) Conductors
(3) Parallel but not uniform field lines? 18

19. To determine electric field from the potential, use
E determined from V
To determine electric field from the potential, use
In differential form:
Component of E in the direction of dl :
Total electric field: 19

20. Gradient → partial derivative in direction V changes most rapidly
Gradient of V V
V+dV 1 2 dl dn  a b
Gradient of V :
Gradient → partial derivative in direction V changes most rapidly 20

21. Determine E from V
Example5: The electric potential in a region of space varies as V = x2yz. Determine E.
Solution: 21

22. Example6: The electric field in space varies as . Determine V.
Determine V from E
Example6: The electric field in space varies as
. Determine V.
Solution:
( C is constant ) 22