1. Ch 29 Potential and Field
講者： 許永昌 老師

2. Contents Connecting Potential and Field Source of Electric Field
E  V
Source of Electric Field
Finding the Electric Field from the Potential
V  E
A Conductor in Electrostatic Equilibrium
Capacitance and Capacitors
The Energy Stored in a Capacitor
Dielectrics

3. Connecting Potential and Field (請預讀P911~P913)
Four key concepts:
The electric potential and electric field are just two different mathematical representations of how source charges alter the space around them.
Force Concept
Energy Concept
Acts locally
Everywhere in space

4. Sources of Electric Potential (請預讀P914~P916)
Important Concept:
We can create a potential difference by creating a charge separation.
電荷分離即產生電位差。
Examples:
Van de Graaff generator:
Work done to lift the positive charge to the hollow metal sphere.
Batteries:
Chemical reactions do work Wchem to move charge q from the negative to the positive terminal. (charge escalator model)
emf Wchem/q =DVbat.

5. Homework
Student Workbook
1, 2

6. Finding the Electric Field from the Potential (請預讀P916~P920)
Since DV=-Edr, we get
Where the direction of dr can be chosen by us.
For instance,
dr=êxdx. 
We get the x component of E.
Gradient:
We get

7. The Geometry of Potential and Field (請預讀P918~P919)
Geometry representation of electric potential:
1D: s vs. V graph
2D: contour line graph
3D: Equipotential surface
The potential on the same contour (2D) and equipotential surface (3D) will be constant, i.e.
Therefore, the E field must be perpendicular to the contour or equipotential surface.

8. Exercises Code:electric_potential.m
Find the electric field or electric potential from these figures.
E(N/C)
V(volt)
x(m)
x(m)

9. Kirchhoff’s Loop Law s

10. Homework
Student Workbook
3, 4, 7, 8, 9, 10

11. A Conductor in Electrostatic Equilibrium (請預讀P921)
Properties:
E inside a conductor is zero.
 V=constant inside the conductor.
An equipotential surface close to an electrode must roughly match the shape of the electrode.
E is large at the sharp corners.

12. Exercise
Two spherical conductors are linked by a metal wire as shown in the figure. Find Q2=?
Find DV12, DV34, DV23 if the system is in electrostatic equilibrium.
DVloop=?
Can we say that these two rods work
as a capacitor?
How much time does these two rods need to be charged?
需要 Ohm’s Law 的概念。
Q1=1.0 nC
R=1.0 cm
Q2=?? nC
R=2.0 cm

13. Homework
Student Workbook
12, 13, 14, 15

14. Capacitance and Capacitors(請預讀P922~P927)
Reference:
It consists of two conductors separated by a non-conductive region (dielectric).
Capacitance:
Q: The charges of these two conductors are +Q and –Q, respectively. i.e. the whole system is in neutral.
DV: The potential difference between these two conductors.
SI Unit: 1 farad = 1 F = 1 C/V.

15. Examples: Keys of Computer Keyboard: Touchscreen of iPod and iPhone:
Soft dielectric
Keys of Computer Keyboard:
Press the key will change its capacitance C.
DV is fixed, so that Q=CDV will change, too.
It need the current I to change Q.
Detect the current.
Touchscreen of iPod and iPhone:
Reference:
Sensor is one of the conductor, and the target object (human) is the other. (AC circuit) DV
Note:這只是示意圖

16. Exercise Find the capacitance of a spherical capacitor:
Q, R2
Find the capacitance of a spherical capacitor:
How about R2?
Find the capacitance of a parallel-plate capacitor.
-Q, R1
Area: A
Spacing: d

17. Combinations of Capacitors (請預讀P924~P926)
Two basic combinations:
Parallel combination:
DV1=DV2=DV3=…DV.
Equivalent capacitance: Ceq=SQi/DV =SCi.
Series combination:
Q1=Q2=Q3=… Q.
Equivalent capacitance: Q/Ceq=DV=SDVi=

18. The energy stored in a Capacitor (請預讀P927~P928)
Goal:
Find the potential energy stored in a capacitor whose charge is Q.
 The work should be done by a battery because DE=W.
Method II:
基於增加dq需要作功多少的概念而得 (dU=dqDV).
基於point charge 間能量儲存的概念而得

19. The energy stored in a Capacitor
For a spring, we can “see” that the potential energy of a stretched spring is in the tension of the coils. But, where is the stored energy for a capacitor?
The energy is stored in the electric field of capacitor in the space.
Uc= ½C(DV)2=
Energy density:
In fact, this equation is correct for any electric field.

20. 補充 The energy stored in a charge distribution is Local Gauss’s Law微分形式
Everywhere

21. Dielectrics (介電質) (請預讀P929~P932)
An insulator in an electric field is called a dielectric.
External E field E0  polarization  induced surface charge densities (hinduced) Einduced.
hinduced is related to the external electric field strength and the properties of the insulator.
Dielectric constant:

22. Dielectric constant, Dielectric strength and the Capacitances
Capacitance of a dielectric-filled capacitor:
Disconnected to the battery.
 pull dielectric into the capacitor.
All materials have a maximum electric field they can sustain without breakdown-the production of a spark.

23. Homework
Student Workbook
16, 17, 19, 21, 22, 23, 24