Chapter 22 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Electric Potential 電位 Pearson Education, Inc.

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1 Chapter 22 Lecture Essential University Physics Richard Wolfson 2 nd Edition Electric Potential 電位 Slide 22-1

2 In this lecture you ll learn 簡介 The concept of electric potential difference 電位差 Including the meaning of the familiar term volt 伏特 To calculate potential difference between two points in an electric field 計算兩點的電位差 To calculate potential differences of charge distributions by summing or integrating over point charges The concept of equipotentials 等電位 How charge distributes itself on conductors 電荷怎麼在導 2012 體表面分佈 Pearson Education, Inc.. Slide 22-2

3 Electric Potential Difference The electric potential difference between two points describes the energy per unit charge involved in moving charge between those two points. 電位差代表單位電荷在兩點之間移動的位能差 B! Mathematically, ΔV AB = ΔU AB q = E d r!, A where V AB is the potential difference between points A and B, and U AB is the change in potential energy of a charge q moved between those points. 數學的表示式如下 : ΔV AB = ΔU AB B! q = E d r!, 其中 V AB 代表 AB 兩點的電位差, 而 U AB 代表位能的改變值 A Slide 22-3

4 庫侖力做的功 W 與電位能 U, 電場產生的電位差 ΔV W elec. =!! B AB F d r = B q E! d r! 庫侖靜電力做的功,, 功與動能定理, 動能的改變量等於 電位能的改變量的負值 因此 A W AB = ΔK, ΔK = ΔU AB, ΔV AB = ΔU AB q = A! B E d r!, A Slide 22-4

5 Electric Potential Difference Potential difference is a property of two points. Since electrostatic field is conservative, it doesn t matter what path is taken between those points. 庫侖力為保守力, 位能差的計算與路徑無關 In a uniform field, the potential difference is 在均勻電場中, 電位能差等於 ΔV AB = E! Δ r.! Slide 22-5

6 The Volt and the Electronvolt 伏特與電子伏特 The unit of electric potential difference is the volt (V). 1 volt is 1 joule per coulomb (1 V = 1 J/C). Example: A 9-V battery supplies 9 joules of energy to every coulomb of charge that passes through an external circuit connected between its two terminals. The volt is not a unit of energy, but of energy per charge that is, of electric potential difference. A related energy unit is the electronvolt (ev), defined as the energy gained by one elementary charge e falling through a potential difference of 1 volt. Therefore, 1 ev = J. Slide 22-6

7 Electric Potential of a Point Charge 點電荷的電位 Since the electric field of a point-charge field varies with position, the potential differences between two points in the point-charge field must be found by integration. The result is # ΔV AB =V A V B = kq% 1 1 $ r A r B & ( ' Taking the zero of potential at infinity gives 取無窮遠處為零電位, 則 r B = and V B = 0, then V A =V r =V (r) = kq r for the potential difference between infinity and any point a distance r from the point charge. Slide 22-7

8 庫侖力做的功 W 與電位能 U, 電場產生的電位差 ΔV 推導上式 : 電位差 ΔV AB =! B E d r!, A where! E = kq r 2 ˆr ΔV AB = B A kq r 2 ˆr d! r = B A kq r dr = ( kq 2 r r r B ) = kq kq, r A r B r A because ΔV AB = V B V A so, V A V B = kq r A kq r B, Slide 22-8

9 Potential Difference of a Charge Distribution If the electric field of the charge distribution is known, potential differences can be found by integration as was done for the point charge on the preceding slide. If the distribution consists of point charges, potential differences can be found by summing point-charge potentials: kqi For discrete point charges, 點電荷的分佈時 : V( P) =, r where V(P) is the potential difference between infinity and a point P in the electric field of a distribution of point charges q 1, q 2, q 3, For a continuous charge distribution, 連續電荷分佈時 k dq V( P) =. r i i Slide 22-9

10 Discrete Charges: The Dipole Potential The potential of an electric dipole follows from summing the potentials of its two equal but opposite point charges: For distances r large compared with the dipole spacing 2a, the result is kp cosθ V(, r θ ) = 2 r where p = 2aq is the dipole moment. -q! p 2a, p=2aq +q Slide 22-10

11 Discrete Charges: The Dipole Potential A 3-D plot of the dipole potential shows a hill for the positive charge and a hole for the negative charge. V=0 Slide 22-11

12 Continuous Distributions: A Ring and a Disk For a uniformly charged ring of total charge Q, integration gives the potential on the ring axis: kdq k kq r r x + a ( ) = = dq= 2 2 V x Integrating the potentials of charged rings gives the potential of a uniformly charged disk: 2kQ V x x a x a ( ) ( ) 2 2 = + 2 This result reduces to the infinitesheet potential close to the disk, and the point-charge potential far from the disk. Slide 22-12

13 如果已經知道了 V, 是否可以反推出電場? 取 B 點在無窮遠處, 則 because V = 0,! so, V A = E d r! A r! then, V = E d r! ΔV AB = V V A =! # E = % V $ x A î + V y! E d! r YES! ĵ + V z & ˆk ( ' Slide 22-13

14 Potential Difference and the Electric Field Potential difference involves an integral over the electric field. So the field involves derivatives of the potential. Specifically, the component of the electric field in a given direction is the negative of the rate of change (the derivative) of potential in that direction. 電場方向是電位減少的方向 Then, given potential V (a scalar quantity) as a function of position, the electric field (a vector quantity) follows from The derivatives here are partial derivatives, expressing the variation with respect to one variable alone. This approach may be used to find the field from the potential.! # E = % V $ x î + V y ĵ + V z & ˆk ( ' Potential is often easier to calculate, since it s a scalar rather than a vector. Slide 22-14

15 Equipotentials 等電位面 An equipotential is a surface on which the potential is constant. 3 維空間的電場分佈, 有等電位面 In two-dimensional drawings, we represent equipotentials by curves similar to the contours of height on a map. 在 2 維平面上, 有等電位線 The electric field is always perpendicular to the equipotentials. 電場永遠與等位面或線垂直 曲面方程式 V(x,y,z)=constant 代表一個等位面, 此面上的電場都是垂直此面的! Slide 22-15

16 Equipotentials 等電位線 Equipotentials for a dipole Equipotential for a point charge Slide 22-16

17 Charged Conductors There s no electric field inside a conductor in electrostatic equilibrium. 靜電力平衡, 導體內部的電場為零 And even at the surface there s no field component parallel to the surface. 表面外的電場也是垂直表面的 Therefore it takes no work to move charge inside or on the surface of a conductor in electrostatic equilibrium. 所以導體表面也是等電位面 So a conductor in electrostatic equilibrium is an equipotential. That means equipotential surfaces near a charged conductor roughly follow the shape of the conductor surface. 因此靠近導體表面的地方, 等電位面與導體的表面的形狀是一樣的 Slide 22-17

18 Charged Conductors That generally makes the equipotentials closer, and therefore the electric field stronger and the charge density higher, where the conductor curves more sharply. 因為在表面彎曲比較大的地方, 要比較靠近才近似於一個平面, 所以電場的變化也比較大, 用來表示電場的電場線的密度也需要比較多, 對應的電荷密度也就比較大了. 表面電荷比較集中於表面的尖銳處 Slide 22-18

19 Summary Electric potential difference describes the work per unit charge involved in moving charge between two points in an electric field: B!! V = E dr The SI unit of electric potential is the volt (V), equal to 1 J/C. Electric potential always involves two points; to say the potential at a point is to assume a second reference point at which the potential is defined to be zero. Electric potential differences of a point charge is V( P) = kq r, where the zero of potential is taken at infinity. This result may be summed or integrated to find the potentials of charge distributions. r Electric field follows from differentiating the potential: V ˆ V ˆ = + + V Equipotentials are surfaces of constant potential. The electric field and the equipotential surfaces are always perpendicular. Equipotentials near a charged conductor approximate the shape of the conductor. A conductor in equilibrium is itself an equipotential. AB A E i j kˆ x y z Slide 22-19

20 Homework problems Slide 22-20

21 Chapter 22 Lecture Essential University Physics Richard Wolfson 2 nd Edition Electric Potential Slide 22-21

22 Clicker Question 1 What would happen to the potential difference between points A and B in the figure if the distance Δr were doubled? A. ΔV would be doubled. B. ΔV would be halved. C. ΔV would be quadrupled D. ΔV would be quartered. Slide 22-22

23 Clicker Question 2 An alpha particle (charge 2e) moves through a 10-V potential difference. How much work, expressed in ev, is done on the alpha particle? A. 5 ev B. 10 ev C. 20 ev D. 40 ev Slide 22-23

24 Clicker Question 3 The figure shows three straight paths AB of the same length, each in a different electric field. Which one of the three has the largest potential difference between the two points? A. (a) B. (b) C. (c) Slide 22-24

25 Clicker Question 4 You measure a potential difference of 50 V between two points a distance 10 cm apart parallel to the field produced by a point charge. Suppose you move closer to the point charge. How will the potential difference over a closer 10 -cm interval be different? A. The potential difference will remain the same. B. The potential difference will increase. C. The potential difference will decrease. D. We cannot find this without knowing how much closer we are. Slide 22-25

26 Clicker Question 5 The figure shows cross sections through two equipotential surfaces. In both diagrams the potential difference between adjacent equipotentials is the same. Which of these two could represent the field of a point charge? A. (a) B. (b) C. neither (a) nor (b) Slide 22-26

27 Clicker Question 6 Which of the following statements is TRUE? A. The electric potential is zero on the surface of a conductor in electrostatic equilibrium. B. When the electric field is zero at a point, the electric potential must also be zero there. C. If the electric potential is constant on a surface, then any electric field present can only be perpendicular to that surface. D. If the electric field is zero everywhere inside a region of space, the electric potential must also be zero in that region. Slide 22-27

28 Clicker Question 7 As an electron moves from higher potential to lower potential, its electrical potential energy A. increases. B. decreases. C. stays the same. Slide 22-28

29 Clicker Question 8 Two conducting spheres initially have a charge +Q uniformly distributed on each of their surfaces. Sphere A has a larger radius than sphere B. The two spheres are brought together so that they touch for a few seconds, and then they are separated. After they are separated, which sphere has a higher charge? A. Sphere A B. Sphere B C. The charge of each sphere is still equal. Slide 22-29