Chapter 3 Electric Potential 3.1 Potential and Potential Energy......... 3-2 3.2 Electric Potential in a Uniform Field........... 3-5 3.3 Electric Potential due to Point Charges......... 3-6 3.3.1 Potential Energy in a System of Charges. 3-8 3.4 Continuous Charge Distribution................... 3-9 3.5 Deriving Electric Field from the Electric Potential .3-10 3.5.1 Gradient and Equipotentials......................... 3-11 Example 3.1:Uniformly Charged Rod.... 3-13 Example 3.2:Uniformly Charged Ring..... .3-15 Example 3.3:Uniformly Charged Disk............ 3-16 Example 3.4:Calculating Electric Field from Electric Potential. .3-18 3.6 Summary… 3-18 3.7 Problem-Solving Strategy:Calculating Electric Potential......3-20 3.8 Solved Problems..................... 3-22 3.8.1 Electric Potential Due to a System of Two Charges. .3-22 3.8.2 Electric Dipole Potential.................. 3-23 3.8.3 Electric Potential of an Annulus........ 3-24 3.8.4 Charge Moving Near a Charged Wire 3-25 3.9 Conceptual Questions....... 3-26 3.10 Additional Problems ........... 3-27 3.10.1Cube.. 3-27 3.10.2 Three Charges........... 3-27 3.10.3 Work Done on Charges. … 3-27 3.10.4 Calculating E from V.... 3-28 3.10.5 Electric Potential of a Rod................ 3-28 3.10.6 Electric Potential......... 3-29 3.10.7 Calculating Electric Field from the Electric Potential.... 3-29 3.10.8 Electric Potential and Electric Potential Energy................. 3-30 3.10.9.Electric Field,Potential and Energy..... .3-30 3-1
Chapter 3 Electric Potential 3.1 Potential and Potential Energy..............................................................................3-2 3.2 Electric Potential in a Uniform Field....................................................................3-5 3.3 Electric Potential due to Point Charges ................................................................3-6 3.3.1 Potential Energy in a System of Charges.......................................................3-8 3.4 Continuous Charge Distribution ...........................................................................3-9 3.5 Deriving Electric Field from the Electric Potential ............................................3-10 3.5.1 Gradient and Equipotentials.........................................................................3-11 Example 3.1: Uniformly Charged Rod .................................................................3-13 Example 3.2: Uniformly Charged Ring................................................................3-15 Example 3.3: Uniformly Charged Disk ................................................................3-16 Example 3.4: Calculating Electric Field from Electric Potential..........................3-18 3.6 Summary.............................................................................................................3-18 3.7 Problem-Solving Strategy: Calculating Electric Potential..................................3-20 3.8 Solved Problems.................................................................................................3-22 3.8.1 Electric Potential Due to a System of Two Charges....................................3-22 3.8.2 Electric Dipole Potential..............................................................................3-23 3.8.3 Electric Potential of an Annulus..................................................................3-24 3.8.4 Charge Moving Near a Charged Wire .........................................................3-25 3.9 Conceptual Questions.........................................................................................3-26 3.10 Additional Problems.........................................................................................3-27 3.10.1 Cube ...........................................................................................................3-27 3.10.2 Three Charges............................................................................................3-27 3.10.3 Work Done on Charges..............................................................................3-27 3.10.4 Calculating E from V .................................................................................3-28 3.10.5 Electric Potential of a Rod .........................................................................3-28 3.10.6 Electric Potential........................................................................................3-29 3.10.7 Calculating Electric Field from the Electric Potential ...............................3-29 3.10.8 Electric Potential and Electric Potential Energy........................................3-30 3.10.9. Electric Field, Potential and Energy ..........................................................3-30 3-1
Electric Potential 3.1 Potential and Potential Energy In the introductory mechanics course,we have seen that gravitational force from the Earth on a particle of mass m located at a distance r from Earth's center has an inverse- square form: F.=-GMmp (3.1.1) where G=6.67x10-1N.m2/kg2 is the gravitational constant and r is a unit vector pointing radially outward.The Earth is assumed to be a uniform sphere of mass M.The corresponding gravitational field g,defined as the gravitational force per unit mass,is given by g= F-_GMi (3.1.2) m r2 Notice that g only depends on M,the mass which creates the field,and r,the distance from M. Figure 3.1.1 Consider moving a particle of mass m under the influence of gravity (Figure 3.1.1).The work done by gravity in moving m from A to B is g-尾-（学-[6 (3.1.3) The result shows that W.is independent of the path taken;it depends only on the endpoints 4 and B.It is important to draw distinction between W.,the work done by the 3-2
Electric Potential 3.1 Potential and Potential Energy In the introductory mechanics course, we have seen that gravitational force from the Earth on a particle of mass m located at a distance r from Earth’s center has an inversesquare form: 2 ˆ g Mm G r F = − r G (3.1.1) where is the gravitational constant and is a unit vector pointing radially outward. The Earth is assumed to be a uniform sphere of mass M. The corresponding gravitational field 11 2 2 G 6.67 10 N m /kg − = × ⋅ rˆ g G , defined as the gravitational force per unit mass, is given by 2 ˆ g GM m r = = − F g r G G (3.1.2) Notice that g G only depends on M, the mass which creates the field, and r, the distance from M. Figure 3.1.1 Consider moving a particle of mass m under the influence of gravity (Figure 3.1.1). The work done by gravity in moving m from A to B is 2 1 1 B B A A r r g g r B A r GMm r W d dr GMm r r GMm r − ⎛ ⎞ ⎛ ⎞ = ⋅ = ⎜ ⎟ = = ⎜ − ⎝ ⎠ ⎝ ⎠ ⎡ ⎤ ∫ ∫ ⎢ ⎥ ⎣ ⎦ F s G G ⎟ (3.1.3) The result shows that Wg is independent of the path taken; it depends only on the endpoints A and B. It is important to draw distinction between , Wg the work done by the 3-2
field and W,the work done by an external agent such as you.They simply differ by a negative sign:W=-West Near Earth's surface,the gravitational field g is approximately constant,with a magnitude g=GM/29.8m/s2,where re is the radius of Earth.The work done by gravity in moving an object from height y to y(Figure 3.1.2)is W=∫厘ds=-∫mg cos6s=-∫mg cods=-∫mg少=-mg0a-y4)(6.14) B dy Figure 3.1.2 Moving a mass m from A to B. The result again is independent of the path,and is only a function of the change in vertical height ye-y In the examples above,if the path forms a closed loop,so that the object moves around and then returns to where it starts off,the net work done by the gravitational field would be zero,and we say that the gravitational force is conservative.More generally,a force F is said to be conservative if its line integral around a closed loop vanishes: ∮F.ds=0 (3.1.5) When dealing with a conservative force,it is often convenient to introduce the concept of potential energy U.The change in potential energy associated with a conservative force F acting on an object as it moves from A to B is defined as: AU=U。-U4=-∫F.ds=-W (3.1.6) where W is the work done by the force on the object.In the case of gravity,W=W and from Eq.(3.1.3),the potential energy can be written as U,=-GMm+U。 (3.1.7) 3-3
field and , the work done by an external agent such as you. They simply differ by a negative sign: . Wext Wg = −Wext Near Earth’s surface, the gravitational field g G is approximately constant, with a magnitude , where is the radius of Earth. The work done by gravity in moving an object from height 2 2 / 9.8m/ E g G= M r ≈ s Er A y to (Figure 3.1.2) is B y cos cos ( ) B A B B y g g B A A A y W = ⋅ d = mg θ φ ds = − mg ds = − mg dy = −mg y − y ∫ ∫ ∫ ∫ F s G G (3.1.4) Figure 3.1.2 Moving a mass m from A to B. The result again is independent of the path, and is only a function of the change in vertical height . B A y y − In the examples above, if the path forms a closed loop, so that the object moves around and then returns to where it starts off, the net work done by the gravitational field would be zero, and we say that the gravitational force is conservative. More generally, a force F G is said to be conservative if its line integral around a closed loop vanishes: ⋅ d = 0 ∫ F s G G v (3.1.5) When dealing with a conservative force, it is often convenient to introduce the concept of potential energy U. The change in potential energy associated with a conservative force F acting on an object as it moves from A to B is defined as: JG B B A A ∆ = U U −U = −∫ F s ⋅ d = −W G G (3.1.6) where W is the work done by the force on the object. In the case of gravity, W = Wg and from Eq. (3.1.3), the potential energy can be written as g 0 GMm U r = − +U (3.1.7) 3-3
where Uo is an arbitrary constant which depends on a reference point.It is often convenient to choose a reference point where U is equal to zero.In the gravitational case,we choose infinity to be the reference point,withU(r=)=0.Since U depends on the reference point chosen,it is only the potential energy differenceAU.that has physical importance.Near Earth's surface where the gravitational field g is approximately constant,as an object moves from the ground to a height h,the change in potential energy is AU=+mgh,and the work done by gravity is W.=-mgh. A concept which is closely related to potential energy is "potential."From AU,the gravitational potential can be obtained as .-g=可e/mds=-d (3.1.8) Physically Al represents the negative of the work done per unit mass by gravity to move a particle from A to B. Our treatment of electrostatics is remarkably similar to gravitation.The electrostatic force F.given by Coulomb's law also has an inverse-square form.In addition,it is also conservative.In the presence of an electric field E,in analogy to the gravitational field g,we define the electric potential difference between two points 4 and Bas Ay=-∫./gds=-∫Eds (3.1.9) where go is a test charge.The potential difference Al represents the amount of work done per unit charge to move a test charge do from point 4 to B,without changing its kinetic energy.Again,electric potential should not be confused with electric potential energy.The two quantities are related by △U=qo△V (3.1.10) The SI unit of electric potential is volt (V): 1volt =1 joule/coulomb (1 V=1 J/C) (3.1.11) When dealing with systems at the atomic or molecular scale,a joule (J)often turns out to be too large as an energy unit.A more useful scale is electron volt (eV),which is defined as the energy an electron acquires(or loses)when moving through a potential difference of one volt: 3-4
where is an arbitrary constant which depends on a reference point. It is often convenient to choose a reference point where is equal to zero. In the gravitational case, we choose infinity to be the reference point, with U0 U0 0 U r( = ∞ =) 0 . Since Ug depends on the reference point chosen, it is only the potential energy difference ∆Ug that has physical importance. Near Earth’s surface where the gravitational field g G is approximately constant, as an object moves from the ground to a height h, the change in potential energy is ∆Ug = +mgh , and the work done by gravity is W m g = − gh . A concept which is closely related to potential energy is “potential.” From , the gravitational potential can be obtained as ∆U ( / ) B g g g A U V m d m ∆ ∆ = = − ⋅ = − ⋅ ∫ F s B A d ∫ g s G G G G (3.1.8) Physically ∆Vg represents the negative of the work done per unit mass by gravity to move a particle from A B to . Our treatment of electrostatics is remarkably similar to gravitation. The electrostatic force given by Coulomb’s law also has an inverse-square form. In addition, it is also conservative. In the presence of an electric field E Fe JG JG , in analogy to the gravitational field g G , we define the electric potential difference between two points A B and as 0 ( / ) B e A ∆ = V q − ⋅ d = − ⋅ ∫ ∫ F s E B A d s G G G G (3.1.9) where is a test charge. The potential difference 0 q ∆V represents the amount of work done per unit charge to move a test charge from point A to B, without changing its kinetic energy. Again, electric potential should not be confused with electric potential energy. The two quantities are related by 0 q ∆U q = ∆0 V (3.1.10) The SI unit of electric potential is volt (V): 1volt =1 joule/coulomb (1 V= 1 J/C) (3.1.11) When dealing with systems at the atomic or molecular scale, a joule (J) often turns out to be too large as an energy unit. A more useful scale is electron volt (eV), which is defined as the energy an electron acquires (or loses) when moving through a potential difference of one volt: 3-4
1eV=(1.6×10-19C)1V)=1.6×10-19J (3.1.12) 3.2 Electric Potential in a Uniform Field Consider a charge +gmoving in the direction of a uniform electric field E=E(-j),as shown in Figure 3.2.1(a). A B (a) (b) Figure 3.2.1(a)A charge g which moves in the direction of a constant electric field E. (b)A mass m that moves in the direction of a constant gravitational field g. Since the path taken is parallel to E,the potential difference between points A and B is given by AV=。-y,=-∫Eds=-Es=-Ed<0 (3.2.1) implying that point B is at a lower potential compared to 4.In fact,electric field lines always point from higher potential to lower.The change in potential energy is AU=U-U=-gEd.Since g>0,we haveAU<0,which implies that the potential energy of a positive charge decreases as it moves along the direction of the electric field. The corresponding gravitational analogy,depicted in Figure 3.2.1(b),is that a mass m loses potential energy (AU =-mgd )as it moves in the direction of the gravitational field g. Figure 3.2.2 Potential difference due to a uniform electric field What happens if the path from 4 to B is not parallel toE,but instead at an angle e,as shown in Figure 3.2.2?In that case,the potential difference becomes 3-5
(3.1.12) 19 19 1eV (1.6 10 C)(1V) 1.6 10 J − = × = × − 3.2 Electric Potential in a Uniform Field Consider a charge +q moving in the direction of a uniform electric field 0 ˆ E = E (−j) JG , as shown in Figure 3.2.1(a). (a) (b) Figure 3.2.1 (a) A charge q which moves in the direction of a constant electric field E JG . (b) A mass m that moves in the direction of a constant gravitational field g G . Since the path taken is parallel to E JG , the potential difference between points A and B is given by 0 0 0 B B B A A A ∆V V= −V = − ⋅ d = −E ds = −E d < ∫ ∫ E s JG G (3.2.1) implying that point B is at a lower potential compared to A. In fact, electric field lines always point from higher potential to lower. The change in potential energy is . Since we have U UB A U 0 ∆ =−= −qE d q > 0, ∆U < 0 , which implies that the potential energy of a positive charge decreases as it moves along the direction of the electric field. The corresponding gravitational analogy, depicted in Figure 3.2.1(b), is that a mass m loses potential energy ( ) as it moves in the direction of the gravitational field ∆ = U −mgd g G . Figure 3.2.2 Potential difference due to a uniform electric field What happens if the path from A to B is not parallel toE JG , but instead at an angle θ, as shown in Figure 3.2.2? In that case, the potential difference becomes 3-5