Chapter 16Electric Potential16.1 A review of potential energy 216.1.1 Energy of a single particle 216.1.2 Potential energy is associated with pairs of interacting objects 416.2 Systems of charged objects 516.3 Electric potential 816.3.1 Electron volt—a unit of energy 1016.4 Sign of potential difference and direction of field 1116.4.1 Direction of path same as direction of electric field: DV is negative 1116.4.2 Direction of path opposite to direction of electric field: DV is positive1116.4.3 Path perpendicular to the electric field 1116.4.4 Indicating the path direction 1216.5 Change of electric potential in a nonuniform field 1216.5.1 Two adjacent regions with different fields 1216.5.2 A region of varying electric field requires an integral 1316.5.3 Limit on mathematical complexity in this introductory course 1516.6 The potential at one location 1516.6.1 Potential at a location near a point charge 1616.6.2 Potential at one location and potential energy 1616.6.3 Application: Potential inside a metal at static equilibrium 1616.6.4 Application: A metal not in static equilibrium 1816.7 Potential difference is independent of path 1916.7.1 Two different paths near a point charge 1916.8 Round trip potential difference is zero 2016.8.1 Reasoning about patterns of electric field 2116.9 Potential difference in an insulator 2216.9.1 Dielectric constant and net field 2316.10 *Energy density associated with electric field 2416.10.1 An electron and a positron 2516.11 *Shifting the zero of potential 2616.12 *Potential of distributed charges 2716.12.1 Potential along the axis of a ring 2716.12.2 Potential along the axis of a uniformly charged disk 2816.13 Reflection: Potential and potential difference 2816.14 *Integrating the spherical shell 2916.15 Summary 3216.16 Example problem: A disk and a spherical shell 3316.17 Review questions 3416.18 Homework problems 3516.19 Answers to exercises 4216-2 Chapter 16: Electric PotentialChapter 16Electric PotentialIn analyzing the dynamics of moving objects, both at the macroscopic andthe microscopic level, we found that it was frequently important to considernot just forces and momenta, but also work and energy, in trying to modelthe behavior of a physical system. Similarly, to complement our use of theconcept of electric field and electric force, we need the concept of electricpotential. Electric potential is defined as electric potential energy per unitcharge. The concept of electric potential is useful for some of the same reasonsthat the concept of electric field is useful. It allows us to reason about energyin a range of situations without having to worry about the details of someparticular distribution of point charges. Electric potential has practical im-portance, in part because batteries and electric generators maintain a po-tential difference across themselves that is nearly independent of what isconnected to them. The concept also provides significant theoretical power,enabling us to draw conclusions about a surprising range of issues, includ-ing, for example, what patterns of electric field in space are possible, andthe magnitude and direction of the average electric field inside an insulatordue to the polarization of molecules in the insulator.16.1 A review of potential energyThis section is a review of the concept of potential energy, which was firstintroduced in Chapter 4. If this concept is very familiar to you, you may wishto skip ahead to Section 16.2, in which we apply the concept to systems ofinteracting charged particles similar to those that we will typically considerin this volume.16.1.1 Energy of a single particleThe energy of a single particle with charge q1 (Figure 16.1) consists solely ofits particle energy. The particle energy is the sum of the rest energy ofthe particle, and its kinetic energy K.Most of the processes we will study in this volume are low energy processesthat do not involve significant changes in rest energy. (In contrast, processessuch as fission and fusion, in which atomic nuclei combine or split apart toform different elements, involve the release of very large amounts of energy.These are the subject of some of the problems in Chapter 4.) Kinetic energy is energy associated with motion. The kinetic energy of aparticle moving at low speed may be approximated as APPROXIMATE KINETIC ENERGY OF A PARTICLE for v << cRecall that energy is a scalar quantity, not a vector, so it depends only on theparticle’s speed, but not the direction of its motion. (For a discussion of therelativistically correct form for particle energy and kinetic energy, see Chap-ter 4.)This draft document is a replacement forChapter 16 of Matter & Interactions VolumeII: Electric & Magnetic Interactions.© Copyright 2005 John Wiley & Sons.Adopters of Matter & Interactions by RuthChabay and Bruce Sherwood may providethis revised material to their students.q1Figure 16.1 No electric potential energy isassociated with a single charged particle.mc2particle energy mc2K+=K12---mv2≈16.1: A review of potential energy 16-3A reminder about the symbol ∆In this chapter we will be discussing changes in energy. Remember that thesymbol (upper case Greek “delta”) indicates a change in a quantity, cal-culated by subtracting the initial value of the quantity from its final value. Example: An electron is traveling at a speed of . Afterpassing through a region in which there is an electric field, itsspeed is . (Note that these speeds are small compared tothe speed of light.) What is the change in kinetic energy of theelectron? Solution: Since the electric force slowed down the electron, it makes sensethat , the electron’s kinetic energy decreased, and. Ex. 16.1 A proton initially travels at a speed of 3000 m/s. After itpasses through a region in which there is an electric field, theproton’s speed is 5000 m/s. (a) What is the initial kinetic energy ofthe proton? (b) What is the final kinetic energy of the proton? (c)What is the change in kinetic energy of the proton? Ex. 16.2 An electron passes through a region in which there is anelectric field, and while it is in the region its kinetic energydecreases by . Initially the kinetic energy of the electronwas . What is the final speed of the electron? Ex. 16.3 A proton that initially is traveling at a speed of 300 m/senters a region where there is an electric field. Under the influenceof the electric field the proton slows down and comes to a stop.What is the change in kinetic energy of the