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1 Ouestions ' ber. A gamma ruy entere the chamber from the bottom an at one point transforme into an electron an a positron. Because those new particles were charge an moving, each left a trail of tiny bubbles. (The trails were curve because a magnetic fiel ha been set up in the chamber.) The gammaray,being electrically neutral, left no trail. Still, you can tell exactly where it unerwent pair prouction -at the tip of the curve V, which is where the trails of the electron an positron begin. Electric Charge The strength of a particle's electrical interaction with objects aroun it epens on its electric charge, which can be either positive or negative. Charges with the same sign repel each other, an charges with opposite signs attract each other. An object with equal amounts of the two kins of charge is electrically neutral, whereas one with an imbalance is electrically charge. Conuctors are materials in which a signiflcant number of charge particles (electrons in metals) are free to move. The charge particles in nonconuctors, or insulators, are not free to move. The Coulomb an Ampere The SI unit of charge is the coulomb (C). It is eflne in terms of the unit of current, the ampere (A), as the charge passing a particular point in 1 secon when there is a current of 1 ampere at that point: 1C : (1 A)(1 r). This is base on the relation between current i an the rate qlt at which charge passes a point:.q i -, (electric current). (2r-3) Coulomb's Law Coulomb's law escribes the electrostatic force between small (point) electric charges q1 an ez at rest (or nearly at rest) an separate by a istance r: F- I I o.,ll a"l ^ + (Coulomb'slaw). 4 7Tt11 f ' (2r-4) Here 80 : 8.85 x 10 -tz gz75 ' mz is the permittivity constant, an Il4rres - k: 8.99 x 10r Y. m2lc2. The force of attraction or repulsion between point charges at rest acts along the line joining the two charges. If more than two charges are present, Eq. 2I-4 hols for each pair of charges. The net force on each charge is then foun, using the superposition principle, as the vector sum of the forces exerte on the charge by all the others. The two shell theorems for electrostatics are A shell of uniform charge attracts or repels a charge ';:::::'*n':,:#;':,":!::*:,::"'if atttheshett'scharge If a charge particle is locate insie a shell of uniform charge, there is no net electrostatic force on the particle from the shell. The Elementary Charge Electric charge is quantize: any charge can be written as ne,whete n is a positive or negative integer an e rs a constant of nature calle the elementary charge (: x 10 -tq C). Electric charge is conserve: the net charge of any isolate system cannot change. 1 Figure 2l-I3 shows four situations in which charge particles are flxe in place on an axis. In which situations is there a point to the left of the particles where an electron will be in equilibrium? +q -q +3q (") cl -5q (b) 3 Figure 2l-I5 shows four situations in which flve charge particles are evenly space along an axis. The charge values are inicate except for the central particle, which has the same charge in all four situations. Rank the situations accoritrg to the magnitue of the net electrostatic force on the central particle, greatest first. +3q -q o -5q +q (') (,) FlG" 2t-13 Question 1. e Figure 2I-L4 shows two charge particles on an axis. The charges are free to move. o -5q -q Flffi, en-14 QuestionZ. However, a thir charge particle can be place at a certain point such that all three particles are then in equilibrium. (a) Is that point to the left of the first two particles, to their right, or between them? (b) Shoul the thir particle be positively or negatively charge? (c) Is the equilibrium stable or unstable? (1) (2) (3) (4) +e +e Flffi" Al"tS Question 3. 4 Figure 2I-16 shows three pairs of ientical spheres that are to be touche together an then separate. The initial charges on them are inicate. Rank the pairs accoring to (a) the mag-

2 ffihmpter RR I Electric Fiels uces a torque on an electric ipole moment F to align 1 with E. gecause the oven's E otiillates, the water molecules continuously nip-nop in a frustrate attempt to align with E. Energy is transferre from the electric fiel to the thermal energy of the water (an thus of the foo) where three water molecules happene to have bone together to form a group.the flip-flop breaks some of the bons. When the molecules reform the bons, energy is transferre to the ranom motion of the group an then to the surrouning molecules. Soon, the thermal energy of the water is enough to cook the foo. Sometimes the heating is surprisitrg. If you heat a jelly onut, for example, the jelly (which hols a lot of water) heats far more than the onut material (which hols much less water). Although the exterior of the onut may not be hot, biting into the jelly can burn you. If water molecules were not electric ipoles, we woul not have microwave ovens. A neuftal water molecule (FI ro) in its vapor state has an electric ipole moment of magnitue 6.2 x C.m. (u) How far apart are the molecule's centers of positive an negative charge? A molecule's ipole moment epens on the magnitue q of the molecule's positive or negative charge an the charge separation. eafeufafons; There are 10 electrons an 10 protons in a neutral water molecule; so the magnitue of its ipole moment is p - q : (10e) (), in whi ch is the separation we are seeking an e is the elementary charge. Thus, p 6.2 x C.m t\e (10)(1.60 x 10-1e C) This istance is not only small, but it is also actually smaller than the raius of a hyrogen atom. (b) If the molecule is place in an electric flel of 1.5 x 104 N/C, what maximum torque can the flel exert on it? (Such a flel can easily be set up in the laboratory.) r - pe sin g ipole is maximum when is 90". 90" in Eq yiels (.) FIow much work must an external agent o to rotate this molecule by 180" in this flel, starting from its fully aligne position, for which 0 : 0? The work one by an external agent (bv means of a torque applie to the molecule) is equal to the change in the molecule's potential energy ue to the change in orientation. Ca faffom; From Eq ,we fln W- (Jtro" Uo -,pe - (2)(6.2 x C.-)(1.5 x 104 N/C) L ' I.9 x I0-2s J. (Answer) Electric Fiel One way to explain the electrostatic force between two charges is to assume that each charge sets up an electric flel in the space aroun it. The electrostatic force acting on any one charge is then ue to the electric flel set up at its location by the other charge visualizing the irection an magnitue of electric flels. The electric fiel vector at any point is tangent to a fiel line through that point. The ensity of flel lines in any region is proportional to the magnitue of the electric fiel in that region. Fiel iines originate on positive charges an terminate on negative charges. Definition of Electric Fiel The electric fiet E at any point is efine in terms of the electrostatic force F that woul be exerte on a positive test charge qsplace there: ) F Fiel Due to a Point Charge The magnitue of the electric fiel E set up by a point rhurg. q at a istan ce r from the charge is (22-3) E-_ (22-t) qo Electric Fiel Lines Electric fiet lines provie a means for The irection of E ir away from the point charge if the charge is positive an towar it if the charge is negative.

3 Questiorrs,, Fiel Due to an Electric Dipole An electric ipote consists of two particles with charges of equal magnitue q bfi opposite sign, separate by a small istance. Their electric ipole moment n has magnitue q an points from the negative charge to the positive charge. The magnitue of the electric fiel set up by the ipole at a istant point on the ipole axis (which runs through both charges) is (22-e) where z is the istance between the point an the center of the ipole. Fiel Due to a Continuous Charge Distribution The electric flel ue to a continuous charge istribution is foun by treating charge elements as point charges an then summing, via integration, the electric fiel vectors prouce by all the charge elements. Force on a Point Charge in an Electric Fiel When a point charge q is qface in an external electric fiel E, th" electrostatic force F that acts on the point charge is F-qE. (22-28) Force F has the same irection as E it q is positive an the opposite irection if q is negative. Dipole in an Electric Fiel When an electric ipole moment p' is place in an electric flel E, exerts a torqu e i on the ipole: i--i*8. ipole of the flel (22-34) The ipole has a potential ener gy U associate with its orientation in the fiel: (Ji (22-38) This potential energy is efine to be zero when F is perpe!_ icular to E;it is least ((I: -pe) when P'is aligne with E an greatest (U - pe) when l is irecte opposite E. T Figure 22-2I shows three affangements of electric fiel lines. In each arrangement, a proton is release from rest at point A an is then accelerate through point B by the electric fiel. Points A an B have equal separations in the three arrangements. Rank the arrangements accoring to the linear momentum of the proton at point B,greatest first. axis (other than at an infinite istance) is there a point at which their net electric flel is zero: between the charges, to their left, or to their right? (b) Is there a point of zero net electric flel anywhere off the axis (other than at aninflnite istance)? 4 Figure shows two square arrays of charge particles. The squares, which are centere on point P, are misaligne. The particles are separate by either or lz along the perimeters of the squares. What are the magnitue an irection of the net electric flel at P? +6q -2q +3q (") (b) Ffrffi, *P-*1 Question 1. (c) o -5q +2rl tr Figure shows four situations in which four charge particles are evenly space to the left an right of a central point. The charge values are inicate. Rank the situations accoring to the magnitue of the net electric fiel at the central point, greatest first. (1) +e -e -e +e -2q +3q +2q -Zq -2q F[G" 2Z"A& Question 4. -q +6q (2) (3) (4) +e +e -e -e -e +e +e +e -e -e +e -e la-l--,++'-) FEffi" 2?-ZA Question2. S Figure shows two charge particles flxe in place on an axis. (u) Where on the +q -3q Ff,G. 2&-fr3 Question 3. S In Fig , two particles of charge - q are arrange symmetrically about the y axis; each prouces an electric flel at point P on that axis. (a) Are the magnitues of the fiels at P equal? (b) Is each electric fiel irecte towar or away from FflG. AA-e5 Question 5. the charge proucing it? (c) Is the magnitue of the net electric flel at P equal to the sum of the magnitues,e of the two fiel vectors (is it equal to 2E)? () Do the x components of those two fiel vectors a or cancel? (e) Do their y components a or cancel? (f ) Is the irection of the net fiel at P that of the canceling components or

4 #ha,pter 2S I Gauss' Law ffiffikp#$rut 4 The figure shows two large, parallel, nonconucting sheets with ientical (positive) uniform surface charge ensities, an a sphere with a uniform (positive) volume charge ensity. Rank the four o1 ol c3 o4 a-_-] '.l--- a-*- a numbere points accoring to the rnagnitue of the net electric fiel there, greatest first. Gauss' Law Gauss' law an Coulomb's law are ifferent ways of escribing the relation between charge an electric fiel in static situations. Gauss' law is eoo : e"n" (Gauss' law), (23-6) in which e"n is the net charge insie an imaginary close surface (a Gaussian surface) an Q is the net flu* of the electric fiel through the surface:.--? (electric flux through a o-? - ^t E'A = Gaussiansurface). Q3-4) Coulomb's law can be erive from Gauss' law. Applications of Gauss' Law Using Gauss' law an, in some cases, symmetry arguments, we can erive several important results in electrostatic situations. Among these are: L. An excess charge on an isolate conuctor ts locate entirely on the outer surface of the conuctor. 2. The external electric fiel near the surface of a charge conuctor is perpenicular to the surface an has magnitue a E - - ^(j0 (conuctingsurface). (23-rr) Within the conuctor, E : The electric fiel at any point ue to an infinite line of charge with uniform linear charge ensity,1, is perpenicular to the line of charge an has magnitue E,- (line of charge), (23-12) 2Tesr where r is the perpenicular istance from the line of charge to the point. 4. The electric fiel ue to an infinite nonconucting sheet with uniform surface charge ensity o is perpenicular to the plane of the sheet an has magnitue a E - 2"0 ^ (sheet of charge). 5. The electric flel outsie a spherical shell of charge with raius R an total charge q is irecte raially an has magnitue E- q- 4nes 12 (23-r3) (spherical shell, for r > R). (23-15) Here r is the istance from the center of the shell to the point at which E is measure. (Th" charge behaves, for external points, as if it were all locate at the center of the sphere.) The fiel insie a uniform spherical shell of charge is exactly zero: E : 0 (spherical shell, for r < R). (23-16) 6. The electric fiel insie a uniform sphere of charge is irecte raially an has magnitue (23-20) $ Figure shows, in cross section, a central metal ball, two spherical metal shells, an three spherical Gaussian surfaces of raii R,2R, an 3R, all with the same center. The uniform charges on the three objects are; ball, Q; smaller shell, 3Q;larger shell, 5Q. Rank the Gaussian surfaces accoring to the magnitue of the Shell Gaussian surface F$ffi" AS-40 Question 1. electric fiel at any point on the surface, greatest first. A Figure 23-2I shows, in cross section, two Gaussian spheres an two Gaussian cubes that are centere on a positively charge particle. (u) Rank the net flux through the four Gaussian surfaces, greatest first. (b) Rank the magnitues of the electric fiels on the surfaces, greatest flrst, an inicate whether the magnitues are uniform or variable along each surface. S A surface has the area vector A- Qi + gi) ttrt. What is the flux of a uniform electric fiel through it if the flel is (q) E - +i Nrc an (b) E - 4K N/C? 4 Figure shows, in cross section, three soli cyliners, each F!ffi. A3-tr1 QuestionZ. of length L an uniform charge Q. Concentric with each cyliner is a cylinrical Gaussian surface, with all three surfaces having the same raius. Rank the Gaussian surfaces accoring to the electric fiel at any point on the surface,greatest first.

5 Review & Summal.y you may have so much excess charge that the potential ifference between your boy an your surrounings is 5 kv or more. If you touch a computer keyboar while charge like this, the excess charge on your boy can flow through the computer's circuit chips, overloaing an ruining them. There are countless examples in which contact between a person an some other type of material leaves the person so highly charge that the person might ischarge with a spark. Chilren sliing own a plastic slie on a ry ay have been measure to have a potential of about 60 kv. If such a chil reaches for any conucting object (such as another person), the chil probably will ischarge to the object with a very painful spark. -Such a spark ischarge woul be isastrous in a hospital operatitrg room where a flammable gas (such as an anesthetic gas) is present. To rain the charge they collect as they move aroun, a surgical team wears conucting shoes an stans on a conucting floor. Spark ischarges have also cause a number of flres at self-serve gasoline stations when a customer has sli back into a car seat to wait for the car's tank to be fille. Contact with the car seat can leave the customer with so much charge that a spark might jump between flngers an pump nozzle when the customer goes back to remove the nozzle from the ear. The spark can ignite the gasoline vapor that surrouns the nozzle. Ffffi. tr4-ffiffi An uncharge conuctor is suspene in an external electric fiel. The free electrons in the conuctor istribute themselves on the surface as shown, so as to reuce the net electric flel insie the conuctor to zero an make the net fiel at the surface perpenicular to the surface. If an isolate conuctor is place in an external electric fiel, as in Fig , all points of the conuctor still come to a single potential regarless of whether the conuctor has an excess charge.the free conuction electrons istribute themselves on the surface in such a way that the electric flel they prouc e at interior points cancels the external electric fiel that woul otherwise be there. Furthermore, the electron istribution causes the net electric fiel at all points on the surface to be perpenicular to the surface. If the conuctor in Fig coul be somehow remove, leaving the surface charges frozen in place, the pattern of the electric fiel woul remain absolutely unchange, for both exterior an interior points. Electric Potential Energy The change LU in the electric potential ener gy U of a point charge as the charge moves from an initial point i to a final point f tn an electric fiel is L(J: Uf - (Jr - -W, (24-r) where W'is the work one by the electrostatic force (ue to the external electric fiel) on the point charge uring the move from i to f.if the potential energy is efine to be zero at inflnity, the electric potential energy U of the point charge at a particular point is (Jt - -w*. (24-2) Here W* is the work one by the electrostatic force on the point charge as the charge moves from inflnity to the particular point. Electric Potential Difference an Electric Potential We efine the potential ifference LV between two points i an f in an electric fiel as V- The SI unit of potential is the volt: I volt - 1 joule per coulomb. Pot*ntial an potential ifference can also be written in terms of the electric potential ener gy U of a particle of charge q rn an electric fiel: v _,, (24_5) LV: V- V: -+ (24-7) q Fining V from i The electric potential ifference between two points i an/is where q is the charge of a pafircle on which work is one by the flel. The potential at a point is v-v- i, (24-rB) W^ Lv: v-v-+-! qq q q (24-B) (24-6) Equipotential Surfaces The points on an equipotential surface all have the same electric potential. The work one on a test charge in moving it from one such surface to another is inepenent of the locations of the initial an final points on these surfaces an of the path that joins the points. The electric flel E is always irecte perpenicularly to corresponing equipotential surfaces. Lu

6 C'hapter 2,& I Electric Potential where the integral is taken over any path connecting the points. If we choose [ : 0, we have, for the potential at a particular point, (24-re) Potential Due to Point Charges The electric potential ue to a single point charge at a istance r from that point charge is V- Iq (24-26) 4rres r where I/ has the same sign as q.the potential ue tion of point charges is v_2u:*t+ Potential Due to j"'el."tr3;; from an electric ipole with ipole moment q,the electric potential of the ipole is to a collec- (24-27) At a istance r magnitue p - Calculating F from V The component tion is the negative of the rat e at which the with istance in that irection: AV DLss The x,!,an e components of E may be foun from of E in any irecpotential changes (24-40) E,: -#i E,: -# Q4-4r) When E is uniform, Eq reuces to AV E_ (24-42) As) where s is perpenicular to the equipotential surfaces. The electric fiel is zero parallel to an equipotential surface. Electric Potential Energy of a System of Point Charges The electric potential energy of a system of point charges is equal to the work neee to assemble the system with the charges initially at rest an infinitely istant from each other. For two charges at separation r, u-w- r QQz 4rres r (24-43) Potential of a Charge Conuctor An excess charge place on a conuctor will, in the equilibrium state, be locate entirely on the outer surface of the conuctor. The charge will istribute itself so that the entire conuctor, incluing interior points, is at a uniform potential. 1 Figure 24-2I shows four pairs of charge particles. For each pair, let V - 0 at infinity an consiet Vn"tat points on the x axis. For which pairs is there a point at which Vn"t: 0 (a) between thegarticles an (b) to the right of the particles? (c) At such a point is En", ue to the particles equal to zero? () For each patr, are there off-axis points (other than at infinity) wherevn"l: 0? -2q +6q +3q (1) (2) +l2q +q -4q -6q -2q (3) (4) Ffffi" &e-a1 Questions I an7. 2 Figure shows four arrangements of charge particles, all the same istance from the origin. Rank the situations accoring to the net electric potential at the origin, most positive first. Take the potential to be zero at infinity. point P at the center of the square if the electric potential is zero at infinity? & Figure shows three sets of cross sections of equipotential surfaces; all three cover the same size region of space. (u) Rank the arrangements accoring to the magnitue of the electric fiel present in the region, greatest first. (b) In which is the electric fiel irecte own the page? (1) -!q -?q +:l +5q o -5o PI -q -2q +4q Flffi. #4-23 Question 3. 20v *--140v -*-10v ** **-50 (2) (3) FlG" Question 4. (a) (b) Ffrffi QuestionZ. 3 In Fig , eight particles form a square, with istance between ajacent particles. What is the electric potential at (c) 5 Figure shows three paths along which we can move the positively charge sphere A closer to positively charge sphere B,which is hel fixe in place. (a) Woul sphere Abe move to a higher or lower electric potential? Is the FIG" A4-e5 Question 5.

7 Ouestions Capacitor; Capacitance A capacitor consists of two isolate conuctors (the plates) with charges + q an - q. Its capacitance C is efine from q-cv, (2s-r) where I/ is the potential ifference between the plates. The SI unit of capacitance is the fara (1. fara : 1 F - 1 coulomb per volt). Determining Capacitance We generally etermine the capacitance of a particular capacitor configuration by (1) assuming a charge q to have been place on the plates, (2) fining the electric fiel E.r" to this charge, (3) evaluating the potential ifference V,, an (4) calculating C from Eq. 25-L Some specific results are the following: A parallel-plate capacitor with flat parallel plates of area A an spacing has capacttance,1 eoa L: (2s-e) A cylinrical capacitor (two long coaxial cyliners) of length L an raii a an b has capacitance A spherical capacitor with concentric spherical plates of raii a an b has capacitance ab C - 4rres D-A If we let b -+ oo an a - R in Eq. 25-t7, we capacitance of an isolate sphere of raius R: C - 4rresR. (2s-r4) (2s-r7) obtain the (2s-18) Capacitors in Parallel an in Series The equivalent capacitances C"n of combinations of iniviual capacitors connecte in parallel an in series can be foun from c"q,: C j (n capacitors in parallel) (25-19) j :1' 1 (1 an (n capacitors in series). (25-20) %-3,1 Equivalent capacrtances can be use to calculate the cap acitances of more complicate series-parallel combinations. Potential Energy an Energy Density The electric potential energy U of a charge capacitor, fr q2 u - b -lcv!, (25-27,25-22) is equal to the work require to charge the capacitor. This energy can be associate with the capuiitor', eleitric flel E By extension we can associate store energy with any electric fiel. In vacuuffi, the energy ensity tt) or potential energy per unit volume, within an electric fiel of magnitue E is given by u - L"oE' (2s-2s) Capacitance with a Dielectric If the space between the plates of a capacitor is completely fille with a ielectric material, the capacitance C is increase by a factor rc, calle the ielectric constant, which is characteristic of the material. In a region that is completely fille by a ielectric, all electrostatic equations containing s6 rnust be moifie by replacing es with Kro. The effects of aing a ielectric can be unerstoo physically in terms of the action of an electric fiel on the permanent or inuce electric ipoles in the ielectric slab. The result is the formation of inuce charges on the surfaces of the ielectric, which results in a weakening of the flel within the ielectric for a given amount of free charge on the plates. Gauss' Law with a Dielectric When a ielectric is present, Gauss'law may be generaltzeto (2s-36) Here q is the free charge;any inuce surface charge is accounte for by incluing the ielectric constant xinsie the integral. 1 Figure shows plots of charge versus potential ifference for three parallelplate capacitors that have the plate areas an separations given in the table. Which plot goes with which capacitor? F$ffi" A$-19 Question L. Capacitor Area Separation T 2 -t J A 2A A 2 Figure shows an open switch, o battery of potential ifference V, current-measuring meter A, an three ientical 2 uncharge capacitors of capacrtance C. When the switch is close an the circuit reaches equilibrium, what are (u) the potential ifference across each capacitor an (b) the charge on the left plate of each capacitor? Fnffi" AS-AS Question2. (c) During charging, what net charge passes through the meter? S For each circuit in Fig. zs-zi,are the capacitors connecte in series, in parallel, or in neither moe? *J-Fl- (a) (b) Flfi. 2S-At Question 3.