Electric Potential Potential Differences in a Uniform Electric Field Electric Potential and Potential Energy Due to Point Charges

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1 P U Z Z L E R Jennife is holding on to an electically chaged sphee that eaches an electic potential of about V. The device that geneates this high electic potential is called a Van de Gaaff geneato. What causes Jennife s hai to stand on end like the needles of a pocupine? Why is she safe in this situation in view of the fact that 110 V fom a wall outlet can kill you? (Heny Leap and Jim Lehman) c h a p t e Electic Potential Chapte Outline 25.1 Potential Diffeence and Electic Potential 25.2 Potential Diffeences in a Unifom Electic Field 25.3 Electic Potential and Potential Enegy Due to Point Chages 25.4 Obtaining the Value of the Electic Field fom the Electic Potential 25.5 Electic Potential Due to Continuous Chage Distibutions 25.6 Electic Potential Due to a Chaged Conducto 25.7 (Optional) The Millikan Oil-Dop Epeiment 25.8 (Optional) pplications of Electostatics 768

2 25.1 Potential Diffeence and Electic Potential 769 T he concept of potential enegy was intoduced in Chapte 8 in connection with such consevative foces as the foce of gavity and the elastic foce eeted by a sping. By using the law of consevation of enegy, we wee able to avoid woking diectly with foces when solving vaious poblems in mechanics. In this chapte we see that the concept of potential enegy is also of geat value in the study of electicity. Because the electostatic foce given by Coulomb s law is consevative, electostatic phenomena can be conveniently descibed in tems of an electic potential enegy. This idea enables us to define a scala quantity known as electic potential. Because the electic potential at any point in an electic field is a scala function, we can use it to descibe electostatic phenomena moe simply than if we wee to ely only on the concepts of the electic field and electic foces. In late chaptes we shall see that the concept of electic potential is of geat pactical value POTENTIL DIFFERENCE ND ELECTRIC POTENTIL When a test chage q 0 is placed in an electic field E ceated by some othe chaged object, the electic foce acting on the test chage is q 0 E. (If the field is poduced by moe than one chaged object, this foce acting on the test chage is the vecto sum of the individual foces eeted on it by the vaious othe chaged objects.) The foce q 0 E is consevative because the individual foces descibed by Coulomb s law ae consevative. When the test chage is moved in the field by some etenal agent, the wok done by the field on the chage is equal to the negative of the wok done by the etenal agent causing the displacement. Fo an infinitesimal displacement ds, the wok done by the electic field on the chage is F ds q 0 E ds. s this amount of wok is done by the field, the potential enegy of the chagefield system is deceased by an amount du q 0 E ds. Fo a finite displacement of the chage fom a point to a point B, the change in potential enegy of the system U U B U is U q 0 B E ds (25.1) The integation is pefomed along the path that q 0 follows as it moves fom to B, and the integal is called eithe a path integal o a line integal (the two tems ae synonymous). Because the foce q 0 E is consevative, this line integal does not depend on the path taken fom to B. Change in potential enegy Quick Quiz 25.1 If the path between and B does not make any diffeence in Equation 25.1, why don t we just use the epession U q 0 Ed, whee d is the staight-line distance between and B? The potential enegy pe unit chage U/q 0 is independent of the value of q 0 and has a unique value at evey point in an electic field. This quantity U/q 0 is called the electic potential (o simply the potential) V. Thus, the electic potential at any point in an electic field is V U q 0 (25.2)

3 770 CHPTER 25 Electic Potential The fact that potential enegy is a scala quantity means that electic potential also is a scala quantity. The potential diffeence V V B V between any two points and B in an electic field is defined as the change in potential enegy of the system divided by the test chage q 0 : Potential diffeence V U q 0 B E ds (25.3) Potential diffeence should not be confused with diffeence in potential enegy. The potential diffeence is popotional to the change in potential enegy, and we see fom Equation 25.3 that the two ae elated by U q 0 V. Electic potential is a scala chaacteistic of an electic field, independent of the chages that may be placed in the field. Howeve, when we speak of potential enegy, we ae efeing to the chagefield system. Because we ae usually inteested in knowing the electic potential at the location of a chage and the potential enegy esulting fom the inteaction of the chage with the field, we follow the common convention of speaking of the potential enegy as if it belonged to the chage. Because the change in potential enegy of a chage is the negative of the wok done by the electic field on the chage (as noted in Equation 25.1), the potential diffeence V between points and B equals the wok pe unit chage that an etenal agent must pefom to move a test chage fom to B without changing the kinetic enegy of the test chage. Just as with potential enegy, only diffeences in electic potential ae meaningful. To avoid having to wok with potential diffeences, howeve, we often take the value of the electic potential to be zeo at some convenient point in an electic field. This is what we do hee: abitaily establish the electic potential to be zeo at a point that is infinitely emote fom the chages poducing the field. Having made this choice, we can state that the electic potential at an abitay point in an electic field equals the wok equied pe unit chage to bing a positive test chage fom infinity to that point. Thus, if we take point in Equation 25.3 to be at infinity, the electic potential at any point P is V P P E ds (25.4) In eality, V P epesents the potential diffeence V between the point P and a point at infinity. (Eq is a special case of Eq ) Because electic potential is a measue of potential enegy pe unit chage, the SI unit of both electic potential and potential diffeence is joules pe coulomb, which is defined as a volt (V): Definition of volt 1 V 1 J C That is, 1 J of wok must be done to move a 1-C chage though a potential diffeence of 1 V. Equation 25.3 shows that potential diffeence also has units of electic field times distance. Fom this, it follows that the SI unit of electic field (N/C) can also be epessed in volts pe mete: 1 N C 1 V m

4 25.2 Potential Diffeences in a Unifom Electic Field 771 unit of enegy commonly used in atomic and nuclea physics is the electon volt (ev), which is defined as the enegy an electon (o poton) gains o loses by moving though a potential diffeence of 1 V. Because 1 V 1 J/C and because the fundamental chage is appoimately C, the electon volt is elated to the joule as follows: 1 ev CV J (25.5) Fo instance, an electon in the beam of a typical television pictue tube may have a speed of m/s. This coesponds to a kinetic enegy of J, which is equivalent to ev. Such an electon has to be acceleated fom est though a potential diffeence of 3.5 kv to each this speed. The electon volt 25.2 POTENTIL DIFFERENCES IN UNIFORM ELECTRIC FIELD Equations 25.1 and 25.3 hold in all electic fields, whethe unifom o vaying, but they can be simplified fo a unifom field. Fist, conside a unifom electic field diected along the negative y ais, as shown in Figue 25.1a. Let us calculate the potential diffeence between two points and B sepaated by a distance d, whee d is measued paallel to the field lines. Equation 25.3 gives V B V V B E ds B E cos 0 ds B E ds Because E is constant, we can emove it fom the integal sign; this gives V E B ds Ed (25.6) The minus sign indicates that point B is at a lowe electic potential than point ; that is, V B V. Electic field lines always point in the diection of deceasing electic potential, as shown in Figue 25.1a. Now suppose that a test chage q 0 moves fom to B. We can calculate the change in its potential enegy fom Equations 25.3 and 25.6: U q 0 V q 0 Ed (25.7) Potential diffeence in a unifom electic field E B q (a) d g B m (b) d Figue 25.1 (a) When the electic field E is diected downwad, point B is at a lowe electic potential than point. positive test chage that moves fom point to point B loses electic potential enegy. (b) mass m moving downwad in the diection of the gavitational field g loses gavitational potential enegy.

5 772 CHPTER 25 Electic Potential QuickLab It takes an electic field of about V/cm to cause a spak in dy ai. Shuffle acoss a ug and each towad a dooknob. By estimating the length of the spak, detemine the electic potential diffeence between you finge and the dooknob afte shuffling you feet but befoe touching the knob. (If it is vey humid on the day you attempt this, it may not wok. Why?) Figue 25.2 unifom electic field diected along the positive ais. Point B is at a lowe electic potential than point. Points B and C ae at the same electic potential. n equipotential suface s B C E 11.9 Fom this esult, we see that if q 0 is positive, then U is negative. We conclude that a positive chage loses electic potential enegy when it moves in the diection of the electic field. This means that an electic field does wok on a positive chage when the chage moves in the diection of the electic field. (This is analogous to the wok done by the gavitational field on a falling mass, as shown in Figue 25.1b.) If a positive test chage is eleased fom est in this electic field, it epeiences an electic foce q 0 E in the diection of E (downwad in Fig. 25.1a). Theefoe, it acceleates downwad, gaining kinetic enegy. s the chaged paticle gains kinetic enegy, it loses an equal amount of potential enegy. If q 0 is negative, then U is positive and the situation is evesed: negative chage gains electic potential enegy when it moves in the diection of the electic field. If a negative chage is eleased fom est in the field E, it acceleates in a diection opposite the diection of the field. Now conside the moe geneal case of a chaged paticle that is fee to move between any two points in a unifom electic field diected along the ais, as shown in Figue (In this situation, the chage is not being moved by an etenal agent as befoe.) If s epesents the displacement vecto between points and B, Equation 25.3 gives V B E ds E B ds Es (25.8) whee again we ae able to emove E fom the integal because it is constant. The change in potential enegy of the chage is U q 0 V q 0 E s (25.9) Finally, we conclude fom Equation 25.8 that all points in a plane pependicula to a unifom electic field ae at the same electic potential. We can see this in Figue 25.2, whee the potential diffeence V B V is equal to the potential diffeence V C V. (Pove this to youself by woking out the dot poduct E s fo s :B, whee the angle between E and s is abitay as shown in Figue 25.2, and the dot poduct fo s :C, whee 0.) Theefoe, V B V C. The name equipotential suface is given to any suface consisting of a continuous distibution of points having the same electic potential. Note that because U q 0 V, no wok is done in moving a test chage between any two points on an equipotential suface. The equipotential sufaces of a unifom electic field consist of a family of planes that ae all pependicula to the field. Equipotential sufaces fo fields with othe symmeties ae descibed in late sections. Quick Quiz 25.2 The labeled points in Figue 25.3 ae on a seies of equipotential sufaces associated with an electic field. Rank (fom geatest to least) the wok done by the electic field on a positively chaged paticle that moves fom to B; fom B to C; fom C to D; fom D to E. 9 V 8 V 7 V C B E D 6 V Figue 25.3 Fou equipotential sufaces.

6 25.2 Potential Diffeences in a Unifom Electic Field 773 EXMPLE 25.1 The Electic Field Between Two Paallel Plates of Opposite Chage battey poduces a specified potential diffeence between conductos attached to the battey teminals. 12-V battey is connected between two paallel plates, as shown in Figue The sepaation between the plates is d 0.30 cm, and we assume the electic field between the plates to be unifom. Figue 25.4 B d 12 V 12-V battey connected to two paallel plates. The electic field between the plates has a magnitude given by the potential diffeence V divided by the plate sepaation d. (This assumption is easonable if the plate sepaation is small elative to the plate dimensions and if we do not conside points nea the plate edges.) Find the magnitude of the electic field between the plates. Solution The electic field is diected fom the positive plate () to the negative one (B ), and the positive plate is at a highe electic potential than the negative plate is. The potential diffeence between the plates must equal the potential diffeence between the battey teminals. We can undestand this by noting that all points on a conducto in equilibium ae at the same electic potential 1 ; no potential diffeence eists between a teminal and any potion of the plate to which it is connected. Theefoe, the magnitude of the electic field between the plates is, fom Equation 25.6, E V B V d 12 V m V/m This configuation, which is shown in Figue 25.4 and called a paallel-plate capacito, is eamined in geate detail in Chapte 26. EXMPLE 25.2 Motion of a Poton in a Unifom Electic Field poton is eleased fom est in a unifom electic field that has a magnitude of V/m and is diected along the positive ais (Fig. 25.5). The poton undegoes a displacement of 0.50 m in the diection of E. (a) Find the change in electic potential between points and B. Solution Because the poton (which, as you emembe, caies a positive chage) moves in the diection of the field, we epect it to move to a position of lowe electic potential. E Fom Equation 25.6, we have (b) Find the change in potential enegy of the poton fo this displacement. Solution V Ed ( V/m)(0.50 m) U q 0 V e V ( C)( V) V J Figue 25.5 v v = 0 B B d poton acceleates fom to B in the diection of the electic field. The negative sign means the potential enegy of the poton deceases as it moves in the diection of the electic field. s the poton acceleates in the diection of the field, it gains kinetic enegy and at the same time loses electic potential enegy (because enegy is conseved). Eecise Use the concept of consevation of enegy to find the speed of the poton at point B. nswe m/s. 1 The electic field vanishes within a conducto in electostatic equilibium; thus, the path integal E ds between any two points in the conducto must be zeo. moe complete discussion of this point is given in Section 25.6.

7 774 CHPTER 25 Electic Potential 25.3 ELECTRIC POTENTIL ND POTENTIL ENERGY DUE TO POINT CHRGES Conside an isolated positive point chage q. Recall that such a chage poduces an electic field that is diected adially outwad fom the chage. To find the electic potential at a point located a distance fom the chage, we begin with the geneal epession fo potential diffeence: V B V B E ds d θ ds B whee and B ae the two abitay points shown in Figue t any field point, the electic field due to the point chage is E k e q ˆ/ 2 (Eq. 23.4), whee ˆ is a unit vecto diected fom the chage towad the field point. The quantity E ds can be epessed as B E ds k q e 2 ˆ ds ˆ q Because the magnitude of ˆ is 1, the dot poduct ˆ ds ds cos, whee is the angle between ˆ and ds. Futhemoe, ds cos is the pojection of ds onto ; thus, ds cos d. That is, any displacement ds along the path fom point to point B poduces a change d in the magnitude of, the adial distance to the chage ceating the field. Making these substitutions, we find that E ds (k e q/ 2 )d; hence, the epession fo the potential diffeence becomes Figue 25.6 The potential diffeence between points and B due to a point chage q depends only on the initial and final adial coodinates and B. The two dashed cicles epesent coss-sections of spheical equipotential sufaces. Electic potential ceated by a point chage V B V E d k e q B V B V k e q 1 1 (25.10) B The integal of E ds is independent of the path between points and B as it must be because the electic field of a point chage is consevative. Futhemoe, Equation epesses the impotant esult that the potential diffeence between any two points and B in a field ceated by a point chage depends only on the adial coodinates and B. It is customay to choose the efeence of electic potential to be zeo at. With this efeence, the electic potential ceated by a point chage at any distance fom the chage is V k e q (25.11) Electic potential is gaphed in Figue 25.7 as a function of, the adial distance fom a positive chage in the y plane. Conside the following analogy to gavitational potential: Imagine tying to oll a mable towad the top of a hill shaped like Figue 25.7a. The gavitational foce epeienced by the mable is analogous to the epulsive foce epeienced by a positively chaged object as it appoaches anothe positively chaged object. Similaly, the electic potential gaph of the egion suounding a negative chage is analogous to a hole with espect to any appoaching positively chaged objects. chaged object must be infinitely distant fom anothe chage befoe the suface is flat and has an electic potential of zeo. d 2 k eq B

8 25.3 Electic Potential and Potential Enegy Due to Point Chages Electic potential (V) y 0 (a) Figue 25.7 (a) The electic potential in the plane aound a single positive chage is plotted on the vetical ais. (The electic potential function fo a negative chage would look like a hole instead of a hill.) The ed line shows the 1/ natue of the electic potential, as given by Equation (b) View looking staight down the vetical ais of the gaph in pat (a), showing concentic cicles whee the electic potential is constant. These cicles ae coss sections of equipotential sphees having the chage at the cente. (b)

9 776 CHPTER 25 Electic Potential Quick Quiz 25.3 spheical balloon contains a positively chaged object at its cente. s the balloon is inflated to a geate volume while the chaged object emains at the cente, does the electic potential at the suface of the balloon incease, decease, o emain the same? How about the magnitude of the electic field? The electic flu? Electic potential due to seveal point chages Electic potential enegy due to two chages We obtain the electic potential esulting fom two o moe point chages by applying the supeposition pinciple. That is, the total electic potential at some point P due to seveal point chages is the sum of the potentials due to the individual chages. Fo a goup of point chages, we can wite the total electic potential at P in the fom q V k e i (25.12) i i whee the potential is again taken to be zeo at infinity and i is the distance fom the point P to the chage q i. Note that the sum in Equation is an algebaic sum of scalas athe than a vecto sum (which we use to calculate the electic field of a goup of chages). Thus, it is often much easie to evaluate V than to evaluate E. The electic potential aound a dipole is illustated in Figue We now conside the potential enegy of a system of two chaged paticles. If V 1 is the electic potential at a point P due to chage q 1, then the wok an etenal agent must do to bing a second chage q 2 fom infinity to P without acceleation is q 2 V 1. By definition, this wok equals the potential enegy U of the two-paticle system when the paticles ae sepaated by a distance 12 (Fig. 25.9). Theefoe, we can epess the potential enegy as 2 U k q 1q 2 e (25.13) 12 Note that if the chages ae of the same sign, U is positive. This is consistent with the fact that positive wok must be done by an etenal agent on the system to bing the two chages nea one anothe (because like chages epel). If the chages ae of opposite sign, U is negative; this means that negative wok must be done against the attactive foce between the unlike chages fo them to be bought nea each othe. If moe than two chaged paticles ae in the system, we can obtain the total potential enegy by calculating U fo evey pai of chages and summing the tems algebaically. s an eample, the total potential enegy of the system of thee chages shown in Figue is U k e q 1 q 2 q 1q 3 q 2q 3 (25.14) Physically, we can intepet this as follows: Imagine that q 1 is fied at the position shown in Figue but that q 2 and q 3 ae at infinity. The wok an etenal agent must do to bing q 2 fom infinity to its position nea q 1 is k e q 1 q 2 / 12, which is the fist tem in Equation The last two tems epesent the wok equied to bing q 3 fom infinity to its position nea q 1 and q 2. (The esult is independent of the ode in which the chages ae tanspoted.) 2 The epession fo the electic potential enegy of a system made up of two point chages, Equation 25.13, is of the same fom as the equation fo the gavitational potential enegy of a system made up of two point masses, Gm 1 m 2 / (see Chapte 14). The similaity is not supising in view of the fact that both epessions ae deived fom an invese-squae foce law.

10 25.3 Electic Potential and Potential Enegy Due to Point Chages q 2 Electic potential (V) q 1 Figue If two point chages ae sepaated by a distance 12, the potential enegy of the pai of chages is given by k e q 1 q 2 / 12. q y q 1 13 q (a) Figue Thee point chages ae fied at the positions shown. The potential enegy of this system of chages is given by Equation Figue 25.8 (a) The electic potential in the plane containing a dipole. (b) Top view of the function gaphed in pat (a). (b)

11 778 CHPTER 25 Electic Potential EXMPLE 25.3 The Electic Potential Due to Two Point Chages chage q C is located at the oigin, and a chage q C is located at (0, 3.00) m, as shown in Figue 25.11a. (a) Find the total electic potential due to these chages at the point P, whose coodinates ae (4.00, 0) m. Solution V P k e q 1 1 q 2 2 Fo two chages, the sum in Equation gives Nm2 C C m V C 5.00 m (b) Find the change in potential enegy of a 3.00-C chage as it moves fom infinity to point P (Fig b). Solution When the chage is at infinity, U i 0, and when the chage is at P, U f q 3 V P ; theefoe, Theefoe, because W U, positive wok would have to be done by an etenal agent to emove the chage fom point P back to infinity. Eecise U q 3 V P 0 ( C)( V) Find the total potential enegy of the system illustated in Figue 25.11b. nswe J J y y 6.00 µc µ 6.00 µc µ 3.00 m 3.00 m P 2.00 µc µ 4.00 m 2.00 µc µ 4.00 m 3.00 µc µ (a) (b) Figue (a) The electic potential at P due to the two chages is the algebaic sum of the potentials due to the individual chages. (b) What is the potential enegy of the thee-chage system? 25.4 OBTINING THE VLUE OF THE ELECTRIC FIELD FROM THE ELECTRIC POTENTIL The electic field E and the electic potential V ae elated as shown in Equation We now show how to calculate the value of the electic field if the electic potential is known in a cetain egion. Fom Equation 25.3 we can epess the potential diffeence dv between two points a distance ds apat as dv Eds (25.15) If the electic field has only one component E, then E ds E d. Theefoe, Equation becomes dv E d, o E dv d (25.16)

12 25.4 Obtaining the Value of the Electic Field fom the Electic Potential 779 That is, the magnitude of the electic field in the diection of some coodinate is equal to the negative of the deivative of the electic potential with espect to that coodinate. Recall fom the discussion following Equation 25.8 that the electic potential does not change fo any displacement pependicula to an electic field. This is consistent with the notion, developed in Section 25.2, that equipotential sufaces ae pependicula to the field, as shown in Figue small positive chage placed at est on an electic field line begins to move along the diection of E because that is the diection of the foce eeted on the chage by the chage distibution ceating the electic field (and hence is the diection of a). Because the chage stats with zeo velocity, it moves in the diection of the change in velocity that is, in the diection of a. In Figues 25.12a and 25.12b, a chage placed at est in the field will move in a staight line because its acceleation vecto is always paallel to its velocity vecto. The magnitude of v inceases, but its diection does not change. The situation is diffeent in Figue 25.12c. positive chage placed at some point nea the dipole fist moves in a diection paallel to E at that point. Because the diection of the electic field is diffeent at diffeent locations, howeve, the foce acting on the chage changes diection, and a is no longe paallel to v. This causes the moving chage to change diection and speed, but it does not necessaily follow the electic field lines. Recall that it is not the velocity vecto but athe the acceleation vecto that is popotional to foce. If the chage distibution ceating an electic field has spheical symmety such that the volume chage density depends only on the adial distance, then the electic field is adial. In this case, E ds E d, and thus we can epess dv in the fom dv E d. Theefoe, E dv (25.17) d Fo eample, the electic potential of a point chage is V k e q/. Because V is a function of only, the potential function has spheical symmety. pplying Equation 25.17, we find that the electic field due to the point chage is E k e q/ 2, a familia esult. Note that the potential changes only in the adial diection, not in q E (a) (b) Figue Equipotential sufaces (dashed blue lines) and electic field lines (ed lines) fo (a) a unifom electic field poduced by an infinite sheet of chage, (b) a point chage, and (c) an electic dipole. In all cases, the equipotential sufaces ae pependicula to the electic field lines at evey point. Compae these dawings with Figues 25.2, 25.7b, and 25.8b. (c)

13 780 CHPTER 25 Electic Potential Equipotential sufaces ae pependicula to the electic field lines any diection pependicula to. Thus, V (like E ) is a function only of. gain, this is consistent with the idea that equipotential sufaces ae pependicula to field lines. In this case the equipotential sufaces ae a family of sphees concentic with the spheically symmetic chage distibution (Fig b). The equipotential sufaces fo an electic dipole ae sketched in Figue 25.12c. When a test chage undegoes a displacement ds along an equipotential suface, then dv 0 because the potential is constant along an equipotential suface. Fom Equation 25.15, then, dv Eds 0; thus, E must be pependicula to the displacement along the equipotential suface. This shows that the equipotential sufaces must always be pependicula to the electic field lines. In geneal, the electic potential is a function of all thee spatial coodinates. If V() is given in tems of the catesian coodinates, the electic field components E, E y, and E z can eadily be found fom V(, y, z) as the patial deivatives 3 E V E y V y E z V z Fo eample, if V 3 2 y y 2 yz, then V (32 y y 2 yz) (32 y) 3y d d (2 ) 6y EXMPLE 25.4 The Electic Potential Due to a Dipole n electic dipole consists of two chages of equal magnitude and opposite sign sepaated by a distance 2a, as shown in Figue The dipole is along the ais and is centeed at the oigin. (a) Calculate the electic potential at point P. Solution Fo point P in Figue 25.13, (How would this esult change if point P happened to be located to the left of the negative chage?) (b) Calculate V and E at a point fa fom the dipole. Solution If point P is fa fom the dipole, such that W a, then a 2 can be neglected in the tem 2 a 2, and V becomes V k e q i i k e q a q a 2k e qa 2 a 2 V 2k e qa 2 ( W a) Using Equation and this esult, we can calculate the electic field at a point fa fom the dipole: y E dv d 4k e qa 3 ( W a) Figue a a P q q n electic dipole located on the ais. (c) Calculate V and E if point P is located anywhee between the two chages. Solution V k e q i i k e q a E dv d d d q a 2k eq 2 a 2 2k eq 2 2 a 2k eq 2 a 2 2 ( 2 a 2 ) 3 In vecto notation, E is often witten whee is called the gadient opeato. E V i j y k z V

14 25.5 Electic Potential Due to Continuous Chage Distibutions 781 We can check these esults by consideing the situation at the cente of the dipole, whee 0, V 0, and E 2k e q/a 2. Eecise Veify the electic field esult in pat (c) by calculating the sum of the individual electic field vectos at the oigin due to the two chages ELECTRIC POTENTIL DUE TO CONTINUOUS CHRGE DISTRIBUTIONS We can calculate the electic potential due to a continuous chage distibution in two ways. If the chage distibution is known, we can stat with Equation fo the electic potential of a point chage. We then conside the potential due to a small chage element dq, teating this element as a point chage (Fig ). The electic potential dv at some point P due to the chage element dq is dv k dq e (25.18) whee is the distance fom the chage element to point P. To obtain the total potential at point P, we integate Equation to include contibutions fom all elements of the chage distibution. Because each element is, in geneal, a diffeent distance fom point P and because k e is constant, we can epess V as V k e dq (25.19) In effect, we have eplaced the sum in Equation with an integal. Note that this epession fo V uses a paticula efeence: The electic potential is taken to be zeo when point P is infinitely fa fom the chage distibution. If the electic field is aleady known fom othe consideations, such as Gauss s law, we can calculate the electic potential due to a continuous chage distibution using Equation If the chage distibution is highly symmetic, we fist evaluate E at any point using Gauss s law and then substitute the value obtained into Equation 25.3 to detemine the potential diffeence V between any two points. We then choose the electic potential V to be zeo at some convenient point. We illustate both methods with seveal eamples. P Figue dq The electic potential at the point P due to a continuous chage distibution can be calculated by dividing the chaged body into segments of chage dq and summing the electic potential contibutions ove all segments. EXMPLE 25.5 Electic Potential Due to a Unifomly Chaged Ring (a) Find an epession fo the electic potential at a point P located on the pependicula cental ais of a unifomly chaged ing of adius a and total chage Q. Solution Let us oient the ing so that its plane is pependicula to an ais and its cente is at the oigin. We can then take point P to be at a distance fom the cente of the ing, as shown in Figue The chage element dq is at a distance! 2 a 2 fom point P. Hence, we can epess V as V k e dq dq k e! 2 a 2 Because each element dq is at the same distance fom point P, we can emove! 2 a 2 fom the integal, and V educes to (25.20) The only vaiable in this epession fo V is. This is not supising because ou calculation is valid only fo points along the ais, whee y and z ae both zeo. (b) Find an epession fo the magnitude of the electic field at point P. Solution V k e! 2 a 2 dq Fom symmety, we see that along the ais E can have only an component. Theefoe, we can use Equak e Q! 2 a 2

15 782 CHPTER 25 Electic Potential tion 25.16: E dv d k eq d d (2 a 2 ) 1/2 k e Q( 1 2 )(2 a 2 ) 3/2 (2) te, V has eithe a maimum o minimum value; it is, in fact, a maimum. dq k e Q ( 2 a 2 ) 3/2 (25.21) a 2 a 2 This esult agees with that obtained by diect integation (see Eample 23.8). Note that E 0 at 0 (the cente of the ing). Could you have guessed this fom Coulomb s law? Eecise What is the electic potential at the cente of the ing? What does the value of the field at the cente tell you about the value of V at the cente? nswe V k e Q /a. Because E dv/d 0 at the cen- Figue unifomly chaged ing of adius a lies in a plane pependicula to the ais. ll segments dq of the ing ae the same distance fom any point P lying on the ais. P EXMPLE 25.6 Electic Potential Due to a Unifomly Chaged Disk Find (a) the electic potential and (b) the magnitude of the electic field along the pependicula cental ais of a unifomly chaged disk of adius a and suface chage density. Solution (a) gain, we choose the point P to be at a distance fom the cente of the disk and take the plane of the disk to be pependicula to the ais. We can simplify the poblem by dividing the disk into a seies of chaged ings. The electic potential of each ing is given by Equation Conside one such ing of adius and width d, as indicated in Figue The suface aea of the ing is d 2 d; Figue a d 2 2 unifomly chaged disk of adius a lies in a plane pependicula to the ais. The calculation of the electic potential at any point P on the ais is simplified by dividing the disk into many ings each of aea 2 d. d = 2πdπ P fom the definition of suface chage density (see Section 23.5), we know that the chage on the ing is dq Hence, the potential at the point P due to this ing is d 2 d. To find the total electic potential at P, we sum ove all ings making up the disk. That is, we integate dv fom 0 to a: V k e a 0 dv This integal is of the fom u n du and has the value u n1 /(n 1), whee n 1 2 and u 2 2. This gives V (25.22) (b) s in Eample 25.5, we can find the electic field at any aial point fom E dv d k e dq! 2 k e 2 d 2! d! 2 k e a ( 2 2 ) 1/2 2 d 2 0 2k e [( 2 a 2 ) 1/2 ] 2k e 1 2! 2 a (25.23) The calculation of V and E fo an abitay point off the ais is moe difficult to pefom, and we do not teat this situation in this tet.

16 25.5 Electic Potential Due to Continuous Chage Distibutions 783 EXMPLE 25.7 Electic Potential Due to a Finite Line of Chage od of length located along the ais has a total chage Q and a unifom linea chage density Q /. Find the electic potential at a point P located on the y ais a distance a fom the oigin (Fig ). Evaluating V, we find that V k e Q ln!2 a 2 a (25.24) Solution The length element d has a chage dq d. Because this element is a distance! 2 a 2 fom point P, we can epess the potential at point P due to this element as To obtain the total potential at P, we integate this epession ove the limits 0 to. Noting that k e and ae constants, we find that V k e This integal has the following value (see ppendi B): dv k e dq 0 d! 2 a k Q 2 e 0 d k e d! 2 a 2! 2 a 2 ln(!2 a 2 ) d! 2 a 2 P a 0 y Figue unifom line chage of length located along the ais. To calculate the electic potential at P, the line chage is divided into segments each of length d and each caying a chage dq d. dq d EXMPLE 25.8 Electic Potential Due to a Unifomly Chaged Sphee n insulating solid sphee of adius R has a unifom positive volume chage density and total chage Q. (a) Find the electic potential at a point outside the sphee, that is, fo R. Take the potential to be zeo at. Solution In Eample 24.5, we found that the magnitude of the electic field outside a unifomly chaged sphee of adius R is E k e Q 2 (fo R ) Because the potential must be continuous at R, we can use this epession to obtain the potential at the suface of the sphee. That is, the potential at a point such as C shown in Figue is V C k e Q R (fo R ) (b) Find the potential at a point inside the sphee, that is, fo R. whee the field is diected adially outwad when Q is positive. In this case, to obtain the electic potential at an eteio point, such as B in Figue 25.18, we use Equation 25.4 and the epession fo E given above: V B E d k e Q d 2 Q R D C B V B k e Q (fo R ) Note that the esult is identical to the epession fo the electic potential due to a point chage (Eq ). Figue unifomly chaged insulating sphee of adius R and total chage Q. The electic potentials at points B and C ae equivalent to those poduced by a point chage Q located at the cente of the sphee, but this is not tue fo point D.

17 784 CHPTER 25 Electic Potential Solution In Eample 24.5 we found that the electic field inside an insulating unifomly chaged sphee is (fo R ) We can use this esult and Equation 25.3 to evaluate the potential diffeence V D V C at some inteio point D: V D V C E d k eq Substituting V C k e Q /R into this epession and solving fo V D, we obtain (fo R ) (25.25) t R, this epession gives a esult that agees with that fo the potential at the suface, that is, V C. plot of V vesus fo this chage distibution is given in Figue Eecise E k eq R 3 R R 3 R V D k eq 2R 3 2 R 2 d k eq 2R 3 (R 2 2 ) What ae the magnitude of the electic field and the electic potential at the cente of the sphee? nswe E 0; V 0 3k e Q /2R. Figue V 0 V 0 V V 0 = 3k e Q 2R V D = k e Q 2R (3 2 R 2 ) R V B = k eq plot of electic potential V vesus distance fom the cente of a unifomly chaged insulating sphee of adius R. The cuve fo V D inside the sphee is paabolic and joins smoothly with the cuve fo V B outside the sphee, which is a hypebola. The potential has a maimum value V 0 at the cente of the sphee. We could make this gaph thee dimensional (simila to Figues 25.7a and 25.8a) by spinning it aound the vetical ais ELECTRIC POTENTIL DUE TO CHRGED CONDUCTOR In Section 24.4 we found that when a solid conducto in equilibium caies a net chage, the chage esides on the oute suface of the conducto. Futhemoe, we showed that the electic field just outside the conducto is pependicula to the suface and that the field inside is zeo. We now show that evey point on the suface of a chaged conducto in equilibium is at the same electic potential. Conside two points and B on the suface of a chaged conducto, as shown in Figue long a suface path connecting these points, E is always pependicula to the displacement ds; thee- B E Figue n abitaily shaped conducto caying a positive chage. When the conducto is in electostatic equilibium, all of the chage esides at the suface, E 0 inside the conducto, and the diection of E just outside the conducto is pependicula to the suface. The electic potential is constant inside the conducto and is equal to the potential at the suface. Note fom the spacing of the plus signs that the suface chage density is nonunifom.

18 25.6 Electic Potential Due to a Chaged Conducto 785 foe E ds 0. Using this esult and Equation 25.3, we conclude that the potential diffeence between and B is necessaily zeo: V B V B E ds 0 This esult applies to any two points on the suface. Theefoe, V is constant eveywhee on the suface of a chaged conducto in equilibium. That is, the suface of any chaged conducto in electostatic equilibium is an equipotential suface. Futhemoe, because the electic field is zeo inside the conducto, we conclude fom the elationship E dv/d that the electic potential is constant eveywhee inside the conducto and equal to its value at the suface. The suface of a chaged conducto is an equipotential suface Because this is tue about the electic potential, no wok is equied to move a test chage fom the inteio of a chaged conducto to its suface. Conside a solid metal conducting sphee of adius R and total positive chage Q, as shown in Figue 25.21a. The electic field outside the sphee is k e Q / 2 and points adially outwad. Fom Eample 25.8, we know that the electic potential at the inteio and suface of the sphee must be k e Q /R elative to infinity. The potential outside the sphee is k e Q /. Figue 25.21b is a plot of the electic potential as a function of, and Figue 25.21c shows how the electic field vaies with. When a net chage is placed on a spheical conducto, the suface chage density is unifom, as indicated in Figue 25.21a. Howeve, if the conducto is nonspheical, as in Figue 25.20, the suface chage density is high whee the adius of cuvatue is small and the suface is conve (as noted in Section 24.4), and it is low whee the adius of cuvatue is small and the suface is concave. Because the electic field just outside the conducto is popotional to the suface chage density, we see that the electic field is lage nea conve points having small adii of cuvatue and eaches vey high values at shap points. Figue shows the electic field lines aound two spheical conductos: one caying a net chage Q, and a lage one caying zeo net chage. In this case, the suface chage density is not unifom on eithe conducto. The sphee having zeo net chage has negative chages induced on its side that faces the (a) R (b) (c) k e Q R V E k e Q k e Q 2 R Electic field patten of a chaged conducting plate placed nea an oppositely chaged pointed conducto. Small pieces of thead suspended in oil align with the electic field lines. The field suounding the pointed conducto is most intense nea the pointed end and at othe places whee the adius of cuvatue is small. Figue (a) The ecess chage on a conducting sphee of adius R is unifomly distibuted on its suface. (b) Electic potential vesus distance fom the cente of the chaged conducting sphee. (c) Electic field magnitude vesus distance fom the cente of the chaged conducting sphee.

19 786 CHPTER 25 Electic Potential Q Q = 0 Figue The electic field lines (in ed) aound two spheical conductos. The smalle sphee has a net chage Q, and the lage one has zeo net chage. The blue cuves ae cosssections of equipotential sufaces. chaged sphee and positive chages induced on its side opposite the chaged sphee. The blue cuves in the figue epesent the coss-sections of the equipotential sufaces fo this chage configuation. s usual, the field lines ae pependicula to the conducting sufaces at all points, and the equipotential sufaces ae pependicula to the field lines eveywhee. Tying to move a positive chage in the egion of these conductos would be like moving a mable on a hill that is flat on top (epesenting the conducto on the left) and has anothe flat aea patway down the side of the hill (epesenting the conducto on the ight). EXMPLE 25.9 Two Connected Chaged Sphees Two spheical conductos of adii 1 and 2 ae sepaated by a distance much geate than the adius of eithe sphee. The sphees ae connected by a conducting wie, as shown in Figue The chages on the sphees in equilibium ae q 1 and q 2, espectively, and they ae unifomly chaged. Find the atio of the magnitudes of the electic fields at the sufaces of the sphees. q 1 1 Solution Because the sphees ae connected by a conducting wie, they must both be at the same electic potential: q 2 2 V k e q 1 1 k e q 2 2 Theefoe, the atio of chages is Figue Two chaged spheical conductos connected by a conducting wie. The sphees ae at the same electic potential V.

20 25.6 Electic Potential Due to a Chaged Conducto 787 (1) q 1 q Taking the atio of these two fields and making use of Equation (1), we find that Because the sphees ae vey fa apat and thei sufaces unifomly chaged, we can epess the magnitude of the electic fields at thei sufaces as E 1 k q 1 e 2 1 and E 2 k q 2 e 2 2 E 1 E Hence, the field is moe intense in the vicinity of the smalle sphee even though the electic potentials of both sphees ae the same. Cavity Within a Conducto Now conside a conducto of abitay shape containing a cavity as shown in Figue Let us assume that no chages ae inside the cavity. In this case, the electic field inside the cavity must be zeo egadless of the chage distibution on the outside suface of the conducto. Futhemoe, the field in the cavity is zeo even if an electic field eists outside the conducto. To pove this point, we use the fact that evey point on the conducto is at the same electic potential, and theefoe any two points and B on the suface of the cavity must be at the same potential. Now imagine that a field E eists in the cavity and evaluate the potential diffeence V B V defined by Equation 25.3: V B V B E ds If E is nonzeo, we can always find a path between and B fo which E ds is a positive numbe; thus, the integal must be positive. Howeve, because V B V 0, the integal of E ds must be zeo fo all paths between any two points on the conducto, which implies that E is zeo eveywhee. This contadiction can be econciled only if E is zeo inside the cavity. Thus, we conclude that a cavity suounded by conducting walls is a field-fee egion as long as no chages ae inside the cavity. B Figue conducto in electostatic equilibium containing a cavity. The electic field in the cavity is zeo, egadless of the chage on the conducto. Coona Dischage phenomenon known as coona dischage is often obseved nea a conducto such as a high-voltage powe line. When the electic field in the vicinity of the conducto is sufficiently stong, electons ae stipped fom ai molecules. This causes the molecules to be ionized, theeby inceasing the ai s ability to conduct. The obseved glow (o coona dischage) esults fom the ecombination of fee electons with the ionized ai molecules. If a conducto has an iegula shape, the electic field can be vey high nea shap points o edges of the conducto; consequently, the ionization pocess and coona dischage ae most likely to occu aound such points. Quick Quiz 25.4 (a) Is it possible fo the magnitude of the electic field to be zeo at a location whee the electic potential is not zeo? (b) Can the electic potential be zeo whee the electic field is nonzeo?

21 788 CHPTER 25 Electic Potential v F D q mg Optional Section 25.7 THE MILLIKN OIL-DROP EXPERIMENT Duing the peiod fom 1909 to 1913, Robet Millikan pefomed a billiant set of epeiments in which he measued e, the elementay chage on an electon, and demonstated the quantized natue of this chage. His appaatus, diagammed in Figue 25.25, contains two paallel metallic plates. Chaged oil doplets fom an atomize ae allowed to pass though a small hole in the uppe plate. hoizontally diected light beam (not shown in the diagam) is used to illuminate the oil doplets, which ae viewed though a telescope whose long ais is at ight angles to the light beam. When the doplets ae viewed in this manne, they appea as shining stas against a dak backgound, and the ate at which individual dops fall can be detemined. 4 Let us assume that a single dop having a mass m and caying a chage q is being viewed and that its chage is negative. If no electic field is pesent between the plates, the two foces acting on the chage ae the foce of gavity mg acting downwad and a viscous dag foce F D acting upwad as indicated in Figue 25.26a. The dag foce is popotional to the dop s speed. When the dop eaches its teminal speed v, the two foces balance each othe (mg F D ). Now suppose that a battey connected to the plates sets up an electic field between the plates such that the uppe plate is at the highe electic potential. In this case, a thid foce qe acts on the chaged dop. Because q is negative and E is diected downwad, this electic foce is diected upwad, as shown in Figue 25.26b. If this foce is sufficiently geat, the dop moves upwad and the dag foce F D acts downwad. When the upwad electic foce q E balances the sum of the gavitational foce and the downwad dag foce F D, the dop eaches a new teminal speed v in the upwad diection. With the field tuned on, a dop moves slowly upwad, typically at ates of hundedths of a centimete pe second. The ate of fall in the absence of a field is compaable. Hence, one can follow a single doplet fo hous, altenately ising and falling, by simply tuning the electic field on and off. (a) Field off Oil doplets tomize qe Battey Pin hole v E q Chaged plate v Chaged plate Telescope mg Figue (b) Field on F D The foces acting on a negatively chaged oil doplet in the Millikan epeiment. Figue Switch Schematic dawing of the Millikan oil-dop appaatus. 4 t one time, the oil doplets wee temed Millikan s Shining Stas. Pehaps this desciption has lost its populaity because of the geneations of physics students who have epeienced hallucinations, nea blindness, migaine headaches, and so foth, while epeating Millikan s epeiment!

22 25.8 pplications of Electostatics 789 fte ecoding measuements on thousands of doplets, Millikan and his cowokes found that all doplets, to within about 1% pecision, had a chage equal to some intege multiple of the elementay chage e : q ne n 0, 1, 2, 3,... whee e C. Millikan s epeiment yields conclusive evidence that chage is quantized. Fo this wok, he was awaded the Nobel Pize in Physics in Optional Section 25.8 PPLICTIONS OF ELECTROSTTICS The pactical application of electostatics is epesented by such devices as lightning ods and electostatic pecipitatos and by such pocesses as eogaphy and the painting of automobiles. Scientific devices based on the pinciples of electostatics include electostatic geneatos, the field-ion micoscope, and ion-dive ocket engines. The Van de Gaaff Geneato In Section 24.5 we descibed an epeiment that demonstates a method fo tansfeing chage to a hollow conducto (the Faaday ice-pail epeiment). When a chaged conducto is placed in contact with the inside of a hollow conducto, all of the chage of the chaged conducto is tansfeed to the hollow conducto. In pinciple, the chage on the hollow conducto and its electic potential can be inceased without limit by epetition of the pocess. In 1929 Robet J. Van de Gaaff ( ) used this pinciple to design and build an electostatic geneato. This type of geneato is used etensively in nuclea physics eseach. schematic epesentation of the geneato is given in Figue Chage is deliveed continuously to a high-potential electode by means of a moving belt of insulating mateial. The high-voltage electode is a hollow conducto mounted on an insulating column. The belt is chaged at point by means of a coona dischage between comb-like metallic needles and a gounded gid. The needles ae maintained at a positive electic potential of typically 10 4 V. The positive chage on the moving belt is tansfeed to the hollow conducto by a second comb of needles at point B. Because the electic field inside the hollow conducto is negligible, the positive chage on the belt is easily tansfeed to the conducto egadless of its potential. In pactice, it is possible to incease the electic potential of the hollow conducto until electical dischage occus though the ai. Because the beakdown electic field in ai is about V/m, a sphee 1 m in adius can be aised to a maimum potential of V. The potential can be inceased futhe by inceasing the adius of the hollow conducto and by placing the entie system in a containe filled with high-pessue gas. Van de Gaaff geneatos can poduce potential diffeences as lage as 20 million volts. Potons acceleated though such lage potential diffeences eceive enough enegy to initiate nuclea eactions between themselves and vaious taget nuclei. Smalle geneatos ae often seen in science classooms and museums. If a peson insulated fom the gound touches the sphee of a Van de Gaaff geneato, his o he body can be bought to a high electic potential. The hai acquies a net positive chage, and each stand is epelled by all the othes. The esult is a Gounded gid Figue B Gound Hollow conducto Belt Schematic diagam of a Van de Gaaff geneato. Chage is tansfeed to the hollow conducto at the top by means of a moving belt. The chage is deposited on the belt at point and tansfeed to the hollow conducto at point B. Insulato

23 790 CHPTER 25 Electic Potential scene such as that depicted in the photogaph at the beginning of this chapte. In addition to being insulated fom gound, the peson holding the sphee is safe in this demonstation because the total chage on the sphee is vey small (on the ode of 1 C). If this amount of chage accidentally passed fom the sphee though the peson to gound, the coesponding cuent would do no ham. QuickLab Spinkle some salt and peppe on an open dish and mi the two togethe. Now pull a comb though you hai seveal times and bing the comb to within 1 cm of the salt and peppe. What happens? How is what happens hee elated to the opeation of an electostatic pecipitato? The Electostatic Pecipitato One impotant application of electical dischage in gases is the electostatic pecipitato. This device emoves paticulate matte fom combustion gases, theeby educing ai pollution. Pecipitatos ae especially useful in coal-buning powe plants and in industial opeations that geneate lage quantities of smoke. Cuent systems ae able to eliminate moe than 99% of the ash fom smoke. Figue 25.28a shows a schematic diagam of an electostatic pecipitato. high potential diffeence (typically 40 to 100 kv) is maintained between a wie unning down the cente of a duct and the walls of the duct, which ae gounded. The wie is maintained at a negative electic potential with espect to the walls, so the electic field is diected towad the wie. The values of the field nea the wie become high enough to cause a coona dischage aound the wie; the dischage ionizes some ai molecules to fom positive ions, electons, and such negative ions as O 2. The ai to be cleaned entes the duct and moves nea the wie. s the electons and negative ions ceated by the dischage ae acceleated towad the oute wall by the electic field, the dit paticles in the ai become chaged by collisions and ion captue. Because most of the chaged dit paticles ae negative, they too ae dawn to the duct walls by the electic field. When the duct is peiodically shaken, the paticles beak loose and ae collected at the bottom. Insulato Clean ai out Dity ai in Weight Dit out (a) Figue (b) (a) Schematic diagam of an electostatic pecipitato. The high negative electic potential maintained on the cental coiled wie ceates an electical dischage in the vicinity of the wie. Compae the ai pollution when the electostatic pecipitato is (b) opeating and (c) tuned off. (c)

24 25.8 pplications of Electostatics 791 In addition to educing the level of paticulate matte in the atmosphee (compae Figs b and c), the electostatic pecipitato ecoves valuable mateials in the fom of metal oides. Xeogaphy and Lase Pintes The basic idea of eogaphy 5 was developed by Cheste Calson, who was ganted a patent fo the eogaphic pocess in The one featue of this pocess that makes it unique is the use of a photoconductive mateial to fom an image. ( photoconducto is a mateial that is a poo electical conducto in the dak but that becomes a good electical conducto when eposed to light.) The eogaphic pocess is illustated in Figue 25.29a to d. Fist, the suface of a plate o dum that has been coated with a thin film of photoconductive mateial (usually selenium o some compound of selenium) is given a positive electostatic chage in the dak. n image of the page to be copied is then focused by a lens onto the chaged suface. The photoconducting suface becomes conducting only in aeas whee light stikes it. In these aeas, the light poduces chage caies in the photoconducto that move the positive chage off the dum. Howeve, positive Lens Light causes some aeas of dum to become electically conducting, emoving positive chage Selenium-coated dum (a) Chaging the dum (b) Imaging the document Negatively chaged tone (c) pplying the tone Intelaced patten of lase lines Lase beam Figue (d) Tansfeing the tone to the pape (e) Lase pinte dum The eogaphic pocess: (a) The photoconductive suface of the dum is positively chaged. (b) Though the use of a light souce and lens, an image is fomed on the suface in the fom of positive chages. (c) The suface containing the image is coveed with a negatively chaged powde, which adhees only to the image aea. (d) piece of pape is placed ove the suface and given a positive chage. This tansfes the image to the pape as the negatively chaged powde paticles migate to the pape. The pape is then heat-teated to fi the powde. (e) lase pinte opeates similaly ecept the image is poduced by tuning a lase beam on and off as it sweeps acoss the selenium-coated dum. 5 The pefi eo- is fom the Geek wod meaning dy. Note that no liquid ink is used anywhee in eogaphy.

25 792 CHPTER 25 Electic Potential chages emain on those aeas of the photoconducto not eposed to light, leaving a latent image of the object in the fom of a positive suface chage distibution. Net, a negatively chaged powde called a tone is dusted onto the photoconducting suface. The chaged powde adhees only to those aeas of the suface that contain the positively chaged image. t this point, the image becomes visible. The tone (and hence the image) ae then tansfeed to the suface of a sheet of positively chaged pape. Finally, the tone is fied to the suface of the pape as the tone melts while passing though high-tempeatue olles. This esults in a pemanent copy of the oiginal. lase pinte (Fig e) opeates by the same pinciple, with the eception that a compute-diected lase beam is used to illuminate the photoconducto instead of a lens. SUMMRY When a positive test chage q 0 is moved between points and B in an electic field E, the change in the potential enegy is U q 0 B E ds (25.1) The electic potential V U/q 0 is a scala quantity and has units of joules pe coulomb ( J/C), whee 1 J/C 1 V. The potential diffeence V between points and B in an electic field E is defined as V U (25.3) q 0 B E ds The potential diffeence between two points and B in a unifom electic field E is V Ed (25.6) whee d is the magnitude of the displacement in the diection paallel to E. n equipotential suface is one on which all points ae at the same electic potential. Equipotential sufaces ae pependicula to electic field lines. If we define V 0 at, the electic potential due to a point chage at any distance fom the chage is V k e q (25.11) We can obtain the electic potential associated with a goup of point chages by summing the potentials due to the individual chages. The potential enegy associated with a pai of point chages sepaated by a distance 12 is U k q 1q 2 e (25.13) 12 This enegy epesents the wok equied to bing the chages fom an infinite sepaation to the sepaation 12. We obtain the potential enegy of a distibution of point chages by summing tems like Equation ove all pais of paticles.

26 Summay 793 TBLE 25.1 Electic Potential Due to Vaious Chage Distibutions Chage Distibution Electic Potential Location Unifomly chaged ing of adius a Unifomly chaged disk of adius a Unifomly chaged, insulating solid sphee of adius R and total chage Q Isolated conducting sphee of adius R and total chage Q V k e V k Q e V k Q e R Q! 2 a 2 V 2k e [( 2 a 2 ) 1/2 ] V k Q e V k eq 2R 3 2 R 2 long pependicula cental ais of ing, distance fom ing cente long pependicula cental ais of disk, distance fom disk cente R R R R If we know the electic potential as a function of coodinates, y, z, we can obtain the components of the electic field by taking the negative deivative of the electic potential with espect to the coodinates. Fo eample, the component of the electic field is E dv (25.16) d The electic potential due to a continuous chage distibution is V k e dq (25.19) Evey point on the suface of a chaged conducto in electostatic equilibium is at the same electic potential. The potential is constant eveywhee inside the conducto and equal to its value at the suface. Table 25.1 lists electic potentials due to seveal chage distibutions. Poblem-Solving Hints Calculating Electic Potential Remembe that electic potential is a scala quantity, so components need not be consideed. Theefoe, when using the supeposition pinciple to evaluate the electic potential at a point due to a system of point chages, simply take the algebaic sum of the potentials due to the vaious chages. Howeve, you must keep tack of signs. The potential is positive fo positive chages, and it is negative fo negative chages. Just as with gavitational potential enegy in mechanics, only changes in electic potential ae significant; hence, the point whee you choose the poten-

27 794 CHPTER 25 Electic Potential tial to be zeo is abitay. When dealing with point chages o a chage distibution of finite size, we usually define V 0 to be at a point infinitely fa fom the chages. You can evaluate the electic potential at some point P due to a continuous distibution of chage by dividing the chage distibution into infinitesimal elements of chage dq located at a distance fom P. Then, teat one chage element as a point chage, such that the potential at P due to the element is dv k e dq/. Obtain the total potential at P by integating dv ove the entie chage distibution. In pefoming the integation fo most poblems, you must epess dq and in tems of a single vaiable. To simplify the integation, conside the geomety involved in the poblem caefully. Review Eamples 25.5 though 25.7 fo guidance. nothe method that you can use to obtain the electic potential due to a finite continuous chage distibution is to stat with the definition of potential diffeence given by Equation If you know o can easily obtain E (fom Gauss s law), then you can evaluate the line integal of E ds. n eample of this method is given in Eample Once you know the electic potential at a point, you can obtain the electic field at that point by emembeing that the electic field component in a specified diection is equal to the negative of the deivative of the electic potential in that diection. Eample 25.4 illustates this pocedue. QUESTIONS 1. Distinguish between electic potential and electic potential enegy. 2. negative chage moves in the diection of a unifom electic field. Does the potential enegy of the chage incease o decease? Does it move to a position of highe o lowe potential? 3. Give a physical eplanation of the fact that the potential enegy of a pai of like chages is positive wheeas the potential enegy of a pai of unlike chages is negative. 4. unifom electic field is paallel to the ais. In what diection can a chage be displaced in this field without any etenal wok being done on the chage? 5. Eplain why equipotential sufaces ae always pependicula to electic field lines. 6. Descibe the equipotential sufaces fo (a) an infinite line of chage and (b) a unifomly chaged sphee. 7. Eplain why, unde static conditions, all points in a conducto must be at the same electic potential. 8. The electic field inside a hollow, unifomly chaged sphee is zeo. Does this imply that the potential is zeo inside the sphee? Eplain. 9. The potential of a point chage is defined to be zeo at an infinite distance. Why can we not define the potential of an infinite line of chage to be zeo at? 10. Two chaged conducting sphees of diffeent adii ae connected by a conducting wie, as shown in Figue Which sphee has the geate chage density? 11. What detemines the maimum potential to which the dome of a Van de Gaaff geneato can be aised? 12. Eplain the oigin of the glow sometimes obseved aound the cables of a high-voltage powe line. 13. Why is it impotant to avoid shap edges o points on conductos used in high-voltage equipment? 14. How would you shield an electonic cicuit o laboatoy fom stay electic fields? Why does this wok? 15. Why is it elatively safe to stay in an automobile with a metal body duing a sevee thundestom? 16. Walking acoss a capet and then touching someone can esult in a shock. Eplain why this occus.

28 Poblems 795 PROBLEMS 1, 2, 3 = staightfowad, intemediate, challenging = full solution available in the Student Solutions Manual and Study Guide WEB = solution posted at = Compute useful in solving poblem = Inteactive Physics = paied numeical/symbolic poblems Section 25.1 Potential Diffeence and Electic Potential 1. How much wok is done (by a battey, geneato, o some othe souce of electical enegy) in moving vogado s numbe of electons fom an initial point whee the electic potential is 9.00 V to a point whee the potential is 5.00 V? (The potential in each case is measued elative to a common efeence point.) 2. n ion acceleated though a potential diffeence of 115 V epeiences an incease in kinetic enegy of J. Calculate the chage on the ion. 3. (a) Calculate the speed of a poton that is acceleated fom est though a potential diffeence of 120 V. (b) Calculate the speed of an electon that is acceleated though the same potential diffeence. 4. Review Poblem. Though what potential diffeence would an electon need to be acceleated fo it to achieve a speed of 40.0% of the speed of light, stating fom est? The speed of light is c m/s; eview Section What potential diffeence is needed to stop an electon having an initial speed of m/s? Section 25.2 Potential Diffeences in a Unifom Electic Field 6. unifom electic field of magnitude 250 V/m is diected in the positive diection C chage moves fom the oigin to the point (, y) (20.0 cm, 50.0 cm). (a) What was the change in the potential enegy of this chage? (b) Though what potential diffeence did the chage move? 7. The diffeence in potential between the acceleating plates of a TV set is about V. If the distance between these plates is 1.50 cm, find the magnitude of the unifom electic field in this egion. 8. Suppose an electon is eleased fom est in a unifom electic field whose magnitude is V/m. (a) Though what potential diffeence will it have passed afte moving 1.00 cm? (b) How fast will the electon be moving afte it has taveled 1.00 cm? 9. n electon moving paallel to the ais has an initial speed of m/s at the oigin. Its speed is educed to m/s at the point 2.00 cm. Calculate the potential diffeence between the oigin and that point. Which point is at the highe potential? 10. unifom electic field of magnitude 325 V/m is diected in the negative y diection as shown in Figue P The coodinates of point ae ( 0.200, 0.300) m, and those of point B ae (0.400, 0.500) m. Calculate the potential diffeence V B V, using the blue path. WEB kg block caying a chage Q 50.0 C is connected to a sping fo which k 100 N/m. The block lies on a fictionless hoizontal tack, and the system is immesed in a unifom electic field of magnitude E V/m, diected as shown in Figue P If the block is eleased fom est when the sping is unstetched (at 0), (a) by what maimum amount does the sping epand? (b) What is the equilibium position of the block? (c) Show that the block s motion is simple hamonic, and detemine its peiod. (d) Repeat pat (a) if the coefficient of kinetic fiction between block and suface is block having mass m and chage Q is connected to a sping having constant k. The block lies on a fictionless hoizontal tack, and the system is immesed in a unifom electic field of magnitude E, diected as shown in Figue P If the block is eleased fom est when the sping is unstetched (at 0), (a) by what maimum amount does the sping epand? (b) What is the equilibium position of the block? (c) Show that the block s motion is simple hamonic, and detemine its peiod.(d) Repeat pat (a) if the coefficient of kinetic fiction between block and suface is k. k y E Figue P25.10 m, Q = 0 Figue P25.11 Poblems 11 and 12. E B

29 796 CHPTER 25 Electic Potential 13. On planet Teha, the acceleation due to gavity is the same as that on Eath but thee is also a stong downwad electic field with the field being unifom close to the planet s suface kg ball having a chage of 5.00 C is thown upwad at a speed of 20.1 m/s and it hits the gound afte an inteval of 4.10 s. What is the potential diffeence between the stating point and the top point of the tajectoy? 14. n insulating od having linea chage density 40.0 C/m and linea mass density kg/m is eleased fom est in a unifom electic field E 100 V/m diected pependicula to the od (Fig. P25.14). (a) Detemine the speed of the od afte it has taveled 2.00 m. (b) How does you answe to pat (a) change if the electic field is not pependicula to the od? Eplain. sting makes an angle 60.0 with a unifom electic field of magnitude E 300 V/m. Detemine the speed of the paticle when the sting is paallel to the electic field (point a in Fig. P25.15). Section 25.3 Electic Potential and Potential Enegy Due to Point Chages Note: Unless stated othewise, assume a efeence level of potential V 0 at. 16. (a) Find the potential at a distance of 1.00 cm fom a poton. (b) What is the potential diffeence between two points that ae 1.00 cm and 2.00 cm fom a poton? (c) Repeat pats (a) and (b) fo an electon. 17. Given two 2.00-C chages, as shown in Figue P25.17, and a positive test chage q C at the oigin, (a) what is the net foce eeted on q by the two 2.00-C chages? (b) What is the electic field at the oigin due to the two 2.00-C chages? (c) What is the electic potential at the oigin due to the two 2.00-C chages? E E y 2.00 µ C q 2.00 µ C = m 0 = m Figue P paticle having chage q 2.00 C and mass m kg is connected to a sting that is L 1.50 m long and is tied to the pivot point P in Figue P The paticle, sting, and pivot point all lie on a hoizontal table. The paticle is eleased fom est when the P θ L λ,, µ Figue P25.14 m q Top View a Figue P25.15 E 18. chage q is at the oigin. chage 2q is at 2.00 m on the ais. Fo what finite value(s) of is (a) the electic field zeo? (b) the electic potential zeo? 19. The Boh model of the hydogen atom states that the single electon can eist only in cetain allowed obits aound the poton. The adius of each Boh obit is n 2 ( nm) whee n 1, 2, 3,... Calculate the electic potential enegy of a hydogen atom when the electon is in the (a) fist allowed obit, n 1; (b) second allowed obit, n 2; and (c) when the electon has escaped fom the atom ( ). Epess you answes in electon volts. 20. Two point chages Q nc and Q nc ae sepaated by 35.0 cm. (a) What is the potential enegy of the pai? What is the significance of the algebaic sign of you answe? (b) What is the electic potential at a point midway between the chages? 21. The thee chages in Figue P25.21 ae at the vetices of an isosceles tiangle. Calculate the electic potential at the midpoint of the base, taking q 7.00 C. 22. Compae this poblem with Poblem 55 in Chapte 23. Fou identical point chages (q 10.0 C) ae located on the cones of a ectangle, as shown in Figue P The dimensions of the ectangle ae L 60.0 cm and W 15.0 cm. Calculate the electic potential enegy of the chage at the lowe left cone due to the othe thee chages.

30 Poblems 797 q 4.00 cm q q 2.00 cm Figue P25.21 collide? (Hint: Conside consevation of enegy and consevation of linea momentum.) (b) If the sphees wee conductos, would the speeds be geate o less than those calculated in pat (a)? 29. small spheical object caies a chage of 8.00 nc. t what distance fom the cente of the object is the potential equal to 100 V? 50.0 V? 25.0 V? Is the spacing of the equipotentials popotional to the change in potential? 30. Two point chages of equal magnitude ae located along the y ais equal distances above and below the ais, as shown in Figue P (a) Plot a gaph of the potential at points along the ais ove the inteval 3a 3a. You should plot the potential in units of k e Q /a. (b) Let the chage located at a be negative and plot the potential along the y ais ove the inteval 4a y 4a. y WEB 23. Show that the amount of wok equied to assemble fou identical point chages of magnitude Q at the cones of a squae of side s is 5.41k e Q 2 /s. 24. Compae this poblem with Poblem 18 in Chapte 23. Two point chages each of magnitude 2.00 C ae located on the ais. One is at 1.00 m, and the othe is at 1.00 m. (a) Detemine the electic potential on the y ais at y m. (b) Calculate the electic potential enegy of a thid chage, of 3.00 C, placed on the y ais at y m. 25. Compae this poblem with Poblem 22 in Chapte 23. Five equal negative point chages q ae placed symmetically aound a cicle of adius R. Calculate the electic potential at the cente of the cicle. 26. Compae this poblem with Poblem 17 in Chapte 23. Thee equal positive chages q ae at the cones of an equilateal tiangle of side a, as shown in Figue P (a) t what point, if any, in the plane of the chages is the electic potential zeo? (b) What is the electic potential at the point P due to the two chages at the base of the tiangle? 27. Review Poblem. Two insulating sphees having adii cm and cm, masses kg and kg, and chages 2.00 C and 3.00 C ae eleased fom est when thei centes ae sepaated by 1.00 m. (a) How fast will each be moving when they collide? (Hint: Conside consevation of enegy and linea momentum.) (b) If the sphees wee conductos would the speeds be lage o smalle than those calculated in pat (a)? Eplain. 28. Review Poblem. Two insulating sphees having adii 1 and 2, masses m 1 and m 2, and chages q 1 and q 2 ae eleased fom est when thei centes ae sepaated by a distance d. (a) How fast is each moving when they a a Q >O 31. In Ruthefod s famous scatteing epeiments that led to the planetay model of the atom, alpha paticles (chage 2e, mass kg) wee fied at a gold nucleus (chage 79e). n alpha paticle, initially vey fa fom the gold nucleus, is fied with a velocity of m/s diectly towad the cente of the nucleus. How close does the alpha paticle get to this cente befoe tuning aound? ssume the gold nucleus emains stationay. 32. n electon stats fom est 3.00 cm fom the cente of a unifomly chaged insulating sphee of adius 2.00 cm and total chage 1.00 nc. What is the speed of the electon when it eaches the suface of the sphee? 33. Calculate the enegy equied to assemble the aay of chages shown in Figue P25.33, whee a m, b m, and q 6.00 C. 34. Fou identical paticles each have chage q and mass m. They ae eleased fom est at the vetices of a squae of side L. How fast is each chage moving when thei distance fom the cente of the squae doubles? Q Figue P25.30

31 798 CHPTER 25 Electic Potential WEB q 2q 2q 35. How much wok is equied to assemble eight identical point chages, each of magnitude q, at the cones of a cube of side s? Section 25.4 Obtaining the Value of the Electic Field fom the Electic Potential 36. The potential in a egion between 0 and 6.00 m is V a b whee a 10.0 V and b 7.00 V/m. Detemine (a) the potential at 0, 3.00 m, and 6.00 m and (b) the magnitude and diection of the electic field at 0, 3.00 m, and 6.00 m. 37. Ove a cetain egion of space, the electic potential is V y 2yz 2. Find the epessions fo the, y, and z components of the electic field ove this egion. What is the magnitude of the field at the point P, which has coodinates (1, 0, 2) m? 38. The electic potential inside a chaged spheical conducto of adius R is given by V k e Q /R and outside the conducto is given by V k e Q /. Using E dv/d, deive the electic field (a) inside and (b) outside this chage distibution. 39. It is shown in Eample 25.7 that the potential at a point P a distance a above one end of a unifomly chaged od of length lying along the ais is V k eq ln!2 a 2 a Use this esult to deive an epession fo the y component of the electic field at P. (Hint: Replace a with y.) 40. When an unchaged conducting sphee of adius a is placed at the oigin of an yz coodinate system that lies in an initially unifom electic field E E 0 k, the esulting electic potential is V(, y, z) V 0 E 0 z b Figue P25.33 E 0 a 3 z ( 2 y 2 z 2 ) 3/2 fo points outside the sphee, whee V 0 is the (constant) electic potential on the conducto. Use this equation to detemine the, y, and z components of the esulting electic field. 3q a Section 25.5 Electic Potential Due to Continuous Chage Distibutions 41. Conside a ing of adius R with the total chage Q spead unifomly ove its peimete. What is the potential diffeence between the point at the cente of the ing and a point on its ais a distance 2R fom the cente? 42. Compae this poblem with Poblem 33 in Chapte 23. unifomly chaged insulating od of length 14.0 cm is bent into the shape of a semicicle, as shown in Figue P If the od has a total chage of 7.50 C, find the electic potential at O, the cente of the semicicle. 43. od of length L (Fig. P25.43) lies along the ais with its left end at the oigin and has a nonunifom chage density (whee is a positive constant). (a) What ae the units of? (b) Calculate the electic potential at. d 44. Fo the aangement descibed in the pevious poblem, calculate the electic potential at point B that lies on the pependicula bisecto of the od a distance b above the ais. 45. Calculate the electic potential at point P on the ais of the annulus shown in Figue P25.45, which has a unifom chage density. a y Figue P25.43 Poblems 43 and 44. b 46. wie of finite length that has a unifom linea chage density is bent into the shape shown in Figue P Find the electic potential at point O. B L b Figue P25.45 P

32 Poblems 799 WEB 2R Section 25.6 Electic Potential Due to a Chaged Conducto 47. How many electons should be emoved fom an initially unchaged spheical conducto of adius m to poduce a potential of 7.50 kv at the suface? 48. Two chaged spheical conductos ae connected by a long conducting wie, and a chage of 20.0 C is placed on the combination. (a) If one sphee has a adius of 4.00 cm and the othe has a adius of 6.00 cm, what is the electic field nea the suface of each sphee? (b) What is the electic potential of each sphee? 49. spheical conducto has a adius of 14.0 cm and chage of 26.0 C. Calculate the electic field and the electic potential at (a) 10.0 cm, (b) 20.0 cm, and (c) 14.0 cm fom the cente. 50. Two concentic spheical conducting shells of adii a m and b m ae connected by a thin wie, as shown in Figue P If a total chage Q 10.0 C is placed on the system, how much chage settles on each sphee? (Optional) Section 25.7 (Optional) Section 25.8 q 2 b q 1 O The Millikan Oil-Dop Epeiment pplications of Electostatics 51. Conside a Van de Gaaff geneato with a 30.0-cmdiamete dome opeating in dy ai. (a) What is the maimum potential of the dome? (b) What is the maimum chage on the dome? 52. The spheical dome of a Van de Gaaff geneato can be aised to a maimum potential of 600 kv; then additional chage leaks off in spaks, by poducing beakdown of the suounding dy ai. Detemine (a) the chage on the dome and (b) the adius of the dome. R Figue P25.46 a Figue P25.50 Wie 2R DDITIONL PROBLEMS 53. The liquid-dop model of the nucleus suggests that high-enegy oscillations of cetain nuclei can split the nucleus into two unequal fagments plus a few neutons. The fagments acquie kinetic enegy fom thei mutual Coulomb epulsion. Calculate the electic potential enegy (in electon volts) of two spheical fagments fom a uanium nucleus having the following chages and adii: 38e and m; 54e and m. ssume that the chage is distibuted unifomly thoughout the volume of each spheical fagment and that thei sufaces ae initially in contact at est. (The electons suounding the nucleus can be neglected.) 54. On a dy winte day you scuff you leathe-soled shoes acoss a capet and get a shock when you etend the tip of one finge towad a metal dooknob. In a dak oom you see a spak pehaps 5 mm long. Make ode-ofmagnitude estimates of (a) you electic potential and (b) the chage on you body befoe you touch the dooknob. Eplain you easoning. 55. The chage distibution shown in Figue P25.55 is efeed to as a linea quadupole. (a) Show that the potential at a point on the ais whee a is V 2k eqa 2 3 a 2 (b) Show that the epession obtained in pat (a) when W a educes to V 2k eqa 2 3 Q 2Q Q (a,0) 56. (a) Use the eact esult fom Poblem 55 to find the electic field at any point along the ais of the linea quadupole fo a. (b) Evaluate E at 3a if a 2.00 mm and Q 3.00 C. 57. t a cetain distance fom a point chage, the magnitude of the electic field is 500 V/m and the electic potential is 3.00 kv. (a) What is the distance to the chage? (b) What is the magnitude of the chage? 58. n electon is eleased fom est on the ais of a unifom positively chaged ing, m fom the ing s y Quadupole Figue P25.55 (a,0)

33 800 CHPTER 25 Electic Potential cente. If the linea chage density of the ing is C/m and the adius of the ing is m, how fast will the electon be moving when it eaches the cente of the ing? 59. (a) Conside a unifomly chaged cylindical shell having total chage Q, adius R, and height h. Detemine the electostatic potential at a point a distance d fom the ight side of the cylinde, as shown in Figue P (Hint: Use the esult of Eample 25.5 by teating the cylinde as a collection of ing chages.) (b) Use the esult of Eample 25.6 to solve the same poblem fo a solid cylinde. b node Cathode a λ λ λ Figue P Two paallel plates having chages of equal magnitude but opposite sign ae sepaated by 12.0 cm. Each plate has a suface chage density of 36.0 nc/m 2. poton is eleased fom est at the positive plate. Detemine (a) the potential diffeence between the plates, (b) the enegy of the poton when it eaches the negative plate, (c) the speed of the poton just befoe it stikes the negative plate, (d) the acceleation of the poton, and (e) the foce on the poton. (f) Fom the foce, find the magnitude of the electic field and show that it is equal to that found fom the chage densities on the plates. 61. Calculate the wok that must be done to chage a spheical shell of adius R to a total chage Q. 62. GeigeMülle counte is a adiation detecto that essentially consists of a hollow cylinde (the cathode) of inne adius a and a coaial cylindical wie (the anode) of adius b (Fig. P25.62). The chage pe unit length on the anode is, while the chage pe unit length on the cathode is. (a) Show that the magnitude of the potential diffeence between the wie and the cylinde in the sensitive egion of the detecto is (b) Show that the magnitude of the electic field ove that egion is given by E h V 2k e ln a b V ln( a / b ) 1 whee is the distance fom the cente of the anode to the point whee the field is to be calculated. R Figue P25.59 d WEB 63. Fom Gauss s law, the electic field set up by a unifom line of chage is E 20 ˆ whee ˆ is a unit vecto pointing adially away fom the line and is the chage pe unit length along the line. Deive an epession fo the potential diffeence between 1 and point chage q is located at R, and a point chage 2q is located at the oigin. Pove that the equipotential suface that has zeo potential is a sphee centeed at ( 4R/3, 0, 0) and having a adius 2R/ Conside two thin, conducting, spheical shells as shown in coss-section in Figue P The inne shell has a adius cm and a chage of 10.0 nc. The oute shell has a adius cm and a chage of 15.0 nc. Find (a) the electic field E and (b) the electic potential V in egions, B, and C, with V 0 at. C B 66. The ais is the symmety ais of a unifomly chaged ing of adius R and chage Q (Fig. P25.66). point chage Q of mass M is located at the cente of the ing. When it is displaced slightly, the point chage accele- 1 2 Figue P25.65

34 Poblems 801 ates along the ais to infinity. Show that the ultimate speed of the point chage is v 2k e Q 2 MR 1/2 y 1 P E E θ Q q R Unifomly chaged ing Figue P25.66 Q v a θ a q n infinite sheet of chage that has a suface chage density of 25.0 nc/m 2 lies in the yz plane, passes though the oigin, and is at a potential of 1.00 kv at the point y 0, z 0. long wie having a linea chage density of 80.0 nc/m lies paallel to the y ais and intesects the ais at 3.00 m. (a) Detemine, as a function of, the potential along the ais between wie and sheet. (b) What is the potential enegy of a 2.00-nC chage placed at m? 68. The thin, unifomly chaged od shown in Figue P25.68 has a linea chage density. Find an epession fo the electic potential at P. P y Figue P25.69 (b) Fo the dipole aangement shown, epess V in tems of catesian coodinates using ( 2 y 2 ) 1/2 and cos y ( 2 y 2 ) 1/2 Using these esults and taking W a, calculate the field components E and E y. 70. Figue P25.70 shows seveal equipotential lines each labeled by its potential in volts. The distance between the lines of the squae gid epesents 1.00 cm. (a) Is the magnitude of the field bigge at o at B? Why? (b) What is E at B? (c) Repesent what the field looks like by dawing at least eight field lines. b a L Figue P dipole is located along the y ais as shown in Figue P (a) t a point P, which is fa fom the dipole ( W a), the electic potential is V k e p cos 2 whee p 2qa. Calculate the adial component E and the pependicula component E of the associated electic field. Note that Do these esults seem easonable fo 90 and 0? fo 0? E (1/)(V/). Figue P disk of adius R has a nonunifom suface chage density C, whee C is a constant and is measued fom the cente of the disk (Fig. P25.71). Find (by diect integation) the potential at P. B

35 802 CHPTER 25 Electic Potential R Figue P solid sphee of adius R has a unifom chage density and total chage Q. Deive an epession fo its total P electic potential enegy. (Hint: Imagine that the sphee is constucted by adding successive layes of concentic shells of chage dq (4 2 d) and use du V dq.) 73. The esults of Poblem 62 apply also to an electostatic pecipitato (see Figs a and P25.62). n applied voltage V V a V b 50.0 kv is to poduce an electic field of magnitude 5.50 MV/m at the suface of the cental wie. The oute cylindical wall has unifom adius a m. (a) What should be the adius b of the cental wie? You will need to solve a tanscendental equation. (b) What is the magnitude of the electic field at the oute wall? NSWERS TO QUICK QUIZZES 25.1 We do if the electic field is unifom. (This is pecisely what we do in the net section.) In geneal, howeve, an electic field changes fom one place to anothe B : C, C : D, : B, D : E. Moving fom B to C deceases the electic potential by 2 V, so the electic field pefoms 2 J of wok on each coulomb of chage that moves. Moving fom C to D deceases the electic potential by 1 V, so 1 J of wok is done by the field. It takes no wok to move the chage fom to B because the electic potential does not change. Moving fom D to E inceases the electic potential by 1 V, and thus the field does 1 J of wok, just as aising a mass to a highe elevation causes the gavitational field to do negative wok on the mass The electic potential deceases in invese popotion to the adius (see Eq ). The electic field magnitude deceases as the ecipocal of the adius squaed (see Eq. 23.4). Because the suface aea inceases as 2 while the electic field magnitude deceases as 1/ 2, the electic flu though the suface emains constant (see Eq. 24.1) (a) Yes. Conside fou equal chages placed at the cones of a squae. The electic potential gaph fo this situation is shown in the figue. t the cente of the squae, the electic field is zeo because the individual fields fom the fou chages cancel, but the potential is not zeo. This is also the situation inside a chaged conducto. (b) Yes again. In Figue 25.8, fo instance, the electic potential is zeo at the cente of the dipole, but the magnitude of the field at that point is not zeo. (The two chages in a dipole ae by definition of opposite sign; thus, the electic field lines ceated by the two chages etend fom the positive to the negative chage and do not cancel anywhee.) This is the situation we pesented in Eample 25.4c, in which the equations we obtained give V 0 and E 0. Electic potential (V) y