Chpter 4 Experiment 2: Equipotentils nd Electric Fields 4.1 Introduction One wy to look t the force between chrges is to sy tht the chrge lters the spce round it by generting n electric field E. Any other chrge plced in this field then experiences Coulomb force. We thus regrd the E field s trnsmitting the Coulomb force. To define the electric field, E, more precisely, consider smll positive test chrge q t given loction. As long s everything else stys the sme, the Coulomb force exerted on the test chrge q is proportionl to q. Then the force per unit chrge, F/q, does not depend on the chrge q, nd therefore cn be regrded meningfully to be the electric field E t tht point. In defining the electric field, we specify tht the test chrge q be smll becuse in prctice the test chrge q cn indirectly ffect the field it is being used to mesure. If, for exmple, we bring test chrge ner the Vn de Grff genertor dome, the Coulomb forces from the test chrge redistribute the chrge on the conducting dome nd thereby slightly chnge the E field tht the dome produces. But secondry effects of this sort hve less nd less effect on the proportionlity between F nd q s we mke q smller. So mny phenomen cn be explined in terms of the electric field, but not nerly s well in terms of chrges simply exerting forces on ech other through empty spce, tht the electric field is regrded s hving rel physicl existence, rther thn being mere mthemticlly-defined quntity. For exmple when collection of chrges in one region of spce move, the effect on test chrge t distnt point is not felt instntneously, but insted is detected with time dely tht corresponds to the chnged pttern of electric field vlues moving through spce t the speed of light. Closely ssocited with the concept of electric field is the pictoril representtion of the field in terms of lines of force. These re imginry geometric lines constructed so tht the direction of the line, s given by the tngent to the line t ech point, is lwys in the direction of the E field t tht point, or equivlently, is in the direction of the force tht would ct on smll positive test chrge plced t tht point. The electric field nd the 41
concept of lines of electric force cn be used to mp out wht forces ct on chrge plced in prticulr region of spce. Figures 4.1() nd 4.1(b) show region of spce round n electric dipole, with the electric field indicted by lines of force. The chrges in Figure 4.1() re identicl but opposite in sign. In Figure 4.1(b) the chrges hve the sme sign. Above ech figure is picture of region round chrges in which grss seed hs been sprinkled on glss plte. The elongted seeds hve ligned themselves with the electric field t ech loction, thus indicting its direction t ech point. A few simple rules govern the behvior of electric field lines. These rules cn be pplied to deduce some properties of the field for vrious geometricl distributions of chrge: 1) Electric field lines re drwn such tht tngent to the line t prticulr point in spce gives the direction of the electricl force on smll positive test chrge plced t the point. 2) The density of electric field lines indictes the strength of the E field in prticulr region. The field is stronger where the lines get closer together. 3) Electric field lines strt on positive chrges nd end on negtive chrges. Sometimes the lines tke long route round nd we cn only show portion of the line within digrm of the kind below. If net chrge in the picture is not zero, some lines will not hve chrge on which to end. In tht cse they hed out towrd infinity, s shown in Figure 4.1(b). () (b) Figure 4.1: Above: Grss seeds lign themselves with the electric field between two chrges. Below: The drwing shows the lines of force ssocited with the electric field between chrges. () shows chrge pir with negtive chrge bove nd positive chrge below. (b) shows two positive chrges. You might think when severl chrges re present tht the electric field lines from two chrges could meet t some loction, producing crossed lines of force. But imgine plcing chrge where the two lines intersect. Chrges re never confused bout the direction of the force cting on them, so long which line would the force lie? In such cse, the electric fields dd vectorilly t ech point, producing single net E field tht lies long one specific line of force, rther thn being t the intersection of two lines of force. Thus, it cn be seen tht none of the lines cross ech other. It cn lso be shown tht two field lines never merge to become one. 42
Figure 4.2: Electric field round group of chrges. Lines of force re shown s solid lines. Equipotentil lines re shown s dshed lines. Under certin circumstnces, the rules defining these field lines cn be used to deduce some generl properties of chrges nd their forces. For exmple, property esily deduced from these rules is tht region of spce enclosed by sphericlly symmetric distribution of chrge hs zero electric field everywhere within tht region (ssuming no dditionl chrges produce electric fields inside). Imgine first sphericlly symmetric thin shell of positive chrge ll t certin distnce from the center. Field lines from the shell would hve to be rdilly outwrd eqully in ll directions. If these outwrd pointing lines continued rdilly inwrd beneth the shell, they would hve nowhere to end. Hence, the field lines must hve ended t the surfce chrge, nd there must be zero field everywhere inside. Next, suppose the sphericlly symmetric distribution of chrge surrounding the unchrged region is not merely thin shell. We cn nevertheless consider the chrge distribution to be divided up into mny thin lyers ech t different rdius. Ech lyer contributes its own field lines tht end t tht lyer, producing none of the field lines in the region enclosed by tht lyer. Then ny point in the region of interest is inside ll of the thin lyers, where ll the field lines hve ended. We cn conclude tht the E field is zero t ny point within region surrounded by the sphericl distribution of chrge. In Figures 4.1() nd 4.1(b) it is seen tht the density of field lines is greter ner the chrges becuse the lines must converge closer together s they pproch prticulr chrge. It cn lso be shown tht the electric field intensity increses ner conducting surfces tht re curved to protrude outwrd, so tht they hve positive curvture. Curvture is defined s the inverse of the rdius. A flt surfce hs zero curvture. A needle point hs very smll rdius nd lrge positive curvture. The lrger the curvture of conducting surfce, the greter the field intensity is ner the surfce. Wht re three properties of electric lines of force? Why do electric lines of force never cross? How do the electric lines of force represent n incresing field intensity? 43
How cn you prove from properties of electric field lines tht sphericl distribution of chrge surrounding n unchrged region produces no electric field nywhere within the region? Power Conducting Electrodes Voltge Sensor Probe Figure 4.3: The pprtus used to mp equipotentil lines nd electric field vectors between two electrodes. Historicl Aside The lightning rod is pointed conductor. An electrified thundercloud bove it ttrcts chrge of the opposite sign to the ner end of the rod, but repels chrge of the sme sign to the fr end. The fr end is grounded, llowing its chrge to move cross the erth. The rod thereby becomes chrged by electric induction. The cloud cn similrly induce chrge of sign opposite to is own in the ground beneth it. The strongest field results ner the point of the lightning rod, nd is intense enough to trnsfer net chrge onto the irborne molecules, thus ionizing them. This produces glow dischrge, in which net electric chrge is crried up on the ionized molecules in the ir to neutrlize prt of the chrge t the bottom of the cloud before it cn produce lightning bolt. The electric field representtion is not the only wy to mp how chrge ffects the spce round it. An equivlent scheme involves the notion of electric potentil. The difference in 44
electric potentil between two points A nd B is defined s the work per unit chrge required to move smll positive test chrge from point A to point B ginst the electric force. For electrosttic forces, it cn be shown tht this work depends only on the loctions of the points A nd B nd not on the pth followed between them in doing the work. Therefore, choosing convenient point in the region nd rbitrrily ssigning its electric potentil to hve some convenient vlue specifies the electric potentil t every other point in the region s the work per unit test chrge done to move test chrge between the points. It is usul to choose either some convenient conductor or else the ground s the reference, nd to ssign it potentil of zero. Generl Informtion This bers some similrity to how the grvittionl potentil energy ws defined. We could hve considered the electric potentil energy of test chrge in nlogy with the grvittionl potentil energy by considering the work, not the work per unit chrge, done in moving test chrge between two points. But just s the grvittionl potentil energy itself cnnot be used to chrcterize the grvittionl field becuse it depends on the test mss used, the electric potentil energy similrly depends on the test chrge used. But the force nd therefore the work to move the test chrge from one loction to nother is proportionl to its chrge. Thus the work per unit chrge, or electric potentil difference, is independent of the test chrge used s long s the field does not vry in time, so tht the electric potentil chrcterizes the electric field itself throughout the region of spce without regrd to the mgnitude of the test chrge used to probe it. It is convenient to connect points of equl potentil with lines in two dimensionl problems; or surfces in the cse of three dimensions. These lines re clled equipotentil lines; these surfces re clled equipotentil surfces; volumes, surfces, or lines whose points ll hve the sme electric potentil re clled equipotentils. If smll test chrge is moved so tht its direction of motion is lwys perpendiculr to the electric field t ech loction, then the electric force nd the direction of motion t ech point re perpendiculr. No work is done ginst the electric force, nd the potentil t ech point trversed is therefore the sme. Hence pth trced out by moving in direction perpendiculr to the electric field t ech point is n equipotentil. Conversely, if the test chrge is moved long n equipotentil, there is no chnge in potentil nd therefore no work done on the chrge by the electric field. For non-zero electric field this cn hppen only if the chrge is being moved perpendiculr to the field t ech point on such pth. Therefore, electric field lines nd equipotentils lwys cross t right ngles. Figure 4.2 shows region of spce round group of chrges. The electric lines of force re indicted with solid lines nd rrows. The electric field cn lso be indicted by equipotentil lines, shown s dshed lines in the figure. The mpping of region of spce 45
with equipotentil lines or, in the cse of 3-D spce, with equipotentil surfces, provides the sme degree of informtion s by mpping out the electric field itself throughout the region. Are the electric field representtion nd the equipotentil line representtion equivlent in terms of how much informtion they contin bout the electric field? 4.2 Theory Recll tht the work done by the electric force, F, in moving the chrge from point to point b is given by W b = F dr, (4.1) where dr is smll piece of the pth trveled from to b. We cn write this in terms of the electric field; if our chrge is q, then F = qe nd W b = qe dr = q E dr. (4.2) Becuse of this we lso find it convenient to tlk bout the work per unit chrge V b = W b q = E dr. (4.3) Since the electric force is conservtive, we lso find it convenient to introduce potentil energy function, U(x, y, z), nd potentil energy per unit chrge function; we cll this the potentil function, V (x, y, z) = U/q, nd define its units to be the Volt (1 V = 1 J/C). We define U(x, y, z) in such wy tht totl energy is conserved. Since work is chnge in kinetic energy, it must correspond to n opposite chnge in potentil energy if the totl is to remin constnt, W b = U = q V = q ( V (x b, y b, z b ) V (x, y, z ) ) = q(v V b ). (4.4) Wht re the units of potentil difference? Wht re units of electric field? 46
4.2.1 Prt 1:Mpping equipotentils between oppositely chrged conductors The equipotentil pprtus is shown in Figure 4.3. The power supply is source of potentil difference (work per unit chrge) mesured in Volts (V). When it is connected to the two conductors, smll mount of chrge is deposited on ech conductor, producing n electric field nd mintining potentil difference, identicl to tht of the power supply, between the two conductors. The blck pper beneth the conductors is wekly conducting to llow smll current to flow. The voltge sensor mesures the potentil difference between the point on the pper where the probe is held nd the power supply s ground (blck) led. The voltge sensor is efficient t determining potentil difference using very smll (but nonzero) current. (We will understnd this better fter we discuss Ohm s lw.) This smll current perturbs the pper s current slightly, but much, much less thn the pper s current itself. Remove the electrodes left behind by the lst clss. Choose the conductor geometry for which you will be mpping the field. Strt with circulr conductor on the terminl post furthest wy from you nd horizontl br on the terminl nerest you. Wipe wy ny erser crumbs from the re of the electrodes. Mount these conductor pieces on the brss bolts which protrude from the blck-coted pper. Ech electrode hs rised lip round its edge on one side. This side must fce down so tht the rised lip mkes good electricl contct with the blck pper. Secure the conductors with the brss nuts. Tighten down the nuts well to ensure good electricl contct between the conductors nd the pper. The bnn jck wy from you is red nd the jck close to you is blck. The positive terminl of the power supply is connected to the red bnn jck nd the negtive terminl to the blck bnn jck. These jcks re connected to the bolt holding the round electrode nd to the center bolt holding the br, respectively, using wires under the pprtus. You will use the red (positive) led of the voltge sensor s n electric potentil probe to mp out V (x, y) in the plne of the pper. The Signl Genertor icon t the left toggles the visibility of the power supply s controls. You cn chnge the disk s potentil by entering different numbers into the signl genertor s control. Before the computer will mke ny mesurements, you must Record on the left end of the toolbr t the bottom of the screen. Choose few points t rndom on the blck pper nd plce the red probe lightly t these points in turn. Notice tht vrying the disk s potentil s described bove cuses the potentils of the rndom points in the blck pper to vry commensurtely. Note this observtion in your Dt. 4.2.2 Setting the potentil difference Adjust the power supply to mintin the desired voltge between the two conductors by following these steps. Touch the red potentil probe to the round electrode nd hold it there. Adjust the power supply voltge to 6.00 Volts, s red by the voltge sensor. Note tht ll points on the round electrode hve the sme voltge. The electrodes re equipotentil volumes nd their surfces re equipotentil surfces. When this djustment is completed, 47
remove the probe from the round electrode. Note the voltge of ech of the electrodes in your Dt. You re now redy to tke dt. 4.2.3 Mpping equipotentil lines Ech equipotentil line or surfce is specified by the sme single vlue of the voltge tht ll its points hve with respect to the br electrode. The gol is to locte points t ech desired potentil in order to trce out the corresponding equipotentil line. Suppose, for exmple, you wnt to find n equipotentil t 5.00 Volts. Lightly plce the red probe on the surfce of the blck pper nd gently move it round until the digitl voltmeter reds 5.00 Volts. This point is then t potentil of 5 Volts bove tht of the br conductor. We need to determine the (x, y) coordintes of this point so tht we cn plot it on the grph pper. The br is inscribed with mrks t every two centimeters. One side of the br is inscribed every two millimeters. These mrks cn be used s our x-coordintes. We lso hve ruler tht we cn use to determine the x, y-coordintes. Plot the point on the grph pper nd drw box, tringle, dimond, str, etc. round the point to distinguish it from dirt or stry toner. An ccepted strtegy is to use different shpes to represent different voltges. Now, gently drg the probe cross the blck pper nd note tht very close to this point is nother point on ech side of the first tht lso hve 5.00 V potentil. It would tke forever to find nd to plot ll of the 5.00 V points becuse these points re rbitrrily close together. The equipotentils re continuous. Move n inch or two wy from your first point, trce n rc round, find nd plot nother point hving 5.00 Volts. Continue until you re confident tht you cn sketch the 5.00 V equipotentil on your grph/mp. Helpful Tip It is not necessry to obtin exctly 5.000 Volts on the meter. We only need to get s close to the 5.000 Volts s we cn trnsfer to the grph pper; get within 1 mm since this is closer thn we cn grph nywy. Note tht the grph pper is hlf s big nd is scled 1:2 with respect to the pprtus. Generl Informtion In science experiments it is often importnt for us to notice symmetry in our pprtus or smple. Tke moment to exmine the pprtus. If we imgine plcing mirror perpendiculr to the pprtus nd pssing through the centers of the two electrodes, we cn see tht 48
the imge in the mirror would be exctly the sme s we see without the mirror. We cll this mirror symmetry (or bilterl symmetry) bout the y-xis becuse of this fct. Exctly the sme stuff is t ( x, y) s is t (x, y) for ll x nd y. Since our pprtus hs mirror symmetry bout the y-xis, we expect tht our observtions will lso hve this symmetry. We need to test enough points to convince ourselves tht our dt is symmetric, but once we re convinced we cn simply plot ech point t ( x, y) nd t (x, y) on the grph pper once its coordintes re determined. If you do not observe mirror symmetry, check for loose nuts, erser crumbs under your electrodes, or torn Teledeltos pper. Correct ny problems before continuing, if possible, or note ny complictions in your Dt. Ask your teching ssistnt to help you if you do not see the problem right wy. Historicl Aside The crbon pper we re using hs trde nme: Teledeltos. It ws developed nd ptented round 1934 by Western Union. It ws originlly used to trnsmit newspper imges over the telegrph lines (s n erly fx mchine). At the receiving end of the Wirephoto, the cylinder of drum ws covered first with sheet of Teledeltos pper, nd then with sheet of white record pper. A pointed electrode triggered by signls trnsmitted over the telegrph lines would then reconstruct the imge by vrying the density of blck dots on the record pper. Now, go fter the 4 Volt equipotentil using the sme technique. Then, do the sme for the 3 Volts, 2 Volts, nd 1 Volt equipotentil lines. For ech cse, drw smooth curve mong the points hving the given potentil. Do not just connect the dots to get segmented line. Remember tht our mesurements nd plots hve experimentl error in them nd tht our gol is to verge out these errors with smooth dt-fitting curve. The curve is intended to fll long the equipotentil between, s well s t, the specific points mrked off, so the points should not be connected by stright line segments. Your equipotentil lines should look like computer fits to mth models. Some dt points will be bove nd some below, but the drwn line will be smooth compred to the dt points. Lbel ech line with its potentil. It is good strtegy to trce the dt points in ink nd to sketch the lines in pencil until they re stisfctory. This llows you to erse erroneous lines without ersing the dt. If you erse pencil lines, plese do so wy from the pprtus so tht the rubber (insulting) crumbs do not degrde its efficiency. Once you re stisfied with the pencil sketches, trce the lines in ink so tht the sme strtegy will pply to the construction of electric field vectors below. 4.2.4 Prt 2: Finding electric field lines Recll Eqution (4.3) nd pply it to n equipotentil line being the pth long which the chrge moves. Since ll points in n equipotentil hve the sme potentil, V = V b for equipotentils, the work done s seen from Eqution (4.4) is zero, nd the work per unit 49
chrge is lso zero. Eqution (4.3) then becomes 0 = (V V b ) = W b q = E dr = E dr cos θ (4.5) for points nd b on the sme equipotentil line, surfce, or volume. If the electrodes hve different potentils, then E 0; n unblnce of chrge will mke n electric force nd field. dr 0 unless nd b re the sme point; if we moved the chrge, then this cnnot be. Only cos θ = 0 or θ = 90 remins s possibility, but this mens tht the electric field, E, must be perpendiculr to the pth tht we trveled, the equipotentil line. Since we now hve set of equipotentil lines, we cn use them to sketch the electric field vectors. In wht wy re equipotentil lines oriented with respect to the electric field lines? The result discussed erlier tht the electric field is everywhere t right ngles to the equipotentil surfces nd the fct tht electric fields strt on positive chrges nd end on negtive chrges cn now be used to drw the field lines in the region where you hve trced the equipotentils. On your drwing, plce your pencil t point representing the br conductor surfce nd drw line perpendiculr to the br going towrd the nerest equipotentil line. As your line pproches the equipotentil, be sure tht it curves to meet the line t right ngle. Proceed similrly to the next equipotentil, nd so on until your line ends on the drwing of the round conductor. Keep in mind tht ech conductor itself is n equipotentil, tht its surfce intersects the pper in n equipotentil line, nd tht the electric field vector must lso be perpendiculr to these lines. Lbel the electrode s imges with their electric potentils. Return to the br in the grph nd construct new line strting t n pproprite distnce (sy 2 cm) from the first line. Construct 6-8 electric field vector lines. Plce n rrow hed t the end of ech line to indicte the correct direction of the vector. Cn you observe n electric field bove nd below the pper using this voltmeter? Does the electric field occupy the spce bove nd below the pper? Why cn t this voltmeter observe electric fields in ir? How else might these fields be observed? 4.3 Prt 3: Finding electric field mgnitude The vectors drwn bove re everywhere prllel to the electric field in the pper. Since we know the directions the E vectors point, we cn imgine constructing coordinte xis,, prllel to segment of one of the field vectors. Figure 4.4 illustrtes this process. Let us 50
consider wht Eqution (4.3) tells us bout this line segment, V b = V V b = = E dr E E dr dr = E r b r. (4.6) 4 V 3 V 2 V 0 V 1 V 3 V 2 V The electric field is not ctully constnt in this intervl but it lso does not chnge much. Additionlly, the Men Vlue Theorem of Integrl Clculus tells us tht t lest one point in this intervl hs electric field equl to this verge (in this cse exctly one point) nd tht the vlue of the integrl is equl to the verge field times the distnce trveled or the intervl s length. Then long the xis we hve -4 V -3 V Figure 4.4: A sketch of coordinte xis constructed prllel to the electric field for the purpose of determining the mgnitude of the electric field. The distnce between the equipotentil lines nd the difference in potentil determines the size of the field. -2 V -1 V E = E = V b V b = V. (4.7) Generl Informtion Students of clculus might recognize tht this (pproximte) derivtive is the inverse of the integrl in Eqution 4.3. This reltionship is complicted by the three dimensions of the field vectors. Why must the E field be perpendiculr to the surfce of n idel conductor? It turns out tht this result cn be generlized. The components of the electric field cn be clculted or mesured using E xi = V x i, (4.8) where V (x, y, z) is the electric potentil function (voltge) s function of position nd the two points whose potentil difference we re clculting re prllel to the x i xis. Normlly, x i is either x, y, or z; we cn lwys find ll three components of E by finding three potentil 51
differences with one prllel to x, nother prllel to y, nd the lst prllel to z. In this cse, however, we hve constructed our xis prllel to E so tht E is the only component nd E = E = E. Additionlly, we hve not collected potentils t points prllel to x, y, or z, so we do not hve the correct informtion to clculte the field components. For the segment of E between the 2 V nd 3 V lines, the 2 V point hs lrger vlue of. In fct, this vlue of is lrger thn on the 3 V line by the distnce between the lines. Since we know the potentils, we cn find the difference. Since we cn mesure the distnce between the lines, we cn mesure : is the distnce between the lines. Actully, the blck pper contining the field lines is twice s big s our mp nd we re mesuring on our mp. To get for the blck pper, we must double our mesurement. Let us suppose tht we mesure = 2.4 cm in Figure 4.4. Then the electric field in the blck pper hs mgnitude E = V = 2 V - 3 V = 0.208 V/cm. (4.9) 2(2.4 cm) Actully, this is the verge electric field strength long this segment of E. To get the electric field t point, we must repet the experiment gin nd gin incresing the number of equipotentil lines ech time. We might imgine finding 0.1 V, 0.2 V,..., 5.9 V equipotentil lines nd then finding 0.01 V, 0.02 V,..., 5.99 V, etc. With ech repetition the lines get closer together nd the verge field strength for ech segment is closer to ll of the points in the segment. In clculus we cll this process tking the limit s V pproches zero. Since the lines get closer together s V decreses, we re lso tking the limit s pproches zero. The rtio, V, gets closer nd closer to some rel number tht is effectively the vlue of the field t the point. Symboliclly, the components of the electric field t ech point in spce re given by V (x + x, y, z) V (x, y, z) E x = lim x 0 x V (x, y + y, z) V (x, y, z) E y = lim y 0 y E z = lim z 0 V (x, y, z + z) V (x, y, z) z Use Eqution (4.7) s illustrted bove to compute the verge electric field for ll six segments of single field vector. Mrk the vector tht you use on your mp so tht your reders cn verify your work. How ccurtely do you know V nd? 4.4 Anlysis Discuss the properties of the equipotentil lines nd the electric field vectors. Do these observtions hve the sme symmetry s the pprtus tht cused them? Wht kind of symmetry is this? Does the pper s potentil chnge when the power supply voltge is chnged? These re indictions tht the pprtus cused the observtions. 52
Were the equipotentil lines continuous s predicted? Are the field lines close together t plces where the field mgnitude is lrge? Are the equipotentil lines more curved t plces where the field mgnitude is lrge? Wht subtle sources of error re present in this experiment? Are these errors lrge enough to explin ny discrepncies between your observtions nd the properties of electric fields? 4.5 Conclusions In science cuse nd its effect lwys hve exctly the sme symmetry. Cn you conclude tht your pprtus cuses your observed equipotentil lines nd electric fields? Is Eqution (4.7) consistent with our dt? If so, include this s prt of your Conclusions nd define ll symbols. Are you confident tht this pprtus nd method revels the electric field round these electrodes? This pprtus hs historicl significnce s design id. Experimenters nd engineers once constructed electrodes hving prticulr shpe in hopes of obtining n electric field suited to specific purpose. For exmple, we might need to design vcuum tube to ct s n mplifier or we might need to focus n electron bem for use in TV s CRT. Tody, we cn simply downlod n electrodynmics simultion progrm to run on our smrtphone; but once upon time the only wy we could view the electric field round our electrodes ws to mesure it using similr pprtus. Wht other purposes cn you imgine using our pprtus to fill? Motivte interest in our work by pointing out how vluble this tool cn be for designing electric fields. 53