Electric & Potential Fields

Transcription

1 Electic & Potential Fields Pupose An electic field suounds any assemblage of chaged objects. To detemine the stength and diection of these fields, it is most convenient to fist map the electic potential of the field, which can be measued with a voltmete. Fom the potential field, the electic field can easily be detemined. In this laboatoy, we will map the electic and potential fields esulting fom thee diffeent configuations of chaged electodes ectangula, concentic, and cicula. Pinciples The electic field at a point in space shows the foce that a unit chage would feel if it wee placed at that point: E E is a vecto uantity: it has both magnitude and diection and has units of newtons pe coulomb. The field is set up by electic chages somewhee in the suounding space. Fo continuous chage distibutions, it is much easie to analyze electic foces using the field concept than using Coulomb s Law fo the foces between point chages. If we measue the field (magnitude and diection) at enough points aound the chage distibution, we could make a map of the electic field lines. These lines show the diection of the electic foce at each point. Howeve, instead of measuing the electic field diectly, we will measue and map the electic potential aound the chage distibution. The electic fields can be detemined fom this. F Electic Potential To place a positive test chage in an electic field we must do wok against the field, since the field ties to push the chage away. Since the electic foce is consevative, the wok we do in placing the chage in the field is stoed as potential enegy. Electic potential is defined as the wok done pe unit chage to move the chage into an electic field. Like potential enegy, electic potential is measued elative to some efeence position, so that we define it as 15

2 Electic & Potential Fields (1) W V whee is the change in electic potential in going fom the efeence point to the point in uestion, W is the wok done (o change in potential enegy) and is the chage. The unit of electic potential is the volt, which is euivalent to a joule pe coulomb. We usually use the Eath itself as ou efeence point, and take the electic potential of the Eath to be zeo. Euipotential Lines Thus a point in an electic field o in an electic cicuit can be chaacteized by the electic potential, o simply the potential, thee. A uantity of chage placed at that point has potential enegy eual to the voltage thee times the amount of chage. We can measue electic potential diectly using a voltmete. We can then map the potential field by connecting points that ae at the same electic potential. Lines between such points ae called euipotential lines. Moving a chage along an euipotential line euies no wok, since the enegy of the chage does not change. To find the diection of the electic field, we make use of the fact that the euipotential lines must always be pependicula to the electic field lines. This is because the electic field lines show the diection of maximum decease in the potential. + E-field lines Euipotential lines Detemining the magnitude of the Electic Field 16

3 Electic & Potential Fields To detemine the electic field fom a seies of voltage measuements, we note that the potential diffeence between two points is the integal of the electic field taken along the path between the two points: W F ds E ds E ds whee ds is an infinitesimal element of the path. The minus sign aises because the extenal agency must exet a foce that is eual and opposite to the foce exeted by the field, E. Convesely, the above expession tells us that E is the negative (vecto) deivative of the potential it gives the ate and diection of maximum decease in V. In situations of symmety, the vecto notation simplifies. Fo instance, suppose E is constant in magnitude and diection, as between 2 oppositely chaged electodes: + E Then the change in potential in moving a distance x away fom the positive plate is W E ds E x Convesely, the magnitude of the electic field can be detemined fom: (2) E dv dx x The minus sign signifies that E points in the diection that V deceases. Note also that the electic field can be expessed as volts pe mete. Now conside a adial electic field in two dimensions, as fom a disk of chage of adius a: 17

4 Electic & Potential Fields Hee, the E-field is the same in all adial diections in the plane. Its magnitude must decease invesely with the adius fom the cente of the disk, since the field lines become spase in popotion to. The magnitude of the field can then be expessed as (3) V E m whee V m is a constant. The diection of the field is adially outwad. The change in potential in going fom adius a to some abitay adius is then Vm (4) V E ds d V m ln( ) a Convesely, if we know the potential has this fom, we can get E by taking the deivative of (4): (5) E dv d V m In moe geneal cases, whee the field is not symmetic, o when the symmety involves moe than one coodinate, the deivative must be taken with espect to each spatial coodinate. Howeve, ove small distances, one can teat the field as appoximately linea and can appoximate: (6) E( x) x whee x is the elevant coodinate. 18

5 Electic & Potential Fields Electodes Sheets Rectangula 12 cm 1 cm 9 cm Cicula 1 cm adius 10 cm cente to cente Concentic 8 cm adius 1 cm adius a adius of inne electode 1 cm b inne adius of oute electode 8 cm 19

6 Electic & Potential Fields LAB 1 Pocedues We will measue the potential and electic fields aound the thee sets of electodes illustated. The electodes ae dawn on sheets of conductive pape with conductive ink. The pocedue will be as follows: Euipment Measue the potential at epesentative points on the electode sheets. (We often speak loosely of measuing the voltage athe than the potential.). Plot the values on gaph pape. Daw the euipotential lines. That is, daw the best smooth cuve though points at the same potential. The esult will be something like a topogaphic map fo the electic field. Daw the electic field lines by stating at the positive (high potential) electode and tacing a path to the negative electode in such a way that the electic field lines always coss the potential field lines at ight angles. Low Voltage Powe Supply (LVPS) Digital Multimete (DMM) Conductive pape electode sets: ectangula, cicula and concentic Metal pushpins (2) Cokboad Banana wies Alligato clips Plastic pushpins Gaph pape 1. Set-up Make electode sheets. If electode sheets ae not aleady made up, you can make them by dawing the electodes on conductive pape using a conductive ink pen. Simply outline the electodes in the pope dimensions and fill in the outlines with the conductive ink. Take cae to make the facing edges of the two electodes as smooth as possible use ules o cicle templates. The exteio edges need not be smooth. Set up a diagam on gaph pape. Daw a hoizontal and vetical axis though the cente of a sheet of centimete gaph pape. Daw the outlines of the electodes on the gaph pape at a 1:1 scale. Place the oigin of the diagam in the cente of the electode aangement. Connect the electodes to the powe souce. 20

7 Electic & Potential Fields LAB 1 Pocedues 1. Tack o pin the electode sheet to a cokboad o othe backing with plastic pushpins. Use the fou cones of the conductive pape. 2. Inset a metallic pushpin o othe conducto into the cente of the positive and negative electodes. Fo the oute cicle in the concentic set, the pin can go anywhee within the oute cicle. 3. With the powe supply off, connect one electode to the gound teminal of the DC output on the powe supply with an alligato clip and banana wie o othe connecto. Connect the othe electode to the positive teminal on the powe supply. 4. Set the cuent knob on the powe supply to one half tun and set its voltage knob to the lowest setting (counteclockwise). Have you instucto inspect you set-up befoe you tun on the powe supply. Review the section on electical measuements in the intoduction befoe poceeding. Make sue you know the pope settings fo the DMM and how to use it. 5. Tun on the powe supply and set the voltage output to 15 VDC. Use the DMM connected acoss the powe supply s teminal to set the souce voltage, not the mete on the powe supply. 6. Take voltage eadings as diected below. Connect the COM teminal on the DMM to gound (the negative teminal) on the powe supply. Touch the DMM voltage pobe to a point on the conductive sheet to measue the voltage thee. It is not necessay to punctue the sheet. Take the eading to the neaest 0.1 volt and mak the coesponding point on you gaph pape diagam. Label the point with the voltage eading. 2. Rectangula electodes 1. Stating at the negative electode (this should be at 0.0 volts) find the euipotential lines at evey 3 volts. That is, find the 3.0 V, 6.0 V, 9.0 V, and 12.0 V euipotential lines. The edge of the positive electode should be at 15.0 V. To find a line, slowly dag the pobe away fom the negative electode until the DMM displays the voltage you seek. Keep the pobe vetical, and allow the DMM enough time to settle down befoe you take the eading. Mak and label the point on you gaph pape diagam. Repeat this pocess fo about 8 evenly spaced points at the same potential, including the egion beyond the ends of the electodes. You should be able to tace the euipotential fom one edge of the pape to the othe. 21

8 Electic & Potential Fields LAB 1 Pocedues Analysis 1. Daw the best smooth cuves though the points at the same potential. These ae the euipotential lines. 2. Daw the electic field lines. Stating at the positive electode on you diagam, daw a line to the negative electode in such a way that the line always cosses an euipotential line at ight angles. Daw 6-9 field lines though the egion between the electodes and the egions above and below thei ends. Use aows on the lines to show the diection of the electic field. 3. Detemine the magnitude of the electic field stength in the cental egion between the electodes by calculating E x fo each adjacent pai of euipotential lines. Let x be the distance fom the positive electode. Find the aveage of the values this is the aveage value of E in this cental egion. 4. Calculate the deviation fo each value and find the aveage deviation. 5. Calculate the standad deviation and epot the value of the field in the fom: 3. Concentic electodes E E aveage ± σ Connect the inne ing electode to the positive teminal on the powe supply and the oute ing to gound. 1. Find the 3.0 V, 6.0 V, 9.0 V and 12.0 V euipotential lines and plot them on you gaph pape diagam. Find 8 ts fo each line in the egion between the electodes. 2. The field should be zeo in the egion outside the lage cicle. Test this by finding the potential at 4 symmetic points outside the oute electode. Analysis 1. Daw the best smooth cuve though the points at the same potential in the inteio egion. 2. Daw 8 electic field lines in the inteio egion. 3. Detemine the aveage adius fo each euipotential line and tabulate the values of the potential with the aveage adii. Measue the adius fom the cente of the electode 22

9 Electic & Potential Fields LAB 1 Pocedues sheet, so that the oute adius of the positive electode is at 1 cm and the inne adius of the gound electode is at 8 cm. 4. Plot /a vesus V on semi-log gaph pape. (Refe to euation 4 above. In ou case, a 1 cm.) See the section on semi-log plots in the appendix. 5. Detemine the slope and the y-intecept and wite down the euation of the line. 6. Solve the euation fo V(). (That is, eaange the euation so that V is on the left and all othe tems on the ight.) Evaluate the numeical tems. You now have an empiical euation fo V as a function of. 7. Euation (4) above can be witten (4) V ( ) V a Vm ln( ) a Compaing you empiical euation with this, detemine the value of V m. Also, find V m diectly fom this expession using V b 0 and compae with you empiical esult. 8. Using you value of V m, calculate the electic field stength at the adii of each of the euipotential lines. 3. Cicula electodes Find the 3.0 V, 6.0 V, 9.0 V and 12.0 V euipotential lines and plot them on the diagam. You plot should fill the diagam take eadings both between and on the fa side of the electodes. Analysis 1. Daw the best smooth cuve though the points at the same potential. 2. Daw 8 electic field lines, stating at 8 symmetic points on the positive electode and ending at the negative electode. 3. Along the cental axis between the electodes, the field should be a staight line, although it is not constant in magnitude. We can find an appoximate value fo the field at points along this cental axis by using: magnitude E fo x closely spaced points. Use this to find the appoximate field stength between each adjacent pai of euipotential lines. (Don t foget that the sufaces of the electodes ae also euipotentials.) 1. Whee is the field stongest and whee weakest along this cental line? 2. Is the field stonge nea the positive o nea the negative electode? 23