I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is


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1 Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes, carbnpaper with traces Objectives In this lab yu will d the fllwing: Analytically relate charge distributin, electric field, and electric ptential fr a rd Cmputatinally map electric field and electric ptential (equiptential lines) fr pairs f charges Experimentally map electric field and electric ptential (equiptential lines) fr varius surce cnfiguratins. Thery In chapter 17, yu learn abut the electric ptential (a.k.a. Vltage), that clsely relates t the wrk r electric ptential energy fr mving a charge in the presence f ther charges. The definitin f electric ptential difference is V E d. Tw imprtant prperties f the electric ptential are: The electric ptential decreases mst rapidly in the directin that the electric field pints. Thrughut a cnductr, if n charge is mving, then the electric ptential is the same. Equiptential Lines. The electric field thrughut a regin f space is ften visualized with arrws representing the magnitude and directin f the field at varius lcatins. The electric ptential thrughut a regin f space is ften visualized with equiptential lines, curves alng which the electric ptential is cnstant. These are analgus t the equielevatin (cntur) lines featured in a tpgraphical map. As with tp maps, where these lines are clsely packed, the ptential changes steeply; where they re spaced ut, the ptential changes mre gradually. I. Analytical Ptential and Field f a Unifrm Rd Unifrm charge Q c On a whitebard, wrk the fllwing prblem (17.P.66). C b B A a The lng rd shwn has a length L, which is much greater than any f the ther lengths, and carries a unifrm charge Q. a. Alng the path A t B t C, draw the electric field at several lcatins. This will prvide yu visual clues as t whether it s varying, whether we re headed with r against the field, etc. b. Draw representative l s alng the path. f c. Add up (r integrate) V E dl frm A t C. (Dn t frget that Ch 16 gives an apprximatin fr the field f a very lng rd.) i alng this path
2 Phys 232 Lab 4 Ch 17 Electric Ptential Difference 2 II. Cmputatinal Ptential and Field f a Charge Pair Yu ll cmplete and run a prgram t determine a pair f pint charges electric ptential and electric field thrughut a regin f space and represent thse with a clr gradient and cntur (r equiptential) lines and vectrs (fr the electric field) a. Equiptential Lines An equiptential is the curve (r 3D surface) that cnnects all bservatin lcatins that have the same electric ptential. The situatin yu ll mdel is illustrated in the figure belw: tw pint charges, q 1 and q 2, bth cntribute t the electric ptential at an bservatin lcatin. (As the text discusses, the ptential at a pint is really the ptential difference between that pint and sme reference lcatin; in this case, the reference lcatin is ut at infinity.) Given the familiar expressin fr the electric field due t a pint charge and given the general relatinship between field and ptential (see the thery sectin at the beginning f this lab), the electric ptential at an bservatin lcatin due t a pint charge is ( ) q V r 4 where r 1 r r q 1. 1 q 2 r 1 Written in terms f vectr cmpnents, the magnitude f this separatin vectr is 2 2 r x x y y z z1 Similar expressins can be written fr the electric ptential due t charge 2. Frm WebAssign, dwnlad the pythn cde Ptential and Field.py (pen a blank VPythn file frm the desktp, pen the Ptential and Field file thrugh WebAssign, cpy the cde, and paste it int the VPythn windw.) Nte: This prgram cntains a mix f functins that yu shuld be getting familiar with (like linspace) and functins that yu needn t wrry abut mastering (like meshgrid and the pltting functins); the latter are useful fr efficiently visualizing vltage and field, but we wn t use again in this curse. In the Calculatins sectin, cmplete the xbs = line s xbs is a list f 100 x crdinates fr bservatin lcatins, running frm 1.5 t 1.5 meters. D the same with the ybs= line (these being the bservatin lcatin s ycrdinates.) Cmplete the r1 =, V1 =, r2=, V2=, and V = lines s they give the magnitudes f the separatin vectrs between the charges and the bservatin lcatin (Xbs, Ybs), each charge s cntributin t the electric ptential there, and the ttal electric ptential there. Recall that in Pythn, x n is written x**n. Square rt in Pythn is either sqrt( ) r ( )**0.5. Nte: The pwer f the meshgrid functin that defined Xbs and Ybs is that yur single line that defines V in terms f these will actually return nt just ne value f electric ptential, but the list f equiptential values fr all bservatin lcatins! 2.
3 Phys 232 Lab 4 Ch 17 Electric Ptential Difference 3 T test that yu ve gtten everything right s far, temprarily add a print statement just belw the V= line, run the prgram, and check in WebAssign the first three V values in the list that s printed. Nte: if yu re wrking n yur wn cmputer and Pythn cmplains that pylab is nt installed, yu ll need t mve yur cde t ne f the lab cmputers t cntinue wrking n it. The prgram is incmplete, s it will crash, but nt befre printing. In the Display sectin at the end f the prgram, uncmment and cmplete the LineVs= line s LineVs is a list f 401 vltages running frm t 10000V. This will set the equiptential levels t be displayed. Run the prgram. The levelspltv= line shuld be respnsible fr creating a cntur plt displaying 401 equiptential curves running frm 10,000V t 10,000V, thugh the curves near the pint charges will be s clse t each ther they blur tgether. b. Clr Gradient Of curse, the 401 curves are just discrete samplings f the smthlyvarying electric ptential; t better visualize the smth variatin, Just belw the levelspltv= line, add a line that defines ClrVs t be a list f 1001 vltages, frm 500V t 500V fr which clr shades will represent vltage (yu re nly clrizing frm 500V t 500V s all the clr gradatin desn t get devted t the space near the charges where the vltages vary steeply.) Remve the # frm befre clrpltv= t activate that line which is respnsible fr creating the clrgradient plt. Run the prgram and enjy. c. Electric Field Accrding t the equatin given at the beginning f this lab, electric field and ptential are intimately related electric ptential difference is the path integral f electric field r, cnversely, electric field is the negative gradient (3D slpe) f the electric ptential. T see the relatinship between electric ptential and field, The prgram already has lines f cde that determine the x and ycmpnents f the electric field at several lcatins. Yu ll ntice sme redundancy: Ex, Ey, and Ex2 and Ey2 are calculated using slightly different lists f bservatin lcatins (XbsE, YbsE and XbsE2, YbsE2); this is t avid calculating E t near the tw charges (where the field is huge) Remve the # frm the beginning f the quiverplte= line and the beginning f the quiverplte2= line. Tgether, they will display the electric field n the same figure as the electric ptential. Run the Prgram. Based n the plt that s generated, in WebAssign cmplete these statements abut the relatinship between field and ptential: The electric field pints. The electric field s magnitude is greatest where. Where the electric ptential is zer, the electric field is. Make sure partner names are written in a cmment at the beginning f the prgram, save it, and uplad it.
4 Phys 232 Lab 4 Ch 17 Electric Ptential Difference 4 d. Varying Surce Charges Electric Ptential and Field T really appreciate hw the Electric Ptential and Field are related t each ther, vary the rati f the tw surce charges. Change q2 t equal q1 and run the prgram t see hw the field and ptential change (yu may want t change the limits in ClrVs= t better visualize the electric ptential.) Change q2 t be 2*q1 and run the prgram t see hw the field and ptential change (yu may want t change the limits in ClrVs= t better visualize the electric ptential.) Feel free t make any ther changes yu d like t explre. III. Experimental Ptential and Field Tw pint charges Experimental Set up Get a sheet f resistance paper (carbn impregnated) with electrdes in the shape f tw small dts drawn in silver cnductive ink. Tack the crners f the paper dwn n a crkbard. Fr the first part f the experiment, attach the pwer supply and the multimeter as shwn in the diagram belw. Tw f the leads shuld be cnnected t the electrdes with pins. Set the dial n the multimeter t DC V s that it will measure differences in electric ptential (als called vltage). Turn n the multimeter and the pwer supply. DC Pwer Supply COM 5 V +5 V Digital Multimeter DC 1000V COM Prbe Experimental Prcedure A. Find Equiptential Lines 1. Carefully sketch the shape f the electrdes n a piece f white grid paper. 2. The shape f an equiptential line is fund by mving the prbe n the paper t find enugh pints where the vltage is the same that yu can be cnfident f the shape f the smth curve that cnnects thse pints. Fr a give equiptential curve, yu shuld
5 Phys 232 Lab 4 Ch 17 Electric Ptential Difference 5 find at least fur pints, mark them n yur white grid paper, and then smthly cnnect the dts. Nte: Since yur hand is abut as cnductive as the paper is, yur measurements wuld be affected if yu rested yur hand n the paper while take measurements, s be careful nt t. 3. Start by finding the curve where the vltage is +3 V. Carefully draw and label the equiptential line n the white grid paper, nt n the black cnducting paper. 4. Repeat fr the vltages +4 V, +2 V, +1 V, 0 V, 1 V, 2 V, 3 V, 4 V. This shuld lk similar t the cntur plt in yur prgram when q2 = q1. B. Find Electric Field Vectrs Suppse the vltage is measured at tw pints A and B that are a shrt distance d apart. Midway between the pints, the size f the cmpnent f the electric field alng the line cnnecting the pints is apprximately E V B V A d. If yu chse pints A and B s that the line cnnecting them is alng the electric field, the ptential difference will be maximum and yu will be measuring the size f the whle electric field. 1. Discnnect the multimeter frm the pwer supply. Als, remve the prbe frm the multimeter. Cnnect the multimeter t the pair f prbes that are held at a fixed distance apart. Place the red wire in the V scket and the black wire in the COM scket. In this cnfiguratin, the meter will display the electric ptential difference between the prbes. A psitive reading will mean that the red prbe is at a pint with a higher electric ptential, which means the cmpnent f the electric field pints tward the black prbe (Since electric field is the negative gradient f electric ptential). 2. Measure the distance between the tips f the prbes. 3. Fr the fllwing fur pints, determine the directin and magnitude f the electric field. Yu ll recrd the magnitude in WebAssign and sketch electric field vectrs n the white grid paper t indicate the apprpriate directin and their relative magnitudes. a. Straddling the +1V curve alng the line directly between the tw pint surces (and dn t frget t sketch an arrw n yur white grid paper) b. Straddling the +2V curve where it passes belw ne f the pint surces (be careful t find the apprpriate directin f the field here and dn t frget t sketch an arrw n yur white grid paper) c. Straddling the +2V curve where it passes abve the same pint surce (be careful t find the apprpriate directin f the field here and dn t frget t sketch an arrw n yur white grid paper) d. Straddling the 3V curve alng the line directly between the tw pint surces. b d a c
6 Phys 232 Lab 4 Ch 17 Electric Ptential Difference 6 When yu draw arrws representing the electric field at these pints, scale them s that the largest is nt t big t draw n the diagram with the equiptential lines. Remember that the tail f a vectr shuld be where the electric field was determined. The directins and relative magnitudes f the electric field at these pints shuld qualitatively agree with what yu d seen in yur prgram fr q2=q1. Pint and Line Electrdes Replace yur black cnductive paper with a cnductive page with ne pint and ne line electrde. In rder t test that gd cntacts are made with the electrdes, tuch varius pints n each f the electrdes with the tip f the prbe. The vltage readings fr all pints n a single electrde shuld be very nearly the same (within abut 0.1 vlts). Fr this, repeat steps A and B. It s alright if yu can t find the equiptential lines t clse t the small electrde. Write yur grup members names n the tw white grid sheets and turn them in.
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