Numerical Solution to Laplace s Equation

Transcription

1 Numercal Soluton to Laplace s Equaton Carleton Unversty Department of Electroncs ELEC 3105 Laboratory Exercse 1 July 8 015

2 PRE-LABORATORY EXERCISE You need to complete the pre-lab and have the TA sgn off your pre-lab wor before startng the computer laboratory exercse. Pre-1: Numercal soluton to Poson s and Laplace s equaton Please refer to the course lecture sldes related to Poson s and Laplace s equatons for addtonal detals on the technque. A summary s provded here. The startng equaton s: (P-1) whch s nown as Poson s equaton. It s a pont functon whch mples that the second dervatve (Gradent squared here) of the potental functon at a partcular pont n space must equal the negatve rato of the charge densty at the pont dvded by the delectrc constant at that same pont. Should the charge densty be zero then the equaton smplfes to Laplace s form: 0 (P-) A consderable amount of effort goes nto solvng ths equaton. For nstance once you solve for the potental you can determne the magntude and drecton of the electrc feld through: E (P-3) Once you now the electrc potental and electrc feld you can pretty well calculate anythng else related to electrostatc. The pre-lab wll examne solvng Laplace s equaton usng two dfferent technques. The frst s a drect approach solvng the second order dfferental equaton. The second nvolves a numercal soluton usng a fnte dfference approach. Both technques are dscussed n detal n class. Pre-1: Solvng the dfferental equaton Laplace s equaton s a second order dfferental equaton. In Cartesan coordnates t s: x y z 0 (P-4) The same functon s subected to dervatves wth respect to x y z and when the second dervatves are formed and then summed the resultant must be zero. Only then can the orgnal functon be a vald soluton to the equaton. Under normal crcumstances fndng the functon that satsfes (P-4) can be dffcult and when ths occurs other approaches are used to solve the equaton (such as numercal ndcated below). For ths pre-lab we wll consder a smple soluton to (P-4). Consder the parallel plate capactor shown n fgure Pre-1. The lower plate s at 0 volts and resdes n the (x y) plane. The upper plate s at 100 volts also resdes n the (x y) plane and ntersects the z axs a

3 dstance d from the orgn. We wll treat d (capactor plate separaton) as small such that we may approxmate the capactor plates as nfnte n extent n the (x y) planes. As a result the potental functon s ndependent of the x and y coordnates. Ths statement has to do wth the translatonal symmetry that s present wth regards to the x and y coordnates. As you move about n the (x y) plane KEEPING z CONSTANT the envronment always loos the same. Thus n equaton (P-4) the dervatves wth respect to x and y are zero as (for ths geometry) the potental s ndependent of x and y. The potental does vary n movng along the z drecton. The potental s 0 volts at z = 0 and s 100 volts at z = d. Queston Pre-1.1: Solve the dfferental equaton (P-4) for the parallel plate capactor of fgure Pre-1. It s a second order dfferental equaton so the general soluton wll have two constants. Determne these constants by mang use of the now voltage values at z = 0 and z = d. Tae d = 1 mm. Plot several equpotental lnes and from these draw n the electrc feld lnes. What s the numercal value (magntude and drecton) of the electrc feld? 1 mar Fgure Pre-1: Parallel plate capactor geometry Queston Pre-1.: Two concentrc metal shells are shown n fgure Pre-: The nner shell has a radus of 1 cm and s at 100 volts the outer shell has a radus of cm and s at 00 volts. The regon between the metal surfaces s charge free and ar. Express Laplace s equaton n sphercal coordnates. Indcate whch dervatves of the potental functon wll be zero and why they are zero. Solve the remanng dfferental equaton and plot several equpotental lnes for the regon between the metal shells. Draw the electrc feld lnes. 1 mar Queston Pre-1.3: What approach would you use to solve the second order dfferental equaton f the geometry of the capactor plates do not conform to the unt vector drectons of a coordnate system? 1 mar

4 Fgure Pre-: Concentrc metal shells geometry Pre-: Fnte dfference soluton to Laplace s equaton n 1-D At ths tme t s a good dea to revew the course lecture sldes related to the numercal soluton to Poson s and Laplace s equaton. A revew of the numercal technque s presented here for a geometry whch results n a 1-D varaton n the potental functon. The parallel plate capactor geometry shown n fgure Pre-1 s such a geometry. The potental vares only the z drecton and s constant n the (x y) plane. Now consder the parallel plate capactor geometry redrawn n fgure Pre-3. The z axs between the capactor plates has been segmented and each pont the z axs s assgned an ndex (). The spacng between grd ponts s unform and equal to h. The capactor plate separaton s d. Fgure Pre-3: Parallel plate capactor geometry for numercal technque Consder now any two adacent grd ponts say ponts 4 and 5. The dfference n voltage between these two ponts s The separaton along the z axs between these ponts s z h. By defnton the frst dervatve of the potental wth respect to the z axs s: z lm h 0 z h z h (P-5)

5 If at the moment we gnore the lm as h0 we see that ( z h) ( z) s the dfference n voltage between adacent grd ponts separated by z h. Thus an approxmaton to the frst dervatve can be obtaned by. So now we have a way to calculate the frst dervatve by examnng voltage z z values of adacent pont. But actually Laplace s equaton s made up of second dervatves. A second dervatve s nothng more than the dervatve of the dervatve. So let s frst obtan the dervatve between each grd pont par as shown n fgure Pre-4. Note that the dervatve ponts are offset from the potental ponts by h/. We can now obtan the dervatve of the dervatve usng the green grd ( z h) ( z) z ponts. z z. The dervatve of the dervatve s also offset by h/ n grd z z z pont locaton. Ths brngs the second dervatve grd pont locaton bac on top of the orgnal grd pont locaton. We are almost there but we wll start all over agan. Let s get the dervatve between ponts 4 and 5 and also between ponts 5 and 6: and 6 h (P-6) z h z Let s get the dervatve of the dervatve between ponts 4 5 and 6: z 65 z z z h 5 4 h h h (P-7) For the parallel plate capactor problem there are no varatons n the potental wth respect to x and y and the regon between the plates s charge free. Thus 0 whch when usng (P-7) gves: z h after rearrangng 5 (P-8) Ths expresson ndcates that the voltage at grd pont 5 s the average value of the voltage one grd pont up and grd pont down. Ths expresson can be turned nto a numercal technque through the followng algorthm: Dvde the space nto an equal number of grd ponts. Mae certan that grd ponts are assgned to surfaces that are at fxed voltages (le the plates of the capactors see fgure Pre-3) Assgn an arbtrary voltage to each grd pont that s not fxed. Try to select voltage values n the range of the fxed values. Update the voltage on each grd pont by formng the average of ts nearest neghbours. Usng the updated values for the voltages update them agan by formng the average of nearest neghbours.

6 Repeat the updatng process untl the voltage values at each grd pont no longer change. Usually you wll specfy the number of decmal ponts for the accuracy and once the requred number of decmal ponts are resolved the updatng process s stopped. The fnal voltage values are the voltage values at the grd ponts. Pre-4: Potental frst dervatve and second dervatve Queston Pre-.1: For the parallel plate capactor gven n fgure Pre-3 use the numercal technque to obtan the voltages at the grd ponts accurate to 1 decmal place. Mae a good startng guess to the voltages. Tae d = 1 mm. 1 mar Queston Pre-.: Develop an XL spread sheet to solve the parallel plate capactor numercally to 3 decmal places. (If you wsh you may wrte a MATLAB program nstead). 1 mar Queston Pre-.3: Instead of usng 1 grd ponts use 10 grd ponts. Modfy your program to solve numercally Laplace s equaton for the parallel plate capactor to 5 decmal places. 1 mar Queston Pre--4: Any numercal technque utlzed requres an estmate of ts accuracy. Examne the course lecture sldes text boos on numercal technques and obtan an estmate for the error nvolved n usng ths approach to solvng Laplace s equaton. 1 mar Pre-3: Fnte dfference soluton to Laplace s equaton n -D and 3-D

7 The numercal approach presented above can be easly extended nto -D and 3-D. We need to develop the fnte dfference approxmatons to each of the second order dervatves n equaton (P-4). We have already wored out the dervatve part for the z drecton. We mposed a grd along the z axs and formed the frst and second dervatve. Now n 3-D we need to establsh grd ponts along the other two axes. We thus end up wth a volume of grd ponts wth each grd pont dentfed by the ndces ( ). We then form the second dervatves for each addtonal drecton. Fgure Pre-5 shows one of the grd ponts extracted (pont ) and ts sx nearest neghbours. Pre-6: 3-D grd ponts about center ( ) pont The resultant combnaton of the three second order dervatves of equaton (P-4) results n the followng expresson: h h h z y x (P-9) When dealng wth Laplace s equaton the above equaton s equal to zero and thus can be smplfed and rearranged to yeld an expresson for the voltage at pont ( ) as the average of ts nearest neghbours (3-D Grd): (P-10) In the stuaton where the geometry can be analysed n -D say x and y the averagng would nvolve only 4 nearest neghbours wth the grd usng ndces and (P-11) The same numercal algorthm presented above can be appled to the -D and 3-D grd. The dffculty n usng ths approach n -D and 3-D comes from the booeepng requred to eep all the grd pont averagng correctly lned.

8 Queston Pre-3.1: For the structure shown n fgure Pre-7 use a -D numercal grd approach to obtan a mappng of the potental nsde the electrode regon. To eep the problem manageable use a grd wth a 10 mm spacng. Obtan the voltages on the grd ponts accurate to 1 decmal place and use ether XL or MATLAB to solve. 1 mar Pre-7: Potental well electrode structure Queston Pre-3.: From the potental values determned above draw n the electrc feld vectors. 1 mar Lab 1: Numercal Soluton of Laplace s Equaton ELEC 3105 July Before You Start Ths lab and all relevant fles can be found at the course webste. You wll need to obtan an account on the networ f you do not already have one from another course. Wrte your name n the sgn n sheet when you arrve for the lab. You can wor alone or wth a partner. One lab wrte-up per person. Show unts n all calculatons all graphs requre a legend.. Obectves

9 The obectve of ths lab s to llustrate the use of a powerful numercal technque nown as the fnte element method to solve Laplace s equaton for selected problems. The lab wll run n the Department of Electroncs undergraduate laboratory room ME475. The software pacage we wll use s Maxwell S from Ansoft Corporaton. Ths software wll enable you to vsualze the electrc feld lnes and equpotental lnes n cross sectons of structures consstng of conductors and nsulators. 3. Bacground The fnte element method (FEM) s a numercal technque for fndng approxmate solutons to partal dfferental equatons [1]. Consder the example of a -D soluton and ts correspondng mesh shown n Fgure 1. The lnes represent the drecton and magntude of flux densty smulated usng FEM n the soluton mage and the trangles (or sub regons) represent a sngle calculated soluton n the mesh mage. As an analogy compare a peg fle wth large pxels mang the mage blurry and a peg fle wth smaller pxels allowng the mage to become sharper. Therefore the smaller the sub regon the more accurate the entre soluton. A numercal soluton s always an approxmaton of an analytcal soluton whch s based on mathematcal theory. Fgure 1: The -D soluton (left) and mesh (rght) [1] Consder Laplace s equaton descrbng the potental n a -D regon: x y 0 (1) A soluton can be found usng FEM by approxmatng the sze of d. Smaller trangles are used where the potental (x y) s rapdly varyng and larger trangles are used where the potental s varyng slowly. The potental s approxmated wthn each trangle as a polynomal expanson n x and y. A numercal algorthm s used to solve for the coeffcents of the polynomal n each trangle such that the nodes of

10 adacent trangles have the same potental. Conductng surfaces are constant potental surfaces - the user ntally sets the value of the potental at the conductor. Electrc energy s stored n the electrc feld. The energy stored s gven by the expresson (unts Joules). W E 1 D Ed where D E s the electrc flux densty (C/m ) E s the electrc feld ntensty (/m) and the dot product s used n the ntegrand. The energy stored n a capactor C s gven by (unts J Joules): () W E 1 C( ) (3) where s the potental dfference between the conductors of the capactor. The capactance of a structure can be evaluated as (unts F Farads): W E C (4) Maxwell D Feld Smulator can calculate the energy W E over the -D cross-secton and then calculate the approxmate value of the capactance C per unt length (F/m) of the structure. You wll be analyzng fve dfferent structures: Problem 1 - Feld at a sharp or rased pont Problem - Feld n a hollow Problem 3 - Parallel wre transmsson lne Problem 4 - Parallel wre transmsson lne wth plastc coatng Problem 5- Rectangular potental well You wll be ased to plot the voltage and electrc feld lnes for these structures. The relaton between electrc feld and voltage s found by usng the relaton below (unts J/C or ). [] (pg.60) B B W AB E dl E dl cos Q (5) unt A whch descrbes the potental of pont A wth respect to pont B defned as the wor done W n movng a unt charge Q unt from A to B. The electrc feld and the potental are perpendcular. In the case of the structures n ths lab equaton 5 can be smplfed by choosng a path ntegral such that cos(θ) = 1. If the electrc feld s constant n the regon of ntegraton then all that s left to calculate s the ntegral wth respect to the dsplacement l. Based on these specal crcumstances the resultng equaton s A E (6) l

11 where s the dfference n potental between two ponts and l s the dstance between the ponts. The structures n ths lab have pre-defned voltages. Keep trac of ther values as you go through the lab. 4. Runnng Maxwell D Feld Smulator 1. Go to the course webste and clc on the ln for the lab materal. Follow the nstructons provded.. Extract the downloaded zp fle Lab1 MaxwellFles.zp to a folder n your H drectory 3. Start Maxwell Control Panel on your destop (go to the ANSOFT drectory f the con sn t avalable). If a message pops up asng f you would le to create a worng drectory say Yes. 4. Choose Proects n the Maxwell Control Panel 5. In the lower left hand corner loo for a Proect Drectores headng. Clc Change Dr Go to Sub Drectores and double clc on../ to go up one level to the Maxwell drectory that the zpped fles were extracted to. 7. You should see the drectory fles named prob1c proba prob3 and prob4 show up under the Proects fle lst. 8. Clc OK to go bac to Maxwell Proects. 9. Agan you should see the drectory fles named prob1c proba prob3 and prob4 show up under the Proects fle lst only ths tme as you sngle clc on each drectory a drawng should be dsplayed n the graphcs wndow to the rght. The drawngs boundary condtons materals and post processes have been completed but mae sure you chec each settng manually. 5. Problem 1: Feld at a Rased Pont Ths problem models a parallel plate capactor n whch one plate s dented toward the other as shown n Fgure. The top plate s at 1 and the bottom plate s at 0. You would set these values by clcng Setup Boundares/Sources but for ths lab the values have been set for you. The materal of both plates s copper. The materal around the plates s ar. You would set these values by clcng Setup Materals but agan ths has been done already.

12 Fgure : Conductor structure for Problem 1 1. Clc on prob1c n the Maxwell Proects wndow and clc open under the graphcs wndow. If there s a verson msmatch between the fle and the software you wll see the message n Fgure 3. Clc OK.. Choose Post Process Fgure 3: erson msmatch message 3. To plot the electrc potental go to Plot n the tool bar and run through Plot Feld Ph - surface - all - all. 4. Ensure the values are smlar to those n the form shown Fgure 4. To modfy ths form anytme go to Plot modfy n the fle menu.

13 Fgure 4: Ph scalar surface plot form for Problem 1 5. To plot the electrc feld: Plot Feld E vector - surface - all - all. 6. Ensure the values are smlar to those n the form shown n Fgure 5.

14 7. Answer the followng questons for Problem 1. Fgure 5: E vector surface plot form for Problem 1 (a) Plot the equpotental lnes (contours of constant voltage) and the electrc feld lnes of your structure together n one prntout or ndvdually. Dsplay at least 10 voltage contours and don t forget to clearly nclude the legends. mars (b) Where s the locaton of the maxmum electrc feld strength? What s the value of the maxmum feld strength? Use the coloured electrc feld ntensty plot and the accompaned legend. Don t forget unts. mars (c) Insulatng materals wll brea down or become conductng f the electrc feld strength exceeds the breadown strength of the materal. For ar the breadown strength s about 3 x 10 6 /m. If the gap s reduced to 1 mm estmate the maxmum voltage that could be appled to the top plate. Answer ths queston usng theory and nclude unts. You may use Maxwell to chec your calculaton (note: Maxwell does not actually smulate the delectrc breadown). 1 mar 6. Problem : Feld n a Hollow Ths problem models a parallel plate capactor wth one plate dented away from the other as shown n Fgure 6. The top plate s at 1 and the bottom plate s at 0 source. The materal of both plates s copper and the delectrc s ar.

15 Fgure 6: Conductor structure for Problem 1. Clc on proba n the Maxwell Proects wndow and clc open under the graphcs wndow.. Manually chec Setup Boundares/Sources and Setup Materals. 3. Choose Post Process 4. To plot the electrc potental go to Plot n the tool bar and run through Plot Feld Ph - surface - all - all. 5. Ensure the values are smlar to those n the form shown n Fgure 7. Fgure 7: Ph scalar surface plot form for Problem

16 6. To plot the electrc feld: Plot Feld E vector - surface - all - all. 7. Ensure the values are smlar to those n the form shown n Fgure 8. Fgure 8: E vector surface plot form for Problem 8. Answer the followng questons for Problem. a) Plot the equpotental and electrc feld lnes of your structure as n Problem 1. mars b) Consder the regon between the two plates. Why s the electrc feld dfferent n the hollow? mars 7. Problem 3: Parallel Wre Transmsson Lne HF and UHF antennas are usually connected to T sets by transmsson lnes consstng of two parallel wres of fxed separaton as shown n Fgure 9. To desgn the transmsson lne we need to fnd the capactance per unt length between the wres. The capactance per unt length s gven analytcally by (unts F/m) C D cosh 1 ( ) a where s the dfference n potental between the two wres s the delectrc constant of the homogeneous materal surroundng the wres D s the center to center wre spacng and a s the radus of the wres as shown n Fgure 9. The delectrc constant of ar s 0 = F/m. For other (7)

17 materals we multply ths value by the relatve delectrc constant of the materal (that s = 0 ). 1 The functon cosh s found usng the hyp button on any scentfc calculator. The obect of problem 3 s to fnd the capactance numercally and compare wth the theoretcal value. We wll assume that the radus of the wre s always 1 mm but wll allow for dfferent spacng between the wres. The basc drawng of the two wres has already been completed and s n drectory prob3. Edt the drawng as explaned below and use a center to center wre spacng of D = 6 mm. The materal of both wres s copper and one wre s at 1 whle the other s at -1. If we assume that the parallel wres can be estmate by two parallel plates then the capactance neglectng frngng can also be wrtten as (unts F) [] (pg. 96) r r A Q Q C 0 (8) l El where A s the area of the plates Q s the charge on the plates l s the dstance between the plates 0 s the delectrc constant of ar and r s the relatve delectrc constant of the materal between the plates. Ths relaton ndcates that the electrc feld s related to the delectrc propertes of the materal n between the plates. r Fgure 9: Transmsson wre structure for Problem 3 1. Clc on prob3 n the Maxwell Proects wndow and clc open under the graphcs wndow.. In the Maxwell S wndow choose Solve. 3. To plot the electrc potental go to Plot n the tool bar and run through Plot Feld Ph - surface - all - all. 4. Ensure the values are smlar to those n the form shown Fgure 10.

18 Fgure 10: Ph scalar surface plot from Problem 3 5. To plot the electrc feld: Plot Feld E vector - surface - all - all. 6. Ensure the values are smlar to those n the form shown n Fgure 11. Fgure 11: E vector surface plot form for Problem 3

19 7. Answer the followng questons for Problem 3. a) Plot the equpotental and electrc feld lnes of your structure. mars b) What do you notce about the drecton of the electrc feld at any pont n relaton to the equpotental lnes? 1 mar c) Specfy the regon at whch the electrc feld s maxmum and state the maxmum value. Use the legend to gude you. Theoretcally you wll fnd that the maxmum should not be one pont but several ponts. 3 mars d) Estmate the capactance per unt length of the transmsson lne usng the Post Processor Feld Calculator and Equaton and 4. You can fnd help n the Maxwell S manual. The ln to the manual s provded n the Obectves secton of ths lab. Hnt: as an example of how to use the calculator the manual calculates the capactance of a test electrode set to 1. U In our case we are usng two wres wth = (1 ( 1 )) =. Therefore C U where = 1 becomes C where =. If you follow the nstructons exactly you must tae the 4 1 fracton nto account n your fnal result. 3 mars e) Calculate the theoretcal value of the capactance per unt length as explaned n the ntroducton to Problem 3. Compare to the estmated value of d) and explan any dscrepancy. Remember that you are comparng dfferent methods of solvng for capactance: numercal and analytcal. 3 mars 8. Problem 4: Transmsson Lne wth Plastc Coatng Now modfy the structure n Problem 3 so that the wres are coated wth a plastc (delectrc) layer of relatve permttvty =.1 and radus.0 mm. The plastc materal s Teflon and when drawng the r plastc layer wth the Obect/Crcle/ Pont tool the center of the plastc should be the same as the center of the copper wre. 1. Clc on prob4 n the Maxwell Proects wndow and clc open under the graphcs wndow.. Choose Post Process 3. Edt the drawng as prevously explaned. 4. Choose Solve

20 5. To plot the electrc potental go to Plot n the tool bar and run through Plot Feld Ph - surface - all - all. 6. Ensure the values are smlar to those n the form shown Fgure 1. Fgure 1: Ph scalar surface plot form for Problem 4 7. To plot the electrc feld: Plot Feld E vector - surface - all - all. 8. Ensure the values are smlar to those n the form shown n Fgure 13. Fgure 13: E vector surface plot form for Problem 4 9. Answer the followng questons for Problem 4. a) Plot the equpotental and electrc feld lnes of your structure. mars b) State the maxmum value of the electrc feld and state why t s greater or less than the maxmum values found n Queston 3. mars

21 c) Estmate the capactance per unt length of the transmsson lne usng the Post Processor Feld Calculator. mars d) Is the capactance greater or less than the one estmated n Problem 3? Explan. 3 mars 9. Problem 5: Rectangular potental well Fgure 14: Conductor Structure for Problem 5 1. Choose Proects n the Maxwell Control Panel. In the Proect menu choose New. In the Alas box enter Prob5 then clc on the Mae New Drectory crcle and ht OK. Specfy Solver Type: Select Electrostatc 3. Specfy Drawng Plane: Select XY Plane 4. Choose Defne Model Draw Model 5. Choose Model Drawng Unts mm 6. Choose Model Drawng Sze and set the Mnma to (0 0) and the Maxma to (15 15) 7. Choose Wndow Grd and set du and d smaller for there to be less space between each pont (1 mm s requred).

22 8. Select Obect Polylne and match the example provded. Hnt: Obects can be moved after they have been drawn by selectng them and then choosng Arrange Move (.e. obects can be centered after drawn). 9. Clc save and ext the wndow (Fle Ext) 10. Choose Setup Materals. Select an obect and materal and press Assgn. The bacground materal wll be ar and the conductng plates wll be copper. 11. Clc save and ext the wndow (Fle Ext) 1. Choose Set Up Boundares Sources 13. Choose Edt Select Obect By Clcng and left clc on an obect so t s hghlghted. Then rght-clc anywhere n the dsplay area to stop selectng obects 14. Choose Assgn Source Sold then select oltage and Assgn a correspondng voltage to the conductor. The top plate has a voltage of 100 and the other plate wll be For the bacground once t s hghlghted choose Assgn Boundary Balloon then select Charge and Assgn t. Selectng the Charge opton for the balloon boundary models an electrcally nsulated system. That s the charge at nfnty balances the charge n the problem regon forcng the net charge to be zero. [The oltage opton models an electrcally grounded system.e. the voltage at nfnty s zero but the charge at nfnty may not exactly balance the charge n the problem regon] 16. When all obects are assgned ther source or boundary save and ext 17. Choose Solve and wat for the Soluton Process s Complete message 18. Choose Post Process... and wat for a D Post Processor wndow 19. To plot the E ector and the oltage together for best dsplay do the followng: choose Plot Feld Select Ph Surface -all- and press OK. In the next screen unselect flled and press OK. To change the contour dvsons choose Plot Modfy select Ph and change the number of Dvsons (0). The values should be smlar to Fgure 14.

23 Fgure 15: Ph scalar surface plot form for Problem 5 1. Then select E ector Surface -all- and press OK. In the arrow optons select Type D and press OK. Snce the pattern of arrows s not too consstent go to Plot Modfy... and n Arrow Optons change Sze and Spacng to approxmately 4 and respectvely. Unchec map sze then press OK. (The sze and spacng values may be adusted to your lng). The values should be smlar to Fgure 15. Fgure 16: E vector surface plot for Problem 5

24 . Answer the followng questons for problem 5. a) Plot the equpotental and electrc feld lnes of your structure. mars b) * 3 Compare results obtaned here wth those calculated n the pre-lab secton. mars References [1] accessed September 008. [] Edmnster J.A. Schaums Outlnes: Electromagnetcs second edton 1993.